Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T15:27:42.794Z Has data issue: false hasContentIssue false
  • English
  • Français

Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

Published online by Cambridge University Press:  16 April 2014

Zhong Zheng ,
Ivan C. Christov Open the ORCID record for Ivan C. Christov [Opens in a new window]  and
Howard A. Stone
Zhong Zheng
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Ivan C. Christov
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We report experimental, theoretical and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (a) a horizontal channel (or crack) whose gap thickness varies as a power-law function of the streamwise coordinate; (b) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviours emerge as a result of horizontal heterogeneity: (a) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (b) a second-kind self-similar solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared with experimental results and numerical solutions of the governing partial differential equation developed under the lubrication approximation. All three results are found to be in good agreement.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 747 , 25 May 2014 , pp. 218 - 246
Copyright
© 2014 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

Anderson, D. M., McLaughlin, R. M. & Miller, C. T. 2003 The averaging of gravity currents in porous media. Phys. Fluids 15, 2810–2829.CrossRefGoogle Scholar
Angenent, S. B. & Aronson, D. G. 1995 Intermediate asymptotics for convergent viscous gravity currents. Phys. Fluids 7, 223–225.CrossRefGoogle Scholar
Barenblatt, G. I. 1952 On some unsteady fluid and gas motions in a porous medium. Prikl. Mat. Mekh. (PMM) 16, 67–78; (in Russian).Google Scholar
Barenblatt, G. I. 1996 Similarity, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.Google Scholar
Barenblatt, G. I. & Zel’dovich, Y. B. 1972 Self-similar solutions as intermediate asymptotics. Annu. Rev. Fluid Mech. 4, 285–312.Google Scholar
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.Google Scholar
Christov, C. I. & Deng, K. 2002 Numerical investigation of quenching for a nonlinear diffusion equation with a singular Neumann boundary condition. Numer. Meth. Partial Differ. Equ. 18, 429–440.Google Scholar
Christov, C. I. & Homsy, G. M. 2009 Enhancement of transport from drops by steady and modulated electric fields. Phys. Fluids 21, 083102.CrossRefGoogle Scholar
Ciriello, V., Di Federico, V., Archetti, R. & Longo, S. 2013 Effect of variable permeability on the propagation of thin gravity currents in porous media. Intl J. Non-Linear Mech. 57, 168–175.CrossRefGoogle Scholar
Class, H., Ebigbo, A., Helmig, R., Dahle, H. K., Nordbotten, J. M., Celia, M. A., Audigane, P., Darcis, M., Ennis-King, J., Fan, Y., Flemisch, B., Gasda, S. E., Jin, M., Krug, S., Labregere, D., Naderi Beni, A., Pawar, R. J., Sbai, A., Thomas, S. G., Trenty, L. & Wei, L. 2009 A benchmark study on problems related to ${\rm CO}_{2}$ storage in geologic formations. Comput. Geosci. 13, 409–434.Google Scholar
Courant, R. & Friedrichs, K. O. 1999 Supersonic Flow and Shock Waves, Applied Mathematical Sciences, vol. 21. Springer, corrected 5th printing.Google Scholar
Crank, J. & Nicolson, P. 1947 A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Proc. Camb. Phil. Soc. 43, 50–67.CrossRefGoogle Scholar
De Loubens, R. & Ramakrishnan, T. S. 2011 Analysis and computation of gravity-induced migration in porous media. J. Fluid Mech. 675, 60–86.CrossRefGoogle Scholar
Detournay, E. 2004 Propagation regimes of fluid-driven fractures in impermeable rocks. Intl J. Geomech. 4, 35–45.Google Scholar
Didden, N. & Maxworthy, T. 1982 Viscous spreading of plane and axisymmetric gravity waves. J. Fluid Mech. 121, 27–42.CrossRefGoogle Scholar
Diez, J. A., Gratton, R. & Gratton, J. 1992 Self-similar solution of the second kind for a convergent viscous gravity current. Phys. Fluids A 6, 1148–1155.CrossRefGoogle Scholar
Diez, J. A., Thomas, L. P., Betelú, S., Gratton, R., Marino, B., Gratton, J., Aronson, D. G. & Angenent, S. B. 1998 Noncircular converging flows in viscous gravity currents. Phys. Rev. E 58, 6182–6187.CrossRefGoogle Scholar
Dullien, F. A. L. 1992 Porous Media: Fluid Transport and Pore Structure. Academic Press.Google Scholar
Eggers, J. & Fontelos, M. A. 2009 The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1–R44.Google Scholar
Flitton, J. C. & King, J. R. 2004 Moving-boundary and fixed-domain problems for a sixth-order thin-film equation. Eur. J. Appl. Maths 15, 713–754.Google Scholar
Golding, M. J., Huppert, H. E. & Neufeld, J. A. 2013 The effects of capillary forces on the axisymmetric propagation of two-phase, constant-flux gravity currents in porous media. Phys. Fluids 25, 036602.Google Scholar
Gratton, J., Mahajan, S. M. & Minotti, F. 1999 Theory of creeping gravity currents of a non-Newtonian liquid. Phys. Rev. E 60, 6090–6097.CrossRefGoogle ScholarPubMed
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase plane formalism. J. Fluid Mech. 210, 155–182.Google Scholar
Hallez, Y. & Magnaudet, J. 2009 A numerical investigation of horizontal viscous gravity currents. J. Fluid Mech. 630, 71–91.CrossRefGoogle Scholar
Hesse, M. A., Tchelepi, H. A., Cantwell, B. J. & Orr, F. M. Jr 2007 Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech. 577, 363–383.CrossRefGoogle Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Hoult, D. P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 4, 341–368.Google Scholar
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.Google Scholar
Huppert, H. E. 2000 Geological fluid mechanics. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 447–506. Cambridge University Press.Google Scholar
Huppert, H. E. 2006 Gravity currents: a personal perspective. J. Fluid Mech. 554, 299–322.CrossRefGoogle Scholar
Huppert, H. E., Neufeld, J. A. & Strandkvist, C. 2013 The competition between gravity and flow focusing in two-layered porous media. J. Fluid Mech. 720, 5–14.CrossRefGoogle Scholar
Huppert, H. E. & Woods, A. W. 1995 Gravity driven flows in porous layers. J. Fluid Mech. 292, 55–69.Google Scholar
King, J. R. & Bowen, M. 2001 Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Maths 12, 321–356.Google Scholar
Kochina, I. N., Mikhailov, N. N. & Filinov, M. V. 1983 Groundwater mound damping. Intl J. Engng Sci. 21, 413–421.CrossRefGoogle Scholar
Lister, J. R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631–653.CrossRefGoogle Scholar
Lyle, S., Huppert, H. E., Hallworth, M., Bickle, M. & Chadwick, A. 2005 Axisymmetric gravity currents in a porous medium. J. Fluid Mech. 543, 293–302.Google Scholar
Monteiro, P. J. M., Rycroft, C. H. & Barenblatt, G. I. 2012 A mathematical model of fluid and gas flow in nanoporous media. Proc. Natl Acad. Sci. USA 109, 20309–20313.CrossRefGoogle ScholarPubMed
Nordbotten, J. M. & Celia, M. A. 2006 Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561, 307–327.Google Scholar
Parker, T. S. & Chua, L. O. 1989 Practical Numerical Algorithms for Chaotic Systems. Springer.Google Scholar
Pattle, R. E. 1959 Diffusion from an instantaneous point source with a concentration-dependent coefficient. Q. J. Mech. Appl. Maths 12, 407–409.CrossRefGoogle Scholar
Philip, J. R. 1970 Flow in porous media. Annu. Rev. Fluid Mech. 2, 177–204.CrossRefGoogle Scholar
Phillips, O. M. 1991 Flow and Reactions in Permeable Rocks. Cambridge University Press.Google Scholar
Pritchard, D. 2007 Gravity currents over fractured substrates in a porous medium. J. Fluid Mech. 584, 415–431.Google Scholar
Saffman, P. G. & Taylor, G. 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
Sedov, L. I. 1993 Similarity and Dimensional Methods in Mechanics. 10th edn. CRC Press.Google Scholar
Simpson, J. E. 1982 Gravity currents in the laboratory, atmosphere, and ocean. Annu. Rev. Fluid Mech. 14, 213–234.CrossRefGoogle Scholar
Spence, D. A. & Sharp, P. 1985 Self-similar solutions for elastohydrodynamic cavity flow. Proc. R. Soc. Lond. A 400, 289–313.Google Scholar
Strikwerda, J. 2004 Finite Difference Schemes and Partial Differential Equations. Society for Industrial and Applied Mathematics.Google Scholar
Takagi, D. & Huppert, H. E. 2007 The effect of confining boundaries on viscous gravity currents. J. Fluid Mech. 577, 495–505.Google Scholar
Vella, D. & Huppert, H. E. 2006 Gravity currents in a porous medium at an inclined plane. J. Fluid Mech. 555, 353–362.CrossRefGoogle Scholar
Witelski, T. P. 1998 Horizontal infiltration into wet soil. Water Resour. Res. 30, 1859–1863.Google Scholar
Woods, A. W. & Farcas, A. 2009 Capillary entry pressure and the leakage of gravity currents through a sloping layered permeable rock. J. Fluid Mech. 618, 361–379.Google Scholar
Yanenko, N. N. 1971 The Method of Fractional Steps (ed. Hoult, M.), Springer, English translation.Google Scholar
Zheng, Z., Larson, E. D., Li, Z., Liu, G. & Williams, R. H. 2010 Near-term mega-scale ${\rm CO}_{2}$ capture and storage demonstration opportunities in China. Energy Environ. Sci. 3, 1153–1169.Google Scholar
Zheng, Z., Soh, B., Huppert, H. E. & Stone, H. A. 2013 Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718, 558–568.Google Scholar