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Annual Review of Fluid Mechanics

Volume 54, 2022

The Influence of Boundaries on Gravity Currents and Thin Films: Drainage, Confinement, Convergence, and Deformation Effects

Abstract

Thin film flows, whether driven by gravity, surface tension, or the relaxation of elastic boundaries, occur in many natural and industrial processes. Applications span problems of oil and gas transport in channels to hydraulic fracture, subsurface propagation of pollutants, storage of supercritical CO in porous formations, and flow in elastic Hele–Shaw configurations and their relatives. We review the influence of boundaries on the dynamics of thin film flows, with a focus on gravity currents, including the effects of drainage into the substrate, and the role of the boundaries to confine the flow, force its convergence to a focus, or deform, and thus feedback to alter the flow. In particular, we highlight reduced-order models. In many cases, self-similar solutions can be determined and describe the behaviors in canonical problems at different timescales and length scales, including self-similar solutions of both the first and second kind. Additionally, the time transitions between different solutions are summarized. Where possible, remarks about various applications are provided.

Keyword(s): boundary influence, confinement, deformable boundaries, permeable boundaries, reduced-order models, self-similar solutions, viscous currents
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2022-01-05
2024-04-26
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Literature Cited

  1. Acton JM, Huppert HE, Worster MG. 2001. Two-dimensional viscous gravity currents flowing over a deep porous medium. J. Fluid Mech. 440:359–80
     [Google Scholar]
  2. Al-Housseiny TT, Tsai PA, Stone HA 2012. Control of interfacial instabilities using flow geometry. Nat. Phys. 8:747–50
     [Google Scholar]
  3. Backholm M, Benzaquen M, Salez T, Raphael E, Dalnoki-Veress K 2014. Capillary levelling of a cylindrical hole in a viscous film. Soft Matter 10:2550–58
     [Google Scholar]
  4. Barenblatt GI 1979. Similarity, Self-Similarity and Intermediate Asymptotics, transl. N Stein M Van Dyke New York: Consult. Bur.
     [Google Scholar]
  5. Barenblatt GI, Zel'dovich YB. 1972. Self-similar solutions as intermediate asymptotics. Annu. Rev. Fluid Mech. 4:285–312
     [Google Scholar]
  6. Bear J. 1972. Dynamics of Fluids in Porous Media. Amsterdam: Elsevier
  7. Bhamidipati N, Woods AW. 2020a. Boundary-induced shear and tracer transport in heterogeneous porous rock. J. Fluid Mech. 908:A45
     [Google Scholar]
  8. Bhamidipati N, Woods AW. 2020b. Shear generation in a confined, composite layer of cross-bedded porous rock. J. Fluid Mech. 899:R3
     [Google Scholar]
  9. Box F, Peng GG, Pihler-Puzovic D, Juel A. 2012. Flow-induced choking of a compliant Hele-Shaw cell. PNAS 117:30228–33
     [Google Scholar]
  10. Burland J. 2002. A Tale of Two Towers: Big Ben and Pisa London: R. Acad. Eng.
  11. Chai Y, Salez T, McGraw JD, Benzaquen M, Dalnoki-Veress K et al. 2014. A direct quantitative measure of surface mobility in a glassy polymer. Science 343:994–99
     [Google Scholar]
  12. Cowton LR, Neufeld JA, White NJ, Bickle MJ, Williams GA et al. 2018. Benchmarking of vertically-integrated CO2 flow simulations at the Sleipner Field, North Sea. Earth Planet. Sci. Lett. 491:121–33
     [Google Scholar]
  13. Craster RV, Matar OK. 2009. Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81:1131
     [Google Scholar]
  14. Dana A, Zheng Z, Peng GG, Stone HA, Huppert HE, Ramon GZ. 2018. Dynamics of viscous backflow from a model fracture network. J. Fluid Mech. 836:828–49
     [Google Scholar]
  15. de Gennes PG. 1985. Wetting: statics and dynamics. Rev. Mod. Phys. 57:827–63
     [Google Scholar]
  16. Detournay E. 2016. Mechanics of hydraulic fractures. Annu. Rev. Fluid Mech. 48:311–39
     [Google Scholar]
  17. Dijksman JA, Mukhopadhyay S, Gaebler C, Witelski TP, Behringer RP. 2015. Obtaining self-similar scalings in focusing flows. Phys. Rev. E 92:043016
     [Google Scholar]
  18. Duffy BR, Wilson SK. 1996. A third-order differential equation arising in thin-film flows and relevant to Tanner's law. Appl. Math. Lett. 10:63–68
     [Google Scholar]
  19. Eggers J, Fontelos MA. 2009. The role of self-similarity in singularities of partial differential equations. Nonlinearity 22:R1–44
     [Google Scholar]
  20. Gasda SE, Nordbotten JM, Celia MA. 2011. Vertically averaged approaches for CO2 migration with solubility trapping. Water Resour. Res. 47:W05528
     [Google Scholar]
  21. Golding MJ, Huppert HE, Neufeld JA. 2017. Two-phase gravity currents resulting from the release of a fixed volume of fluid in a porous medium. J. Fluid Mech. 832:550–77
     [Google Scholar]
  22. Gratton J, Minotti F. 1990. Self-similar viscous gravity currents: phase-plane formalism. J. Fluid Mech. 210:155–82
     [Google Scholar]
  23. Gray JMNT 2018. Particle segregation in dense granular flows. Annu. Rev. Fluid Mech. 50:407–33
     [Google Scholar]
  24. Guo B, Bandilla KW, Nordbotten JM, Celia MA, Keilegavlen E, Doster F 2016a. A multiscale multilayer vertically integrated model with vertical dynamics for CO2 sequestration in layered geological formation. Water Resour. Res. 52:6490–505
     [Google Scholar]
  25. Guo B, Zheng Z, Bandilla KW, Celia MA, Stone HA 2016b. Flow regime analysis for geologic CO2 sequestration and other subsurface fluid injections. Int. J. Greenh. Gas Cont. 53:284–91
     [Google Scholar]
  26. Guo B, Zheng Z, Celia MA, Stone HA 2016c. Axisymmetric flows from fluid injection into a confined porous medium. Phys. Fluids 28:022107
     [Google Scholar]
  27. Hallez Y, Magnaudet J. 2009. A numerical investigation of horizontal viscous gravity currents. J. Fluid Mech. 630:71–91
     [Google Scholar]
  28. Hesse MA, Tchelepi HA, Cantwell BJ, Orr FM Jr. 2007. Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech. 577:363–83
     [Google Scholar]
  29. Hesse MA, Woods AW. 2010. Buoyant disposal of CO2 during geological storage. Geophys. Res. Lett. 37:L01403
     [Google Scholar]
  30. Hewitt IJ, Balmforth NJ, De Bruyn JR. 2015. Elastic-plated gravity currents. Eur. J. Appl. Math. 26:1–31
     [Google Scholar]
  31. Higuera FJ. 1995. Steady creeping flow down a slope. Phys. Fluids 7:2918–20
     [Google Scholar]
  32. Hinch EJ. 1991. Perturbation Methods Cambridge, UK: Cambridge Univ. Press
  33. Hinton EM, Woods AW. 2018. Buoyancy-driven flow in a confined aquifer with a vertical gradient of permeability. J. Fluid Mech. 848:411–29
     [Google Scholar]
  34. Hinton EM, Woods AW. 2019. The effect of vertically varying permeability on tracer dispersion. J. Fluid Mech. 860:384–407
     [Google Scholar]
  35. Horsley MC, Woods AW. 2017. Gravity-driven flow in a horizontal annulus. J. Fluid Mech. 830:479–93
     [Google Scholar]
  36. Howell PD, Kim H, Stone HA 2016. Rivulet flow over a flexible beam. J. Fluid Mech. 796:285–305
     [Google Scholar]
  37. Howell PD, Robinson J, Stone HA 2013. Gravity-driven thin-film flow on a flexible substrate. J. Fluid Mech. 732:190–213
     [Google Scholar]
  38. Huh C, Scriven LE. 1971. Hydrodynamics model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interf. Sci. 35:85–101
     [Google Scholar]
  39. Huppert HE, Neufeld JA. 2014. The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46:255–72
     [Google Scholar]
  40. Huppert HE, Woods AW. 1995. Gravity-driven flows in porous layers. J. Fluid Mech. 292:55–69
     [Google Scholar]
  41. Juel A, Pihler-Puzovic D, Heil M. 2018. Instabilities in blistering. Annu. Rev. Fluid Mech. 50:691–714
     [Google Scholar]
  42. Kang M, Kanno CM, Reid MC, Zhang X, Mauzerall DL et al. 2014a. Direct measurements of methane emissions from abandoned oil and gas wells in Pennsylvania. PNAS 111:18173–77
     [Google Scholar]
  43. Kang M, Nordbotten JM, Doster F, Celia MA. 2014b. Analytical solutions for two-phase subsurface flow to a leaky fault considering vertical flow effects and fault properties. Water Resour. Res. 50:3536–52
     [Google Scholar]
  44. King SE, Woods AW. 2003. Dipole solutions for viscous gravity currents: theory and experiments. J. Fluid Mech. 483:91–109
     [Google Scholar]
  45. Kochina IN, Mikhailov NN, Filinov MV. 1983. Groundwater mound damping. Int. J. Eng. Sci. 21:413–21
     [Google Scholar]
  46. Kowal KN, Worster MG. 2015. Lubricated viscous gravity currents. J. Fluid Mech. 766:626–55
     [Google Scholar]
  47. Kowal KN, Worster MG. 2019. Stability of lubricated viscous gravity currents. Part 1. Internal and frontal analyses and stabilisation by horizontal shear. J. Fluid Mech. 871:970–1006
     [Google Scholar]
  48. Lai C-Y, Kingslake J, Wearing MG, Chen P-HC, Gentine P et al. 2020. Vulnerability of Antarctica's ice shelves to meltwater-driven fracture. Nature 584:574–78
     [Google Scholar]
  49. Lai C-Y, Rallabandi B, Perazzo A, Zheng Z, Smiddy SE, Stone HA. 2018. Foam-driven fracture. PNAS 115:8082–86
     [Google Scholar]
  50. Lai C-Y, Zheng Z, Dressaire E, Ramon GZ, Huppert HE, Stone HA. 2016a. Elastic relaxation of fluid-driven cracks and the resulting backflow. Phys. Rev. Lett. 117:268001
     [Google Scholar]
  51. Lai C-Y, Zheng Z, Dressaire E, Stone HA. 2016b. Fluid-driven cracks in an elastic matrix in the toughness-dominated limit. Philos. Trans. R. Soc. A 374:20150425
     [Google Scholar]
  52. Lai C-Y, Zheng Z, Dressaire E, Wexler JS, Stone HA. 2015. Experimental study on penny-shaped fluid-driven cracks in an elastic matrix. Proc. R. Soc. A 471:20150255
     [Google Scholar]
  53. Linden PF 2012. Gravity currents—theory and laboratory experiments. Buoyancy-Driven Flows EP Chassignet, C Cenedese, J Verron 13–51 Cambridge, UK: Cambridge Univ. Press
     [Google Scholar]
  54. Lister JR. 1989. The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface. J. Fluid Mech. 203:215–49
     [Google Scholar]
  55. Lister JR. 1992. Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242:631–53
     [Google Scholar]
  56. Lister JR, Peng GG, Neufeld JA. 2012. Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111:154501
     [Google Scholar]
  57. Liu Y, Zheng Z, Stone HA 2017. The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium. J. Fluid Mech. 817:514–59
     [Google Scholar]
  58. Longo S, Di Federico V, Chiapponi L. 2015. A dipole solution for power-law gravity currents in porous formations. J. Fluid Mech. 778:534–51
     [Google Scholar]
  59. Lyle S, Huppert HE, Hallworth M, Bickle M, Chadwick A. 2005. Axisymmetric gravity currents in a porous medium. J. Fluid Mech. 543:293–302
     [Google Scholar]
  60. MacMinn CW, Szulczewski ML, Juanes R. 2011. CO2 migration in saline aquifers. Part 2. Capillary and solubility trapping. J. Fluid Mech. 688:321–51
     [Google Scholar]
  61. Mathunjwa JS, Hogg AJ. 2007. Freely draining gravity currents in porous media: dipole self-similar solutions with and without capillary retention. Euro. J. Appl. Math. 18:337–62
     [Google Scholar]
  62. Michaut C. 2011. Dynamics of magmatic intrusions in the upper crust: theory and applications to laccoliths on Earth and the Moon. J. Geophys. Res. 116:B05205
     [Google Scholar]
  63. Momen M, Zheng Z, Bou-Zeid E, Stone HA. 2017. Inertial gravity currents produced by fluid drainage from an edge. J. Fluid Mech. 827:640–63
     [Google Scholar]
  64. Nordbotten JM, Celia MA. 2006. Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561:307–27
     [Google Scholar]
  65. Nordbotten JM, Celia MA. 2012. Geological Storage of CO2: Modeling Approaches for Large-Scale Simulation Hoboken, NJ: Wiley
  66. O'Keeffe NJ, Zheng Z, Huppert HE, Linden PF. 2018. Symmetric coalescence of two hydraulic fractures. PNAS 115:10228–32
     [Google Scholar]
  67. Oron A, Davis SH, Bankoff SG. 1997. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69:931
     [Google Scholar]
  68. Pegler SS, Huppert HE, Neufeld JA. 2014. Fluid injection into a confined porous layer. J. Fluid Mech. 745:592–620
     [Google Scholar]
  69. Peng GG, Lister JR. 2020. Viscous flows under an elastic sheet. J. Fluid Mech. 905:A30
     [Google Scholar]
  70. Pihler-Puzović D, IIlien P, Heil M, Juel A. 2012. Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Phys. Rev. Lett. 108:074502
     [Google Scholar]
  71. Pihler-Puzović D, Peng GG, Lister JR, Heil M, Juel A. 2018. Viscous fingering in a radial elastic-walled Hele-Shaw cell. J. Fluid Mech. 849:163–91
     [Google Scholar]
  72. Postma TJW, Bandilla KW, Celia MA. 2019. Estimates of CO2 leakage along abandoned wells constrained by new data. Int. J. Greenh. Gas. Cont. 84:164–79
     [Google Scholar]
  73. Pritchard D. 2007. Gravity currents over fractured substrates in a porous medium. J. Fluid Mech. 584:415–31
     [Google Scholar]
  74. Pritchard D, Woods AW. 2002. Drainage viscous gravity currents in a vertical fracture. J. Fluid Mech. 459:207–16
     [Google Scholar]
  75. Pritchard D, Woods AW, Hogg AJ. 2001. On the slow drainage of a gravity current moving through a layered permeable medium. J. Fluid Mech. 444:23–47
     [Google Scholar]
  76. Rice JR 1968. Mathematical analysis in the mechanics of fracture. Fracture: An Advanced Treatise H Liebowitz 191–311 New York: Academic
     [Google Scholar]
  77. Saffman PG, Taylor GI. 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
     [Google Scholar]
  78. Savitski AA, Detournay E. 2002. Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions. Int. J. Solids Struct. 39:6311–37
     [Google Scholar]
  79. Spence DA, Sharp P. 1985. Self-similar solutions for elastohydrodynamic cavity flow. Proc. R. Soc. Lond. A 400:289–313
     [Google Scholar]
  80. Szulczewski ML, Hesse MA, Juanes R. 2013. Carbon dioxide dissolution in structural and stratigraphic traps. J. Fluid Mech. 736:287–315
     [Google Scholar]
  81. Taghavi SM, Martinez DM, Frigaard IA. 2009. Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639:1–35
     [Google Scholar]
  82. Thomas LP, Marino BM, Linden PF. 1998. Gravity currents over porous substrates. J. Fluid Mech. 366:239–58
     [Google Scholar]
  83. Thomas LP, Marino BM, Linden PF. 2004. Lock-release inertial gravity currents over a thick porous layer. J. Fluid Mech. 503:299–319
     [Google Scholar]
  84. Ungarish M. 2009. An Introduction to Gravity Currents and Intrusions Boca Raton, FL: CRC
  85. Ungarish M, Zhu L, Stone HA. 2019. Inertial gravity current produced by the drainage of a cylindrical reservoir from an outer or inner edge. J. Fluid Mech. 874:185–209
     [Google Scholar]
  86. Vella D, Huppert HE. 2006. Gravity currents in a porous medium at an inclined plane. J. Fluid Mech. 555:353–62
     [Google Scholar]
  87. Witelski TP, Bernoff AJ. 1999. Stability of self-similar solutions for van der Waals driven thin film rupture. Phys. Fluids 11:2443–45
     [Google Scholar]
  88. Woods AW. 2015. Flow in Porous Rocks Cambridge, UK: Cambridge Univ. Press
  89. Woods AW, Farcas A. 2009. Capillary entry pressure and the leakage of gravity currents through a sloping layered permeable rock. J. Fluid Mech. 618:361–79
     [Google Scholar]
  90. Woods AW, Mason R. 2000. The dynamics of two-layer gravity-driven flows in permeable rock. J. Fluid Mech. 421:83–114
     [Google Scholar]
  91. Yu YE, Zheng Z, Stone HA 2017. Flow of a gravity current in a porous medium accounting for the drainage from a permeable substrate and an edge. Phys. Rev. Fluids 2:074101
     [Google Scholar]
  92. Zheng Z, Christov IC, Stone HA. 2014a. Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents. J. Fluid Mech. 747:218–46
     [Google Scholar]
  93. Zheng Z, Fontelos MA, Shin S, Dallaston MC, Tseluiko D et al. 2018a. Healing capillary films. J. Fluid Mech. 838:404–34
     [Google Scholar]
  94. Zheng Z, Fontelos MA, Shin S, Stone HA 2018b. Universality in the nonlinear leveling of capillary films. Phys. Rev. Fluids 3:032001(R)
     [Google Scholar]
  95. Zheng Z, Ghodgaonkar AA, Christov IC 2021. Shape of spreading and leveling gravity currents in a Hele-Shaw cell with flow-wise width variation. Phys. Rev. Fluids 6:094101
     [Google Scholar]
  96. Zheng Z, Griffiths I, Stone HA. 2015a. Propagation of a viscous thin film over an elastic membrane. J. Fluid Mech. 784:443–64
     [Google Scholar]
  97. Zheng Z, Guo B, Christov IC, Celia MA, Stone HA 2015d. Flow regimes for fluid injection into a confined porous medium. J. Fluid Mech. 778:881–909
     [Google Scholar]
  98. Zheng Z, Neufeld JA. 2019. Self-similar dynamics of two-phase flows injected into a confined porous layer. J. Fluid Mech. 877:882–921
     [Google Scholar]
  99. Zheng Z, Rongy L, Stone HA. 2015b. Viscous fluid injection into a confined channel. Phys. Fluids 27:062105
     [Google Scholar]
  100. Zheng Z, Shin S, Stone HA. 2015c. Converging gravity currents over a permeable substrate. J. Fluid Mech. 778:669–90
     [Google Scholar]
  101. Zheng Z, Soh B, Huppert HE, Stone HA. 2013. Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718:558–68
     [Google Scholar]
/content/journals/10.1146/annurev-fluid-030121-025957
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The Influence of Boundaries on Gravity Currents and Thin Films: Drainage, Confinement, Convergence, and Deformation Effects
Annual Review of Fluid Mechanics 54, 27 (2022); https://doi.org/10.1146/annurev-fluid-030121-025957
/content/journals/10.1146/annurev-fluid-030121-025957
/content/journals/10.1146/annurev-fluid-030121-025957
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