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

Zheng, Zhong; Christov, Ivan  C.; Stone, Howard  A.

doi:10.1017/jfm.2014.148

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

Journal Article · Thu May 01 00:00:00 EDT 2014 · Journal of Fluid Mechanics

DOI:https://doi.org/10.1017/jfm.2014.148· OSTI ID:1200619

Zheng, Zhong ^[1];

^[2]; Stone, Howard A. ^[1]

Princeton Univ., NJ (United States)
Princeton Univ., NJ (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

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.

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Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-06NA25396

OSTI ID:: 1200619

Report Number(s):: LA-UR-13-29555

Journal Information:: Journal of Fluid Mechanics, Vol. 747, Issue 1; ISSN 0022-1120

Country of Publication:: United States

Language:: English

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Cited by: 32 works

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References (42)

Horizontal infiltration into wet soil Witelski, T. P. Water Resources Research, Vol. 34, Issue 7 https://doi.org/10.1029/98WR00775	journal	July 1998
Near-term mega-scale CO2 capture and storage demonstration opportunities in China Zheng, Zhong; Larson, Eric D.; Li, Zheng Energy & Environmental Science, Vol. 3, Issue 9 https://doi.org/10.1039/b924243k	journal	January 2010
The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface Huppert, Herbert E. Journal of Fluid Mechanics, Vol. 121, Issue -1 https://doi.org/10.1017/S0022112082001797	journal	August 1982
Self‐similar solution of the second kind for a convergent viscous gravity current Diez, Javier Alberto; Gratton, Roberto; Gratton, Julio Physics of Fluids A: Fluid Dynamics, Vol. 4, Issue 6 https://doi.org/10.1063/1.858233	journal	June 1992
Flow in Porous Media Philip, J. R. Annual Review of Fluid Mechanics, Vol. 2, Issue 1 https://doi.org/10.1146/annurev.fl.02.010170.001141	journal	January 1970
Axisymmetric gravity currents in a porous medium Lyle, Sarah; Huppert, Herbert E.; Hallworth, Mark Journal of Fluid Mechanics, Vol. 543, Issue -1 https://doi.org/10.1017/S0022112005006713	journal	November 2005
Theory of creeping gravity currents of a non-Newtonian liquid Gratton, Julio; Minotti, Fernando; Mahajan, Swadesh M. Physical Review E, Vol. 60, Issue 6 https://doi.org/10.1103/PhysRevE.60.6960	journal	December 1999
Oil Spreading on the Sea Hoult, D. P. Annual Review of Fluid Mechanics, Vol. 4, Issue 1 https://doi.org/10.1146/annurev.fl.04.010172.002013	journal	January 1972
Intermediate asymptotics for convergent viscous gravity currents Angenent, S. B.; Aronson, D. G. Physics of Fluids, Vol. 7, Issue 1 https://doi.org/10.1063/1.868722	journal	January 1995
Noncircular converging flows in viscous gravity currents Diez, J. A.; Thomas, L. P.; Betelú, S. Physical Review E, Vol. 58, Issue 5 https://doi.org/10.1103/PhysRevE.58.6182	journal	November 1998
Diffusion from an Instantaneous Point Source with a Concentration-Dependent Coefficient Pattle, R. E. The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 12, Issue 4 https://doi.org/10.1093/qjmam/12.4.407	journal	January 1959
The role of self-similarity in singularities of partial differential equations Eggers, Jens; Fontelos, Marco A. Nonlinearity, Vol. 22, Issue 1 https://doi.org/10.1088/0951-7715/22/1/R01	journal	December 2008
The effects of capillary forces on the axisymmetric propagation of two-phase, constant-flux gravity currents in porous media Golding, Madeleine J.; Huppert, Herbert E.; Neufeld, Jerome A. Physics of Fluids, Vol. 25, Issue 3 https://doi.org/10.1063/1.4793748	journal	March 2013
Capillary entry pressure and the leakage of gravity currents through a sloping layered permeable rock Woods, Andrew W.; Farcas, Adrian Journal of Fluid Mechanics, Vol. 618 https://doi.org/10.1017/S0022112008004527	journal	January 2009
A numerical investigation of horizontal viscous gravity currents Hallez, Yannick; Magnaudet, Jacques Journal of Fluid Mechanics, Vol. 630 https://doi.org/10.1017/S0022112009006454	journal	July 2009
Viscous flows down an inclined plane from point and line sources Lister, John R. Journal of Fluid Mechanics, Vol. 242 https://doi.org/10.1017/S0022112092002520	journal	September 1992
Gravity currents over fractured substrates in a porous medium Pritchard, David Journal of Fluid Mechanics, Vol. 584 https://doi.org/10.1017/S0022112007006623	journal	July 2007
Self-Similar Solutions as Intermediate Asymptotics Barenblatt, G. I.; Zel'dovich, Y. B. Annual Review of Fluid Mechanics, Vol. 4, Issue 1 https://doi.org/10.1146/annurev.fl.04.010172.001441	journal	January 1972
Gravity Currents in the Laboratory, Atmosphere, and Ocean Simpson, J. E. Annual Review of Fluid Mechanics, Vol. 14, Issue 1 https://doi.org/10.1146/annurev.fl.14.010182.001241	journal	January 1982
Numerical investigation of quenching for a nonlinear diffusion equation with a singular Neumann boundary condition Christov, C. I.; Deng, K. Numerical Methods for Partial Differential Equations, Vol. 18, Issue 4 https://doi.org/10.1002/num.10013	journal	June 2002
A mathematical model of fluid and gas flow in nanoporous media Monteiro, P. J. M.; Rycroft, C. H.; Barenblatt, G. I. Proceedings of the National Academy of Sciences, Vol. 109, Issue 50 https://doi.org/10.1073/pnas.1219009109	journal	November 2012
Effect of variable permeability on the propagation of thin gravity currents in porous media Ciriello, Valentina; Di Federico, Vittorio; Archetti, Renata International Journal of Non-Linear Mechanics, Vol. 57 https://doi.org/10.1016/j.ijnonlinmec.2013.07.003	journal	December 2013
The effect of confining boundaries on viscous gravity currents Takagi, Daisuke; Huppert, Herbert E. Journal of Fluid Mechanics, Vol. 577 https://doi.org/10.1017/S0022112007005174	journal	April 2007
The competition between gravity and flow focusing in two-layered porous media Huppert, Herbert E.; Neufeld, Jerome A.; Strandkvist, Charlotte Journal of Fluid Mechanics, Vol. 720 https://doi.org/10.1017/jfm.2012.623	journal	February 2013
Groundwater mound damping Kochina, I. N.; Mikhailov, N. N.; Filinov, M. V. International Journal of Engineering Science, Vol. 21, Issue 4 https://doi.org/10.1016/0020-7225(83)90124-6	journal	January 1983
Gravity currents in horizontal porous layers: transition from early to late self-similarity Hesse, M. A.; Tchelepi, H. A.; Cantwel, B. J. Journal of Fluid Mechanics, Vol. 577 https://doi.org/10.1017/S0022112007004685	journal	April 2007
A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type Crank, J.; Nicolson, P. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 43, Issue 1 https://doi.org/10.1017/S0305004100023197	journal	January 1947
The viscous spreading of plane and axisymmetric gravity currents Didden, N.; Maxworthy, T. Journal of Fluid Mechanics, Vol. 121, Issue -1 https://doi.org/10.1017/S0022112082001785	journal	August 1982
Practical Numerical Algorithms for Chaotic Systems Parker, Thomas S.; Chua, Leon O. Springer New York https://doi.org/10.1007/978-1-4612-3486-9	book	January 1989
Enhancement of transport from drops by steady and modulated electric fields Christov, C. I.; Homsy, G. M. Physics of Fluids, Vol. 21, Issue 8 https://doi.org/10.1063/1.3179555	journal	August 2009
Gravity-driven flows in porous layers Huppert, Herbert E.; Woods, Andrew W. Journal of Fluid Mechanics, Vol. 292 https://doi.org/10.1017/S0022112095001431	journal	June 1995
Gravity currents in a porous medium at an inclined plane Vella, Dominic; Huppert, Herbert E. Journal of Fluid Mechanics, Vol. 555 https://doi.org/10.1017/S0022112006009578	journal	May 2006
Viscous Fingering in Porous Media Homsy, G. M. Annual Review of Fluid Mechanics, Vol. 19, Issue 1 https://doi.org/10.1146/annurev.fl.19.010187.001415	journal	January 1987
Self-similar viscous gravity currents: phase-plane formalism Gratton, Julio; Minotti, Fernando Journal of Fluid Mechanics, Vol. 210 https://doi.org/10.1017/S0022112090001240	journal	January 1990
Moving boundary problems and non-uniqueness for the thin film equation King, J. R.; Bowen, M. European Journal of Applied Mathematics, Vol. 12, Issue 3 https://doi.org/10.1017/S0956792501004405	journal	June 2001
Analysis and computation of gravity-induced migration in porous media De Loubens, R.; Ramakrishnan, T. S. Journal of Fluid Mechanics, Vol. 675 https://doi.org/10.1017/S0022112010006440	journal	March 2011
A benchmark study on problems related to CO2 storage in geologic formations: Summary and discussion of the results Class, Holger; Ebigbo, Anozie; Helmig, Rainer Computational Geosciences, Vol. 13, Issue 4 https://doi.org/10.1007/s10596-009-9146-x	journal	July 2009
Fluid drainage from the edge of a porous reservoir Zheng, Zhong; Soh, Beatrice; Huppert, Herbert E. Journal of Fluid Mechanics, Vol. 718 https://doi.org/10.1017/jfm.2012.630	journal	February 2013
Propagation Regimes of Fluid-Driven Fractures in Impermeable Rocks Detournay, E. International Journal of Geomechanics, Vol. 4, Issue 1 https://doi.org/10.1061/(ASCE)1532-3641(2004)4:1(35)	journal	March 2004
Gravity currents: a personal perspective Huppert, Herbert E. Journal of Fluid Mechanics, Vol. 554, Issue -1 https://doi.org/10.1017/S002211200600930X	journal	April 2006
The averaging of gravity currents in porous media Anderson, Daniel M.; McLaughlin, Richard M.; Miller, Cass T. Physics of Fluids, Vol. 15, Issue 10 https://doi.org/10.1063/1.1600733	journal	January 2003
Moving-boundary and fixed-domain problems for a sixth-order thin-film equation Flitton, J. C.; King, J. R. European Journal of Applied Mathematics, Vol. 15, Issue 6 https://doi.org/10.1017/S0956792504005753	journal	January 1999

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Dispersive entrainment into gravity currents in porous media Sahu, Chunendra K.; Neufeld, Jerome A. Journal of Fluid Mechanics, Vol. 886 https://doi.org/10.1017/jfm.2019.1049	journal	January 2020
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Title: Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

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