Transport in Porous Media

, Volume 25, Issue 3, pp 313–334 | Cite as

Criterion for instability of steady-state unsaturated flows

  • Vivek Kapoor


The stability of steady-state solutions to the unsaturated flow equation is examined. Conditions under which infinitesimal disturbances are amplified are determined by linear stability analysis. Uniform suction head profiles are shown to be linearly stable to three-dimensional disturbances. The stability of nonuniform suction head profiles to planar (gc1χ2) disturbances is examined. When the steady-state suction head solution (Ψ) increases with depth, χ3, (dΨ/dχ3 > 0), a condition for the amplification of infinitesimal planar disturbances is identified as
$$\frac{{d^2 K(\Psi )}}{{d\Psi ^2 }} > \frac{{\left( {\frac{{dK(\Psi )}}{{d\Psi }}} \right)^2 }}{{K(\Psi )}},$$
, where K(Ψ) is the hydraulic conductivity versus suction head characteristic of the porous medium. The same condition applies when dΨ/dχ3 < -1. Therefore when the rate of change of the slope of the K - Ψ characteristic curves is larger than the squared slope divided by K, even small disturbances can be amplified exponentially. The smallest wavelength of unstable planar perturbations is shown to be inversely related to the coarseness of the soil. Conditions under which the instability criterion is met are delineated for some commonly employed K - Ψ curves.

Key words

unsaturated flow instabilities fingering linear stability analysis cutoff wavelength instability criterion nonlinearity spatial-temporal complexity exponential growth moisture profiles 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Babel, M. S., Loof, R. and Gupta, A. D.: 1995, Fingered and preferential flow in unsaturated homogeneous coarse sands, Hydrol. Sci. J. 40(1), 1–17.Google Scholar
  2. Baker, R. S. and Hillel, D.: 1990, Laboratory tests of a theory of fingering during infiltration into layered systems, Soil Sci. Soc. Am. J. 54, 20–30.Google Scholar
  3. Bear, J.: 1972, Dynamics of Fluids in Porous Media, Dover, New York.Google Scholar
  4. Bouwer, H.: 1964, Unsaturated flow in ground-water hydraulics, J. Hydraul. Div., Am. Soc. Civ. Eng. 90(HY5), 121–144.Google Scholar
  5. Bouwer, H.: 1978, Groundwater Hydrology, McGraw-Hill, New York.Google Scholar
  6. Diment, G. E., Watson, K. K. and Blennerhassett, P. J.: 1982, Stability analysis of water movement in unsaturated porous materials, 1. Theoretical considerations, Water Resour. Res. 18(4), 1248–1254.Google Scholar
  7. Diment, G. E. and Watson, K. K.: 1983, Stability analysis of water movement in unsaturated porous materials, 2. Numerical studies, Water Resour. Res. 19(4), 1002–1010.Google Scholar
  8. Diment, G. E. and Watson, K. K.: 1985, Stability analysis of water movement in unsaturated porous materials, 3. Experimental studies, Water Resour. Res. 21(7), 979–984.Google Scholar
  9. Drazin, P. G. and Reid, W. H.: 1981, Hydrodynamic Stability, Cambridge University Press, New York, NY.Google Scholar
  10. Gardner, W. R.: 1958, Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table. Soil Sci. 85(4), 228–232.Google Scholar
  11. Glass, R. J., Parlange, J. Y. and Steenhuis, T. S.: 1989(a), Wetting front instability, 1, Theoretical discussion and dimensional analysis, Water Resour. Res. 25(6), 1187–1194.Google Scholar
  12. Glass, R. J., Parlange, J. Y. and Steenhuis, T. S.: 1989(b), Wetting front instability, 2, Experimental determination of relationships between system parameters and two dimensional unstable flow field behavior in initially dry porous media, Water Resour. Res. 25(6), 1195–1207.Google Scholar
  13. Glass, R. J., Cann, S., King, J., Bailey, N., Parlange, J. Y. and Steenhuis, T. S.: 1990, Wetting front instability in unsaturated porous media: A three-dimensional study in initially dry sand, Transport in Porous Media 5: 247–268.CrossRefGoogle Scholar
  14. Hardy, G., Littlewood, L. E. and Polya, G.: 1952, Inequalities, 2nd edn., Cambridge University Press.Google Scholar
  15. Hill, D. E. and Parlange, J. Y.: 1972, Wetting front instability in layered soils, Soil. Sci. Sic. Am. Proc. 36, 697–702.Google Scholar
  16. Hillel, D.: 1980, Introduction to Soil Physics, Academic Press, San Diego, CA.Google Scholar
  17. Hillel, D. and Baker, R. S.: 1988, A descriptive theory of fingering during infiltration into layered soils, Soil Sci. 146(1), 51–56.Google Scholar
  18. Homsy, G. M.: 1987, Viscous fingering in porous media, Ann. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
  19. Kapoor, V. and Kitanidis, P. K.: Concentration fluctuations and dilution in two-dimensionally periodic heterogeneous porous media, Transport in Porous Media 22, 91–119.Google Scholar
  20. Miller, D. E. and Gardner, W. H.: 1962, Water infiltration into stratified soils, Soil Sci. Soc. Am. Proc. 26, 115–119.Google Scholar
  21. Miller, E. E. and Miller, R. D.: 1965, Physical theory for capillary flow phenomena, J. Appl. Phys. 27, 324–332.Google Scholar
  22. Parlange, J.-Y. and Hill, D. E.: 1976, Theoretical analysis of wetting front instability in soils, Soil Sci. 122, 236–239.Google Scholar
  23. Peck, A. J.: 1965, Moisture profile development and air compression during water uptake by bounded porous bodies, 3, Vertical columns, Soil Sci. 100, 44–51.Google Scholar
  24. Phillip, J. R.: 1975, Stability analysis of infiltration, Soil Sci. Soc. Am. Proc. 39, 1042–1049.Google Scholar
  25. Raats, P. A. C.: 1973, Unstable wetting fronts in uniform and non-uniform soils, Soil Sci. Soc. Am. Proc. 36, 681–685.Google Scholar
  26. Richards, L. A.: 1931, Capillary conduction of liquids in porous mediums, Physics 1, 318–333.CrossRefGoogle Scholar
  27. Tchelepi, H. A., Orr, F. M., Rakotomalala, N., Salin, D. and Woumeni, R.: 1993, Dispersion, permeability heterogeneity, and viscous fingering: Acoustic experimental observations and particle-tracking simulations, Phys. Fluids A 5(7), 1558–1574.CrossRefGoogle Scholar
  28. van Genuchten, M. Th.: 1980, A closed form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J. 44, 892–898.Google Scholar
  29. White, I., Colombera, P. M. and Phillip, J. R.: 1976, Experimental study of wetting front instability induced by sudden change of pressure gradient, Soil Sci. Soc. Am. J. 40, 824–829.Google Scholar
  30. Yeh, T.-C. J. and Harvey, D. J.: 1990, Effective unsaturated hydraulic conductivity of layered sands, Water Resour. Res. 26(6), 1271–1279.Google Scholar
  31. Zimmerman, W. B. and Homsy, G. M.: 1992, Three-dimensional viscous fingering: A numerical study, Phys. Fluids A 4(9), 1901–1914.CrossRefGoogle Scholar
  32. Zaslavsky, D.: 1986, Theory of unsaturated flow in a non-uniform soil profile, Soil Sci. 97, 400–410.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Vivek Kapoor
    • 1
  1. 1.School of Civil and Environmental Engineering, Georgia Institute of TechnologyAtlantaUSA

Personalised recommendations