Skip to main content
SpringerLink
Log in

A variational principle for problems of steady-state ground water flow with a free surface

  • Published:
Fluid Dynamics Aims and scope Submit manuscript

An Erratum to this article was published on 01 September 2015

Abstract

The problem of steady-state free surface ground water flow through a porous medium is represented in the form of a variational principle for a functional dependent on the flow domain. It is proved that on the region to be found this functional takes a minimum value. This variational principle is generalized to include the flow models that allow the presence of partially saturated zones in the medium. On simple examples it is shown how the variational formulation can be used for proving the existence or absence of solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Ya. Polubarinova-Kochina. Theory of Ground Water Movement (Nauka, Moscow, 1977 [in Russian]; Princeton Univ. Press, Princeton, 1962).

    Google Scholar 

  2. A. G. Egorov, R. Z. Dautov, J. L. Nieber, and A. Y. Sheshukov, “Stability Analysis of Gravity-Driven Infiltrating Flow,” Water Resour. Res. 39 (9), 1886–1928 (2003).

    Google Scholar 

  3. J. R. Philip, “Stability Analysis of Infiltration,” Soil Sci. Soc. Amer. Proc. 39 (6), 1042–1049 (1975).

    Article  Google Scholar 

  4. I. I. Lyashko, I. M. Velikoivanenko, V. I. Lavrik, and G E. Mistetskii, Method of Majorizing Domains in the Theory of Flow through Porous Media (Naukova Dumka, Kiev, 1974) [in Russian].

    Google Scholar 

  5. J. Bear, D. Zaslavsky, and S. Irmay, Physical Principles of Water Percolation and Seepage (UNESCO, Paris, 1968)

    Google Scholar 

  6. B. Schweizer, “Regularization of Outflow Problems in Unsaturated Porous Media with Dry Regions,” J. Differential Equations 237 (2), 278–306 (2007).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. S. N. Antontsev, J. I. Diaz, and S. I. Shmarev, Energy Methods for Free Boundary Problems: With Applications to Nonlinear PDEs and Fluid Mechanics (Springer, New York, 2002).

    Book  Google Scholar 

  8. C. Baiocchi, V. Comincioli, E. Magenes, and G. A. Pozzi, “Free Boundary Problems in the Theory of Fluid Flow through Porous Media: Existence and Uniqueness Theorems,” Ann. Mat. Pura Appl. 97 (4), 1–82 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Duvaut, J.-L. Lions, Les Inéquations en Mécanique et Physique (Dunod, Paris, 1972).

    MATH  Google Scholar 

  10. A. Friedman, Variational Principles and Free Boundary Problems (Willey, New York, 1982).

    MATH  Google Scholar 

  11. C. Baiocchi and A. Friedman, “A Filtration Problem in a Porous Medium with Variable Permeability,” Ann. Mat. Pura Appl. 114 (1), 377–393 (1977).

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Chipot, Variational Inequalities and Flow through Porous Media (Springer, New York, 1984).

    Book  Google Scholar 

  13. H. Brézis, D. Kinderlehrer, and G. Stampacchia, “Sur une Nouvelle Formulation du Problème d’Eqoulement Travers une Digue,” C. R. Acad. Sci. Paris Ser. A 287, 711–714 (1978).

    MATH  MathSciNet  Google Scholar 

  14. H. W. Alt, “Strömungen durch inhomogene poröse Medien mit freiem Rand”, J. Reine Angew. Math. 305, 89–115 (1979).

    MATH  MathSciNet  Google Scholar 

  15. W. H. Green and G. A. Ampt, “Studies in Soil Physics. 1. The Flow of Air and Water through Soils,” J. Agricult. Sci. 4, 1–24 (1911).

    Article  Google Scholar 

  16. J. R. Philip, “An Infiltration Equation with Physical Significance,” Soil Sci. 77, 153–157 (1954).

    Article  Google Scholar 

  17. M. Lenzinger and B. Schweizer, “Two-Phase Flow Equations with Outflow Boundary Conditions in the Hydrophobic-Hydrophilic Case,” Nonlinear Analysis TMA 73 (4), 840–853 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  18. F. G. McCaffery and D. W. Bennion, “The Effect of Wettability on Two-Phase Relative Permeabilities,” J. Can. Pet. Technol. 13 (4), 42–53 (1974).

    Article  Google Scholar 

  19. P. Ustohal, F. Stauffer, and T. Dracos, “Measurement and Modelling of Hydraulic Characteristics of Unsaturated Porous Media with MixedWettability,” J. Contaminant Hydrology 33, 5–37 (1998).

    Article  ADS  Google Scholar 

  20. A. T. Il’ichev and G. G. Tsypkin, “Catastrophic Transition to Instability of Evaporation Front in a Porous Medium,” Eur. J. Mech. B/Fluids 27 (6), 665–677 (2008).

    Article  MATH  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Water Problems Institute, Russian Academy of Sciences, ul. Gubkina 3, Moscow, 119333, Russia

    A. I. Belyaev & I. O. Yushmanov

Authors
  1. A. I. Belyaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. O. Yushmanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. I. Belyaev.

Additional information

Original Russian Text © A.I. Belyaev, I.O. Yushmanov, 2015, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2015, Vol. 50, No. 4, pp. 92–109.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belyaev, A.I., Yushmanov, I.O. A variational principle for problems of steady-state ground water flow with a free surface. Fluid Dyn 50, 543–557 (2015). https://doi.org/10.1134/S0015462815040092

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0015462815040092

Keywords

Search

Navigation