Advertisement

Mathematical modeling of flows from canals

  • É. N. Bereslavskii
Article

In the hydrodynamic formulation, two-dimensional steady filtration in homogeneous isotropic ground from canals through a soil layer to the underlying highly permeable pressure water-bearing stratum is considered in the presence of the ground capillarity and evaporation from the free surface. To study filtration, a combined multiparametric boundary-value problem of the theory of analytical functions is formulated, which is solved using the P. Ya. Polubarinova-Kochina method and procedures of conformal mapping of regions of a special kind that are characteristic of the problems of subsurface hydromechanics. On the basis of this model an algorithm of calculating the capillary water spreading and the filtration discharge is developed for the situations where in water filtration from canals provision is made for the ground capillarity, evaporation from the free-surface of groundwater, and the additional pressure from the side of water of the underlying wellpermeable bed. With the aid of the obtained accurate analytical relations and numerical calculations a hydrodynamic analysis is made of the structure and character of specific features of the modeled process as well as of the effect of all physical parameters of the scheme on the filtration characteristics.

Keywords

filtration groundwater ground capillarity evaporation pressure complex velocity 

References

  1. 1.
    V. V. Vedernikov, Influence of capillary rise on filtration from canals, Gidrotekh. Stroit., No. 3, 20–27 (1935).Google Scholar
  2. 2.
    B. K. Riesenkampf, Hydraulics of Groundwater, Pt. 3, in: Trans. Saratov Univ., Ser. Hydraulics, 15, No. 5, 3–93 (1940).Google Scholar
  3. 3.
    P. Ya. Polubarinova-Kochina, The Theory of Motion of Groundwater [in Russian], Gostekhizdat, Moscow (1952), 2nd ed., Nauka, Moscow (1977).Google Scholar
  4. 4.
    V. I. Aravin and S. N. Numerov, The Theory of Motion of Liquids and Gases in an Undeformable Porous Medium [in Russian], Gostekhizdat, Moscow (1953).Google Scholar
  5. 5.
    G. K. Mikhailov and V. N. Nikolaevskii, Motion of liquids and gases in porous media, in: Mechanics in the USSR in 50 Years [in Russian], Vol. 2, Nauka, Moscow (1970), pp. 585–648. Google Scholar
  6. 6.
    N. N. Verigin, Water filtration from the irrigation canal, Dokl. Akad. Nauk SSSR, 66, No. 4, 589–593 (1949).MathSciNetGoogle Scholar
  7. 7.
    S. N. Numerov, A method of solving filtration problems, Izv. Akad. Nauk SSSR, OTN, No. 4, 133–139 (1954).Google Scholar
  8. 8.
    V. A. Vasil’ev, Filtration from a canal with a shallow layer of water with account for capillarity, in: Proc. Middle-Asian Univ., 83, No. 14, 43–57 (1958).Google Scholar
  9. 9.
    N. B. Ilinskii and A. R. Kasimov, The inverse problem of filtration from a canal in the presence of the additional pressure, in: Proc. of Seminar on Inverse Problems [in Russian], Izd. Kazansk. Univ., Issue 20, Kazan’ (1983), pp. 104–115.Google Scholar
  10. 10.
    V. A. Baron, Filtration from a canal with a shallow layer of water in the presence of a well permeable layer at the finite depth and with account for infiltration, Prikl. Mekh. Tekh. Fiz., No. 1, 101–105 (1961).Google Scholar
  11. 11.
    V. N. Émikh, On the regime of groundwater in an irrigated soil layer with an underlying highly permeable pressurized stratum, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 2, 168–174 (1979).Google Scholar
  12. 12.
    É. N. Bereslavskii, On the problem of filtration from an irrigation canal, Izv. Akad. Nauk USSR, Mekh. Zhidk. Gaza, No. 2, 105–109 (1987).Google Scholar
  13. 13.
    É. N. Bereslavskii and V. V. Matveev, Filtration from small-depth and irrigation canals, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1, 96–102 (1989).Google Scholar
  14. 14.
    É. N. Bereslavskii, On the regime of groundwater in filtration from an irrigation canal, Prikl. Mekh. Tekh. Fiz., No. 5, 88–91 (1989).Google Scholar
  15. 15.
    V. V. Golubev, Lectures on the Analytical Theory of Linear Differential Equations [in Russian], Gostekhizdat, Moscow–Leningrad (1950).Google Scholar
  16. 16.
    É. N. Bereslavskii, On the Fuchs-class differential equations related to conformal mapping of circular polygons in polar grids, Differ. Uravn., 33, No. 3, 296–301 (1997).MathSciNetGoogle Scholar
  17. 17.
    É. N. Bereslavskii, On the closed-form integration of some Fuchs-class differential equations encountered in hydro- and aeromechanics, Dokl. Ross. Akad. Nauk, 428, No. 4, 439–443 (2009).Google Scholar
  18. 18.
    É. N. Bereslavskii, On closed-form integration of some Fuchs-class differential equations related to conformal mapping of circular pentagons with a cut, Differ. Uravn., 46, No. 4, 459–466 (2010).MathSciNetGoogle Scholar
  19. 19.
    É. N. Bereslavskii, On accounting for infiltration or evaporation from a free surface by the method of circular pentagons, Prikl. Mat. Mekh., 74, Issue 2, 239–251 (2010).MathSciNetGoogle Scholar
  20. 20.
    É. N. Bereslavskii and P. Ya. Kochina, On some Fuchs-class equations in hydro- and aeromechanics, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 3–7 (1992).Google Scholar
  21. 21.
    P. Ya. Kochina, É. N. Bereslavskii, and N. N. Kochina, The Analytical Theory of Fuchs-Class Linear Differential Equations and Some Problems of Underground Hydromechanics, Pt. 1, Preprint No. 567 of the Institute for Problems of Mechanics, Russian Academy of Sciences, Moscow (1996).Google Scholar
  22. 22.
    É. N. Bereslavskii and P. Ya. Kochina, On the Fuchs-class differential equations encountered in some problems of the mechanics of liquids and gases, Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 9–17 (1997).Google Scholar
  23. 23.
    É. N. Bereslavskii and N. V. Zakharenkova, Influence of the ground capillarity and of evaporation from the free groundwater surface on filtration from canals, Inzh.-Fiz. Zh., 83, No. 3, 470–477 (2010).Google Scholar
  24. 24.
    I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Nauka, Moscow (1971).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.State University of Civil AviationSt. PetersburgRussia

Personalised recommendations