Differential Equations

, Volume 54, Issue 9, pp 1215–1224 | Cite as

Study of Three-Dimensional Boundary Value Problems of Fluid Filtration in an Anisotropic Porous Medium

  • V. F. Piven’Email author
Integral Equations


Three-dimensional boundary value problems (the first and second boundary value problems and the conjugation problem) of stationary filtration of fluids in anisotropic (orthotropic) and inhomogeneous porous media are posed and studied. A medium is characterized by a symmetric permeability tensor whose components generally depend on the coordinates of points of the space. A nonsingular affine transformation of coordinates is used and the problems are stated in canonical form, which dramatically simplifies their study. In the case of orthotropic and piecewise orthotropic homogeneous medium, the solution of the problem with canonical boundaries (plane and ellipsoid surfaces) can be obtained in finite form. In the general case, where the orthotropic medium is inhomogeneous and the boundary surfaces are arbitrary and smooth, the problem can be reduced to singular and hypersingular integral equations. The problems are topical, for example, in the practice of fluid (water, oil) recovery from natural anisotropic and inhomogeneous soil strata.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Radygin, V.M. and Golubeva, O.V., Primenenie funktsii kompleksnogo peremennogo v zadachakh fiziki i tekhniki (Application of Functions of Complex Variable in Problems of Physics and Technology), Moscow: Vysshaya Shkola, 1983.zbMATHGoogle Scholar
  2. 2.
    Piven’, V.F., Statement of basic boundary value problems of filtration in an anisotropic porous medium, in Proc. XIII Intern. Symp. “Methods of Discreet Singularities in Problems of Mathematical Physics” (MDOZMF-2007), Kharkiv, Kherson, 2007, pp. 239–243.Google Scholar
  3. 3.
    Piven’, V.F., Teoriya i prilozheniya matematicheskikh modelei fil’tratsionnykh techenii zhidkosti (Theory and Applications of Mathematical Models of Liquid Filtration Flows), Orel: Orel Gos. Univ., 2006.Google Scholar
  4. 4.
    Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1967.Google Scholar
  5. 5.
    Mikhlin, S.G., Lektsii po lineinym integral’nym uravneniyam (Lectures on Linear Integral Equations), Moscow: Gos. Izd. Fiz. Mat. Lit., 1959.Google Scholar
  6. 6.
    Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, New Haven, CI: Yale Univ. Press, 1923; reprinted, New York: Dover, 1952. Translated under the title Zadachi Koshi dlya lineinykh uravnenii s chastnymi proizvodnymi giperbolicheskogo tipa, Moscow: Nauka, 1978.zbMATHGoogle Scholar
  7. 7.
    Lifanov, I.K., Metod singulyarnykh integral’nykh uravnenii i chislennyi eksperiment (Method of Singular Integral Equations and Numerical Experiment), Moscow: Yanus, 1995.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Turgenev Orel State UniversityOrelRussia

Personalised recommendations