THERMAL SCIENCE

International Scientific Journal

Thermal Science - Online First

online first only

Numerical research of non-isothermal filtration process in fractal medium with non-locality in time

ABSTRACT
The difference scheme for the numerical solution of boundary problem for a system of equations for non-isothermal filtration with a Caputo derivative of fractional order on time is developed. Stability of the differential scheme is proved. Computational experiment in the analysis of solutions obtained has been done. Physical processes pass slowly in the fractal medium with non-locality in time. It is explained by the fact the occasionally wandering particle is being eliminated from the start place slowly, since not all directions of the movement become available for it. Values of pressure and temperature depending on the coordinate of layer radius and time calculated, and graphs of the dynamics pressure and temperature changes according to the layer radius and in depending on the time are built. Deceleration of the processes with time in the solutions for fractional derivatives which is characteristic for such medium has been established.
KEYWORDS
PAPER SUBMITTED: 2019-02-23
PAPER REVISED: 2019-07-25
PAPER ACCEPTED: 2019-08-01
PUBLISHED ONLINE: 2019-09-15
DOI REFERENCE: https://doi.org/10.2298/TSCI190223328B
REFERENCES
  1. M. G. Alishaev, M. D. Rozenberg, and E. V. Tesluk. Non-isothermal filtration in the development of oil fields. Nedra, Moscow, 1985. (In Russian).
  2. L. S. Leibenzon. The movement of natural liquids and gases in a porous medium. OGIZ, Gostekhizdat, 1947. (In Russian).
  3. F. M. Bochever, I. V. Garmonov, A. V. Lebedev, and V. M. Shestokov. Basics of hydrogeological calculations. Nedra, Moscow, 1965. (In Russian).
  4. R. P. Meilanov and M. R. Shabanova. Peculiarities of non-isothermal filtration accounting memory effects and spatial correlations. Vozobnovlyaemaya energetika: Problemi i perspektivi, 3:96-99, 2014. (In Russian).
  5. E. N. Akhmedov and R. R. Meilanov. Peculiarities of temperature and pressure distribution in the reservoir with non-local non-isothermal filtration. Vozobnovlyaemaya energetika: Problemi i perspektivi, 4:314-319, 2015. (In Russian).
  6. R. R. Meilanov, E. N. Akhmedov, V. D. Beybalaev, R. A. Magomedov, G. B. Ragimkhanov, and A. A. Aliverdiev. To the theory of non-local non-isothermal filtration in porous medium. J. Phys.: Conf. Ser., 946:012076, 2018.
  7. R. P. Meilanov. To the theory of filtration in porous media with a fractal structure. Tech. Phys. Lett, 22(23):40-42, 1996.
  8. A. M. Nakhushev. Fractional calculus and its application. FIZMATLIT, Moscow, 2003. (In Russian).
  9. Jordan Hristov. Approximate solutions to fractional subdiffusion equations. The European Physical Journal Special Topics, 193(1):229-243, 2011.
  10. F. Mainardi, Y. Luchko, and G. Pagnini. The fundamental solutions of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal., 4(2):153-192, 2011.
  11. V. D. Beybalaev and M.R. Shabanova. A finite-difference scheme for solution a fractional heat diffusion-wave equation conditions. Thermal science, 19(2):531-536, 2015.
  12. V. D. Beybalaev, I. A. Abdullaev, K. A. Navruzova, and T. Yu. Gadjieva. On difference methods for solving the Cauchy problem for an ode with a fractional differentiation operator. Vestnik DGU, 6:53-61, 2014. (In Russian).
  13. V. D. Beybalaev, A. A. Aliverdiev, R. A. Magomedov, R. R. Meilanov, and E. N. Akhmedov. Modeling of freezing processes by an one-dimensional thermal conductivity equation with fractional differentiation operators. Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya: Fizikomatematicheskie nauki, 21(2):378-387, 2017. (In Russian).
  14. V. E. Tarasov. Fractional ntegro-differential equations for electromagnetic waves in dielectric media. Theoret. and Math. Phys., 158(3):355-359, 2009.
  15. V. E. Tarasov. Fractional generalization of the quantum markovian master equation. Theoret. and Math. Phys., 158(2):179-195, 2009.
  16. V. D. Beybalaev and R. R. Meilanov. Dirihlet problem for the fractional poisons equation with Caputo derivatives: A finite difference approximation and a numerical solution. Thermal Science, 16(2):385- 394, 2012.
  17. A. A. Samarskiy and A. B. Gulin. Numerical methods. Nauka, Moscow, 1989. (In Russian).