Journal of Physical Studies 27(3), Article 3801 [8 pages] (2023)
DOI: https://doi.org/10.30970/jps.27.3801

ATTENUATION OF VELOCITY FIELDS DURING NON-EQUILIBRIUM FILTRATION IN A HALF-SPACE MEDIUM FOR HARMONIC ACTION ON IT

I. I. Denysiuk , I. A. Skurativska , I. M. Hubar 

Subbotin Institute of Geophysics of the NAS of Ukraine, Division of Geodynamics of Explosion,
63G, Bohdan Khmelnytskyi St., Kyiv, UA–01054, Ukraine,
e-mail: inna.skurativska@gmail.com

Received 20 June 2023; in final form 25 July 2023; accepted 18 August 2023; published online 04 September 2023

When developing methods of wave action on porous media for the purpose of intensifying filtration processes, it becomes necessary to evaluate the attenuation of filtration oscillations in porous strata under conditions of intense high-frequency disturbances. To investigate this problem, the methods of mathematical modeling of non-stationary non-equilibrium filtration and the solving of a one-dimensional linear boundary value problem in a half-space with a harmonic disturbance at its boundary are used. The non-equilibrium nature of filtration processes in reservoirs is taken into account in the dynamic law of filtration, which is a generalization of the classical Darcy's law, by including the effects of velocity and pressure relaxation. In this research, filtering equations are considered with one and two relaxation processes, which were described by exponential relaxation kernels.

The exact stationary solutions of boundary value problems were calculated by using the method of separation of variables and the damping coefficients of oscillations in space were determined. Considering the reduced damping coefficients as the ratio of the “non-equilibrium” damping coefficient to the equilibrium one, the dependences of the reduced coefficients on the disturbance frequency, on the ratio of the equilibrium and frozen filtering coefficients, and on the ratio of the relaxation times (in the case of the dynamic Darcy equation with two relaxation processes) were analyzed. In particular, the existence of a frequency that provides a minimum of damping coefficients is shown, and the displacement of the minimum point of the damping coefficient towards lower frequencies with an increase in the ratio of the filtering coefficients is established.

In the case of using Darcy's law with two relaxation processes, depending on the ratio of relaxation times, the curve of the reduced damping coefficient reaches asymptotic values that differ from those corresponding to Darcy's law with one relaxation process. In addition, the points of intersection of the curves corresponding to Darcy's laws with different number of relaxation processes lie on different sides of the minimum point of the curves. Such features of the curves can be useful in the verification of mathematical filtering models. The problem is solved both for a homogeneous and for a non-homogeneous porous media.

Key words: non-equilibrium filtration, generalized Darcy's law, porous medium, wave action, attenuation.

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References
  1. М. В. Худін, О. М. Карпаш, Наук. вісн. ІФНТУНГ 1(34), 89 (2013).
  2. В. В. Яковлев та ін., Акуст. вiсн. 17(3), 32 (2015); http://hydromech.org.ua/content/pdf/av/av-17-3(32-42).pdf
  3. О. І. Гутак, Наук. вісн. ІФНТУНГ 3(29) 53 (2011).
  4. В. М. Казанцев, та ін., Нафтова і газова промисловість 1, 39 (2003).
  5. I. I. Денисюк, В. А. Лемешко, Т. С. Поляковська, Вч. зап. ТНУ ім. В. І. Вернадського. Сер. Техн. науки 30, 25 (2019);
    Crossref
  6. Ю. В. Човнюк, В. Б. Довгалюк, О. М. Скляренко, І. О. Пефтєва, Суч. техн. будів. 27, 96 (2019);
    Crossref
  7. H. Hamidi et al., Ultrasonics 54, 655 (2014);
    Crossref
  8. N. Khan et al., Ultrason. Sonochem. 38, 381 (2017);
    Crossref
  9. A. Dollah et al., Int. J. Eng. Technol. 7(3.11), 232 (2018);
    Crossref
  10. H. Hamidi et al., J. Petrol Explor. Prod. Technol. 2, 29 (2012);
    Crossref
  11. А.В. Кучернюк, Нафтова і газова промисловість 5, 23 (2003).
  12. Ya. M. Bazhaluk et al., Nauk. Visn. Nat. Hirn. Univ. 5, 16 (2016).
  13. C. Pérez-Arancibia, E. Godoy, M. Durán, Wave Motion 77, 214 (2018);
    Crossref
  14. M. Manga et al., Rev. Geophys. 50, RG2004 (2012);
    Crossref
  15. X. Hu, J. H. Cushman, Stochastic Hydrol. Hydraul. 8, 109 (1994);
    Crossref
  16. M. G. Alishaev, A. H. Mirzadzhanzade, Izv. Vuzov. Neft. Gaz No. 6, 71 (1975); (see also https://www.researchgate.net/publication/323487115 ).
  17. B. Khuzhayorov, J.-L. Auriault, P. Royer, Int. J. Eng. Sci. 38(5), 487 (2000);
    Crossref
  18. Yu. M. Molokovich, Nonequilibrium Filtration and its Application in Oilfield Practice (Regular and chaotic dynamics, Moscow, 2006).
  19. N. Rudraiah, P.N. Kaloni, P. V. Radhadevi, Rheol. Acta 28, 48 (1989);
    Crossref
  20. G. I. Barenblatt, T. W. Patzek, D. B. Silin, SPE J. 8(04), 409 (2003);
    Crossref
  21. H. Shokri, M. H. Kayhani, M. Norouzi, Phys. Fluids 29, 033101 (2017);
    Crossref
  22. N. Chemetov, W. Neves, in: \textit{Hyperbolic Conservation Laws and Related Analysis with Applications. Springer Proceedings in Mathematics \& Statistics 49}, edited by G.-Q. G. Chen et al. (Springer-Verlag, Berlin, Heidelberg, 2014), p. 87;
    Crossref
  23. G. I. Barenblatt, V. M. Entov, V. M. Ryzhik, \textit {Theory of Unsteady Liquid and Gas Filtration} (Nedra, Moscow, 1972).
  24. E. Diamantopoulos, W. Durner, Vadose Zone J. 11(3), 1 (2012);
    Crossref
  25. M. Ciarletta, B.Straughan, V.Zampoli, Int. J. Non-Linear Mech. 52, 8 (2013);
    Crossref
  26. G. Rehbinder, Transp. Porous Med. 8, 263 (1992);
    Crossref
  27. B. Kh. Khuzhayorov, E. Ch. Kholiyarov, Sh. Mamatov, AIP Conf. Proc. 2637, 040009 (2022);
    Crossref
  28. V. A. Danylenko, T. B. Danevych, O. S. Makarenko, S. I. Skurativskyi, V. A. Vladimirov, Self-organization in Nonlocal Non-Equilibrium Media (Subbotin Institute of Geophysics of NANU, Kyiv, 2011).
  29. Yu. A. Buevich, G. P. Yasnikov, J. Eng. Phys. 44(3), 340 (1983);
    Crossref
  30. S. I. Skurativskyi, I. A. Skurativska, Ukr. Fiz. Zh. 64(1), 19 (2019);
    Crossref
  31. A. V. Kosterin, J. Eng. Phys. 39, 769 (1980);
    Crossref
  32. R. Helmig, Multiphase Flow and Transport Processes in the Subsurface (Springer, Berlin, 1997).
  33. Y. P. Zheltov, V. S. Kutlyarov, J. Appl. Mech. Tech. Phys. 6, 45 (1965);
    Crossref
  34. A. D. Polianin, V. F. Zaitsev, Handbook of Ordinary Differential Equations. Exact Solutions, Methods, and Problems (CRC Press, 2018).