Numerical modeling of the seismic waves propagation in a porous medium from singular sources
https://doi.org/10.25587/SVFU.2023.92.13.007
Abstract
A linear two-dimensional problem in the form of dynamic equations of porous media for the components of velocities, stresses and pressure is considered. The dynamic equations are based on conservation laws and consistent with the thermodynamics conditions. The medium is considered to be ideal (there is no energy loss in the system) isotropic and two-dimensional inhomogeneous with respect to space. For the numerical solution of the problem posed, the method of integrating the integral Laguerre transform with respect to time with finite-difference approximation in spatial coordinates is used. The solution algorithm employed makes it possible to efficiently carry out simulations in a complex porous medium and to study the wave effects arising in such media.
About the Authors
Kh. Kh. Imomnazarov
Institute of Conputational Mathematics and Mathematical Physics
Russian Federation
Russian Federation
Kholmatzhon Kh. Imomnazarov
6 Lavrentiev Avenue, Novosibirsk 630090
A. A. Mikhailov
Institute of Conputational Mathematics and Mathematical Physics
Russian Federation
Russian Federation
Aleksandr A. Mikhailov
6 Lavrentiev Avenue, Novosibirsk 630090
A. T. Omonov
Tashkent State University of Economics
Uzbekistan
Uzbekistan
Alisher T. Omonov
49 Islom Karimov Street, Tashkent 100066
S. Tordeux
Universit´e de Pau et des Pays de l’Adour
France
France
Sebastien Tordeux
BP 1155, 64013 Pau Cedex
References
1. Chen X., Zhang Y., and Yan S., “Two-dimensional simulations of resin flow in dual-scale fibrous porous medium under constant pressure,” J. Reinforced Plastics Compos., 32, No. 22, 1757–1766 (2013).
2. Gantois R., Cantarel A., Dusserre G., Felices J. N., and Schmidt F., “Mold filling simulation of resin transfer molding combining BEM and level set method,” Appl. Mech. Mater., 62, 57–65 (2011).
3. Loudad R., Saouab A., Beauchene P., Agogue R., and Desjoyeaux B., “Numerical modeling of vacuum-assisted resin transfer molding using multilayer approach,” J. Compos. Mater., 51, No. 24, 3441–3452 (2017).
4. Song Y. S., “Mathematical analysis of resin flow through fibrous porous media,” Appl. Compos. Mater., 13, No. 5, 335–343 (2006).
5. Yang B., Tang Q., Wang S., Jin T., and Bi F., “Three-dimensional numerical simulation of the filling stage in resin infusion process,” J. Compos. Mater., 50, No. 29, 4171–4186 (2016).
6. Darcy H., Les Fontaines Publiques de la Ville de Dijon, V. Dalmont, ed., Paris (1856).
7. Darcy H., Recherches Exp´erimentales Relatives au Mouvement de L’eau Dans les Tuyaux, Mallet-Bachelier, Paris (1857).
8. Frenkel Ya. I., “On the theory of seismic and seismoelectric phenomena in a moist soil,” J. Phys. USSR, No. 8, 230–241 (1944).
9. Biot M. A., “Theory of propagation of elastic waves in fluid-saturated porous solid. I. Low-frequincy range,” J. Acoust. Soc. Amer., 28, 168–178 (1956).
10. Dorovsky V. N., “Continual theory of filtration,” Sov. Geol. Geofiz., No. 7, 34–39 (1989).
11. Dorovsky V. N., Perepechko Yu. V., and Romensky E. I., “Wave processes in saturated porous elastically deformed media,” Comb. Explos. Shock Waves, No. 1, 100–111 (1993).
12. Blokhin A. M. and Dorovsky V. N., Mathematical Modelling in the Theory of Multivelocity Continuum, Nova Sci., New York (1995).
13. Lake L., Enhanced Oil Recovery, Prentice Hall, Englewood Cliffs (1989).
14. Vafai K., Porous Media: Applications in Biological Systems and Biotechnology, Taylor&Francis, London (2011).
15. Aker E., Maloy K. J., and Hansen A., “Simulating temporal evolution of pressure in two-phase flow in porous media,” Phys. Rev., E 58, 2217–2226 (1998).
16. Zhao B. et al., “Comprehensive comparison of pore-scale models for multiphase flow in porous media,” PNAS, No. 116, 13799–13806 (2019).
17. Imomnazarov Kh. Kh., “Characteristics of interference wave fields in the presence of a porous layer [in Russian],” Dokl. RAN, 352, No. 1, 105–108 (1997).
18. Imomnazarov Kh. Kh., “Numerical simulation of some problems of filtration theory for porous media [in Russian],” Sib. Zh. Ind. Mat., 4, No. 2, 154–165 (2001).
19. Alekseev A. S., Imomnazarov Kh. Kh., Grachev E. V., Rakhmonov T. T., and Imomnazarov B. Kh., “Direct and inverse dynamic problems for a system of equations of homogeneous elastic-porous media,” Appl. Math. Lett., 17, No. 9, 1097–1103 (2004).
20. Imomnazarov Kh. Kh. and Zhapbasbay B. M., “Optimization method to solve the inverse problem of electrokinetics for vertically inhomogeneous media,” J. Inverse Ill-Posed Probl., 12, No. 5, 481–491 (2004).
21. Imomnazarov Kh. Kh. and Kholmurodov A. E., “Direct and inverse dynamic problems for SH-waves in porous media,” Math. Comput. Model., 45, No. 3-4, 270–280 (2007).
22. Sorokin K. E. and Imomnazarov Kh. Kh., “Numerical solution of a linear two-dimensional dynamic problem for porous media [in Russian],” Zh. Sib. Fed. Univ., Ser. Mat. Fiz., 3, No. 2, 256–261 (2010).
23. Imomnazarov Kh. Kh., Imomnazarov Sh. Kh., Rakhmonov T. T., and Yangiboev Z. Sh., “Regularization in inverse dynamic problems for the equation of SH waves in a porous medium [in Russian],” Vladikavkaz. Mat. Zh., 15, No. 2, 46–58 (2013).
24. Imomnazarov Kh. Kh. and Mikhailov A. A., “Application of a spectral method to numerical modeling of the propagation of seismic waves in porous media with energy dissipation,” Numer. Anal. Appl., 7, No. 2, 117–123 (2014).
25. Imomnazarov Kh. Kh., Imomnazarov Sh. Kh., Korobov P. V., and Kholmurodov A. E., “Direct and inverse problems for nonlinear one-dimensional poroelasticity equations,” Dokl. Math., 89, No. 2, 250–252 (2014).
26. Priimenko V. I. and Vishnevskii M. P., “An identification problem related to the Biot system,” J. Inverse Ill-Posed Probl., 23, No. 3, 219–230 (2015).
27. Imomnazarov Kh. Kh., Mikhailov A. A., and Rakhmonov T. T., “Simulation of the seismic wave propagation in porous media described by three elastic parameters,” Sib. Elektron. Mat. Izv., 16, 591–599 (2019).
28. Vishnevskii M. P. and Priimenko V. I., “On the solvability of some dynamic poroelastic problems,” Sib. Math. J., 60, No. 3, 429–449 (2019).
29. Imomnazarov Kh. Kh. and Jabborov N. M., “Application of A-analytic functions to the investigation of the Cauchy problem for a stationary poroelasticity system,” J. Contemp. Math., Fundamental Directions, 65, No. 1, 33–43 (2019).
30. Vicente B. J., Priimenko V. I., and Pires A. P., “Mathematical model of water alternated polymer injection,” Transp. Porous Media, 135, 431–456 (2020).
31. , “A system of equations of the two-velocity hydrodynamics without pressure,” AIP Conf. Proc., 2365, 070002 (2021).
32. Imomnazarov Kh., Kholmuradov A., and Dilmuradov N., “Direct and inverse dynamic quasilinear problems of poroelasticity,” AIP Conf. Proc., 2365, 070020 (2021).
33. Da Silva R. B. and Priimenko V. I., “An analytical solution of the saturated and incompressible poroelastic model for transient wave propagation,” Euaras. J. Math. Comput. Appl., 9, No. 2, 12–31 (2021).
34. Imomnazarov Kh. Kh., “A mathematical model for the movement of a conducting liquid through a conducting porous medium: I. Excitation of oscilations of the magnetic field by the surface Rayleigh wave,” Math. Comput. Model., 24, No. 1, 79–84 (1996).
35. Imomnazarov Kh. Kh., “Some remarks on the system of Biot equations [in Russian],” Dokl. Akad. Nauk, 373, No. 4, 536–537 (2000).
36. Imomnazarov Kh. Kh., “Some remarks on the Biot system of equations describing wave propagation in a porous medium,” Appl. Math. Lett., 13, No. 3, 33–35 (2000).
37. Mikhailenko B. G., Mikhailov A. A., and Reshetova G. V., “Numerical modeling of transient seismic fields in viscoelastic media based on the Laguerre spectral method,” Pure Apll. Geophys., No. 160, 1207–1224 (2003).
38. Mikhailenko B. G., Mikhailov A. A., and Reshetova G. V., “Numerical viscoelastic modeling by the spectral Laguerre method,” Geophys. Prospect., No. 51, 37–48 (2003).
39. Levander A. R., “Fourth order velocity-stress finite-difference scheme,” Proc. 57th SEG Annu. Meeting, pp. 234-245, New Orleans (1987).
Review
For citations:
Imomnazarov Kh.Kh., Mikhailov A.A., Omonov A.T., Tordeux S. Numerical modeling of the seismic waves propagation in a porous medium from singular sources. Mathematical notes of NEFU. 2023;30(1):89-100. (In Russ.) https://doi.org/10.25587/SVFU.2023.92.13.007
Views:
5
JATS XML
Email this article 