CC BY
4The cases of different velocity structure of the medium are considered: homogeneous, gradient, etc., as
well as various types of the initial wave front position surface. The results of numerical experiments are pre-
sented.
References
1. Gjoystdal H., Iversen E., Laurain R. et al. Review of ray theory applications in modelling and imaging of seismic data
// Stud. Geoph. Geod. 2002. V. 46. P. 113-164.
2. Cerveny V. Seismic Ray Theory. New York: Cambridge University Press, 2001.
3. Rabiner L. R., Gold B. Theory and Application of Digital Signal Processing. Prentice Hall Inc, 1975. Ch. 5. P. 300-302.
Exact Navier � Stokes solutions for oil and gas problems
V. A. Galkin1,2, A. O. Dubovik1,2
1Surgut Branch of Scientific Research Institute for System Analysis RAS
2Surgut State University
Email: alldubovik@gmail.com
DOI 10.24412/cl-35065-2021-1-01-02
The system of Navier-Stokes equations is considered, which describes the flow of a viscous incompressible
fluid in a porous medium. The simplest model of a porous medium consists of a discrete set of points - grid
nodes, which is the boundary of the flow region, in which the adhesion boundary condition is satisfied. Classes
of exact solutions corresponding to vortex flow are presented [1]. The study of exact solutions is necessary for
the development of a core simulator and simulation of fluid dynamics in a porous medium, its response to dy-
namic effects of various types and the creation of a domestic technology "digital field" [2].
This work was supported by the Russian Foundation for Basic Research (grants 18-01-00343, 18-47-860005).
References
1. Betelin V.B., Galkin V.A., Dubovik A.O. Exact Solutions of Incompressible Navier�Stokes Equations in the Case of Oil
and Gas Industrial Problems // Doklady Mathematics. 2020. V.102. No 3. pp.456�459. DOI: 10.1134/S1064562420060071.
2. Betelin V.B., Galkin V.A. Control of Incompressible Fluid Parameters in the Case of Time-Varying Flow Geometry//
Doklady Mathematics. 2015. V. 92. No 1. pp. 511-513. DOI: 10.1134/S1064562415040067.
Investigation of the propagation of nonlinear waves in a two-fluid medium caused by an analogue
of the Darcy coefficient
B. Kh. Imomnazarov1, B. B. Khudainazarov2
1Novosibirsk State University
2National University of Uzbekistan named after Mirzo Ulugbek
Email: imom@omzg.sscc.ru
DOI 10.24412/cl-35065-2021-1-01-03
The extreme difficulty of analyzing nonlinear waves, especially strong turbulence, led to another tendency
in the development of their theory - the transition from complex equations of nonlinear random waves to
simpler model equations [1, 2]. In this paper, a system of quasi-linear equations of hyperbolic type is obtained
from a system of non-stationary equations of two-velocity hydrodynamics [3-5]. It is believed that the energy
dissipation occurs due to the analogue of the Darcy. We study the Cauchy problem for a given system of equa-
tions in the class of bounded measurable functions based on the Kruzhkov method.
The support of the Russian Science Foundation under grant � 21-51-15002 is gratefully acknowledged.
References
1. Gurbatov S.N., Saichev A.I., Yakushkin I.G. Nonlinear waves and one-dimensional turbulence in media without
dispersion // UFN, 1983, vol. 141, pp. 221-255.
2. Erkinova D.A., Imomnazarov B.Kh., Imomnazarov Kh.Kh. A one-dimensional system of equations of the Hopf type
// Regional scientific and practical. conf. "TOGU-Start: fundamental and applied research of young people", April 12-16,
2021, Khabarovsk, pp. 61-69.
3. Imomnazarov Sh., Imomnazarov Kh., Kholmurodov A., Dilmuradov N., Mamatkulov M. On a Problem Arising in a
Two-Fluid Medium // International Journal of Mathematical Analysis and Applications, 2018, No. 5(4), pp. 95-100.
4. Imomnazarov Kh.Kh., Mikhailov A.A. Rakhmonov T.T. Simulation of the seismic wave propagation in porous media
described by three elastic parameters // SEMI, 2019, �. 16, pp. 591-599. DOI 10.33048/semi.2019.16.037
5. Baishemirov Z., Tang, J.-G., Imomnazarov K., Mamatqulov M. Solving the problem of two viscous incompressible
fluid media in the case of constant phase saturations // Open Engineering, 2020, 6(1), pp. 742�745.
Three-dimensional stationary flows of viscous fluids of a two-phase continuum with phase equilibrium
with respect to pressure with a singular source in the disipative case
Sh. Kh. Imomnazarov1, B. Kh. Imomnazarov2, B. B. Khudainazarov3
1Institute of Computational Mathematics and Mathematical Geophysics, SB RAS
2Novosibirsk State University
3National University of Uzbekistan named after Mirzo Ulugbek, Tashkent
Email: imom@omzg.sscc.ru
DOI 10.24412/cl-35065-2021-1-01-04
In this paper, an overdetermined system of equations is obtained from the system of non-stationary equa-
tions of two-velocity hydrodynamics in the dissipative case [1-4]. It is believed that the energy dissipation oc-
curs due to the analogue of the Darcy. Construction of a solution for describing three-dimensional stationary
flows of viscous fluids of a two-phase continuum with phase equilibrium with respect to pressure with a singu-
lar source in the dissipative case.
The support of the Russian Science Foundation under grant � 21-51-15002 is gratefully acknowledged.
References
1. Imomnazarov Kh.Kh., Imomnazarov Sh.Kh., Mamatkulov, M.M., Chernykh, E.G. Fundamental solution for a
stationary equation of two-velocity hydrodynamicswith one pressure // Sib. Zh. Ind. Mat, 2014, v. 17, pp. 60-66.
2. Imomnazarov Sh., Imomnazarov Kh., Kholmurodov A., Dilmuradov N., Mamatkulov M. On a Problem Arising in a
Two-Fluid Medium // International Journal of Mathematical Analysis and Applications, 2018, No. 5(4), pp. 95-100.
3. Imomnazarov Kh.Kh., Mikhailov A.A. Rakhmonov T.T. Simulation of the seismic wave propagation in porous media
described by three elastic parameters // SEMI, 2019, �. 16, pp. 591-599. DOI 10.33048/semi.2019.16.037
4. Baishemirov Z., Tang, J.-G., Imomnazarov K., Mamatqulov M. Solving the problem of two viscous incompressible
fluid media in the case of constant phase saturations // Open Engineering, 2020, 6(1), pp. 742�745.
Numerical modeling and physical effects of interwave interactions
M. S. Khairetdinov, G. M. Shimanskaya
Institute of Computational Mathematics and Mathematical Geophysics SB RAS
E-mail: marat@opg.sscc.ru
DOI 10.24412/cl-35065-2021-1-02-89
The problems of studying the interaction of conjugate geophysical wave fields of different nature, arising
from natural and man-made sources simultaneously in different environments: seismic in the lithosphere,
acoustic in the atmosphere, hydroacoustic in the hydrosphere, optical, meteorological in the atmosphere are
