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УДК 550.34
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ РАСПРОСТРАНЕНИЯ СЕЙСМИЧЕСКИХ ВОЛН С УЧЕТОМ СНЕЖНОГО ПОКРОВА
Сайера Тахировна Туйчиева
Ташкентский институт инженеров железнодорожного транспорта, 100167, Республика Узбекистан, г. Ташкент, ул. Адилходжаева, 1, старший преподаватель, тел. (371)276-88-71, е-mail: icenter@tashiit.uz
Гюльнара Маратовна Воскобойникова
Институт вычислительной математики и математической геофизики СО РАН, 630090, Россия, г. Новосибирск, пр. Академика Лаврентьева, 6, кандидат технических наук, научный сотрудник, тел. (383)330-83-52, е-mail: gulya@opg.sscc.ru
Холматжон Худайназарович Имомназаров
Институт вычислительной математики и математической геофизики СО РАН, 630090, Россия, г. Новосибирск, пр. Академика Лаврентьева, 6, доктор физико-математических наук, ведущий научный сотрудник, тел. (383)330-83-52, е-mail: imom@omzg.sscc.ru
Жиан-Ган Тан
Илийский педагогический университет, Китай, г. Кульджа, e-mail: tjg@ylsy.edu.cn Александр Анатольевич Михайлов
Институт вычислительной математики и математической геофизики СО РАН, 630090, Россия, г. Новосибирск, пр. Академика Лаврентьева, 6, кандидат физико-математических наук, научный сотрудник, тел. (383)330-83-52, е-mail: alex_mikh@omzg.sscc.ru
В статье на основе теории динамической пороупругости проводится математическое моделирование распространения сейсмических волн в снежной среде. Снежная среда аппроксимируется как пористая среда, где три упругих параметра выражаются тремя скоростями упругих волн. В свою очередь эти скорости пересчитываются через скорости упругих волн согласно теории Био, которые выражены через упругие параметры снежной среды (Johnson J.B., 1982). Отметим, что полученные решения позволяют изучить особенности распространения сейсмических волн в насыщенных жидкостью (воздухом) снежных средах. При этом полученные формулы позволяют моделировать скорости смещений пористого каркаса и насыщающей жидкости в нем, а также порового давления и компоненты тензора напряжений, как при заданных упругих параметрах среды, так и при заданных скоростях распространений поперечных и продольных волн в пористой среде.
Ключевые слова: снежная среда, медленная волна, математическая модель, зондирующий сигнал.
MATHEMATICAL MODELING OF SEISMIC WAVE PROPAGATION WITH ALLOWANCE FOR THE SNOW COVER
Sayyora T. Tuychieva
Tashkent Institute of Railway Transport Engineers, 100167, Republic of Uzbekistan, Tashkent, Adilkhodjaev st., 1, senior lecturer, tel. (371)276-88-71, e-mail: icenter@tashiit.uz
Gulnara M. Voskoboynikova
Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of RAS, 630090, Russia, Novosibirsk, Lavrentiev Ave, 6, Candidate of Science, Leading Researcher, tel. (383)330-83-52, e-mail: gulya@opg.sscc.ru
Kholmatzhon Kh. Imomnazarov
Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of RAS, 630090, Russia, Novosibirsk, Lavrentiev Ave, 6, Doctor of Science, Leading Researcher, tel. (383)330-83-52, e-mail: imom@omzg.sscc.ru
Jian-Gang Tang
YiLi Normal University, 448 Jiefang Road, Yinning Xinjiang, P.R.of China, e-mail: tjg@ylsy.edu.cn Aleksander A. Mikhailov
Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of RAS, 630090, Russia, Novosibirsk, Lavrentiev Ave, 6, Candidate of Science, Researcher, tel. (383)330-83-52, e-mail: alex_mikh@omzg.sscc.ru
In this paper, based on the theory of dynamic poroelasticity, mathematical modeling of seismic wave propagation in the snow medium is considered. The snow medium is approximated as a porous medium, where the three elastic parameters are expressed by the three elastic wave velocities. In turn, these velocities are recalculated using the elastic waves velocities according to the Biot theory. Such velocities are expressed through the elastic parameters of the snow media (J. B. Johnson, 1982). Note that the solutions obtained allow the study of peculiarities of the seismic wave propagation in the water (air) -saturated snow medium. In this case, the obtained formulas allow us to simulate the displacement velocity of the porous skeleton and the saturating fluid in it, as well as the pore pressure and the stress tensor components with given elastic parameters of the medium and the velocities of propagation of transverse and longitudinal waves in a porous medium.
Key words: snow medium, slow wave, mathematical model, sounding signal.
The understanding of the propagation of seismic waves in the snow has been the object of theoretical and experimental studies many researchers. Such an interest is the result of a desire to use acoustic techniques for: non-destructive methods of the snow texture classification; determination of mechanical parameters for the snow; developing effective methods of explosive control for snow slopes; and monitoring acoustic emissions.
The development of textural classification techniques and determination of the appropriate mechanical parameters for the snow require an accurate acoustic propagation model. Pre-existing seismic wave propagation models made use of either porous media representations, which assumed a rigid ice skeleton, or continuum elastic or inelastic models [1-6]. These models do not adequately explain the wave propagation phenomena observed in the snow. The air pressure waves, propagating in the interstitial pore space, and dilatational and shear stress waves, propagating in the ice skeleton, were detected in the snow [7-10]. However, neither the porous medium or the continuum models can explain all the three wave propagation modes. In [11], based on the Biot model the propagation of seismic waves in the snow as a porous material with an elastic skeleton saturated with a compressible viscous fluid (air) was
discussed. The characteristics of the solutions describing acoustic waves in the snow are discussed in detail and compared with the experimental results [8, 9, 12, 13]. In this paper, three elastic parameters [14-16] for the propagation of seismic waves in the snow using dynamic theory poroelasticity are described.
Equations of motion
In poroelasticity theory [14], the stress-strain relations for a porous aggregate including the effects of fluid pressure and dilatation are considered. As well as in [11, 17] we consider the dynamics of a material and the coupling between a fluid and a solid provided that the material is statistically homogeneous and isotropic in the region of interest, behaves in a linearly elastic manner and that thermoelastic effects are negligible. The macroscopic stress-strain relation for the medium was derived by assuming the isotropic medium. The coupling effect between the elastic skeleton and the compressible fluid was taken into account by introducing a kinetic coefficient into the dissipation function of the system [14]. Dissipation of energy by the viscous fluid was expressed in terms of a relative velocity between the fluid and the solid.
The constitutive relations describing the porous material by the tree parameters are the following [16]:
( к Л
^ij ~ + AskkStj - 1-— p8ip (1)
V (-' у
p = (K- apps )skk - aPPlekk, (2)
_ 1 /дщ диЛ _ 1 /dUt dUj\
where cr is the stress in the solid framework, p is the fluid pore pressure, % and e,, are the dilatations of the solid and the fluid, ¿r„ is the strain in the solid, and 4 is
K-K- IJ IJ
the Kronecker delta, и = (щ, u2, u3) and U = (Ult U2, U3) are the displacement vectors of the elastic matrix and the saturating fluid with the respective partial densities
ps and pi, p = pi + ps, x = Я-(ар2у1к2, к = X + -ju, X,ii,ap2 are elastic parameters
of the porous medium [14].
The dynamic poroelasticity theory for the porous media is derived using the method of conservation laws, which have the forms [18]
on 1 daik p, cp p, dv I dp , ч „
dt Ps dxk PsPdx, Ps dt Рдхг
d<Tik -- + U
dt
(л, я-,, Л i ^ 1 Л
8u. du —- + —:
V. ^ ^ j
p 3
dikdivu - — Kdikdivv = 0,
P
dp
--К - apps divu + appl divv = 0.
dt
where / is the friction coefficient.
Determining the mechanical parameters for the snow
The dynamic nature of a porous material is determined by the three elastic parameters K,//, and a. These parameters are expressed in terms of the velocity of propagation of the shear wave cs and of the velocities of the longitudinal waves Vcn [12, 13].
,, - n r2 Y - P Ps 2 p,
f
+c2 _»Ac2_ a a 3 p '
2 2 c -c
Pi Pi
64 p,p:
9 p
s 4
—c
2 s
1
2p2
+c2 _!Ac 2 +
A ft 3 p ' <
c -c
Pi Pi
2 2 64 p,ps
P
2
4
«J =
The velocities cs and , are determined by the Biot parameters A, N, R, and
Q. These parameters are calculated from the four measurable coefficients for the snow: Table 1 [11]. The friction coefficient is expressed by the dissipation coefficient from [11, 17].
In this paper, an algorithm to numerically solve the 2D dynamic problem of seismic wave propagation for the porous medium with allowance for the energy dissipation is considered. To numerically solve this problem, a method for combining the Laguerre integral transform with respect to time with a finite-difference approximation along the spatial coordinates is used. The proposed method of the solution can be considered as analog to the known spectral-difference method based on Fourier transform, only instead of frequency we have a parameter m, i.e. the degree of the Laguerre polynomials. However, unlike Fourier transform, application of the La-guerre integral transform with respect to time allows us to reduce the initial problem to solving a system of equations in which the parameter of division is present only in the right-hand side of the equations and has a recurrent dependence. The algorithm used for the solution makes it possible to perform efficient calculations when simulating a complicated porous medium and studying wave effects emerging in such media.
This work was supported by the Russian Foundation for Basic Research (grant № 16-07-01052).
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© С. Т. Туйчиева, Г. М. Воскобойникова, Х. Х. Имомназаров, Жиан-Ган Тан, А. А. Михайлов, 2016
