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УДК 537.874.4
ЭФФЕКТИВНЫЕ КОЭФФИЦИЕНТЫ В ЗАДАЧЕ
ЭЛЕКТРОМАГНИТНОГО КАРОТАЖА В МНОГОМАСШТАБНОЙ СРЕДЕ С ЛОГАРИФМИЧЕСКИ-УСТОЙЧИВОЙ ПРОВОДИМОСТЬЮ
Ольга Николаевна Соболева
Институт вычислительной математики и математической геофизики, 630090, Россия, г. Новосибирск, пр. Академика Лаврентьева, 6; Новосибирский государственный университет -лаборатория многомасштабной физики и механики фирмы Бейкер Хьюз, 630090, Россия, г. Новосибирск, ул. Пирогова, 2; Новосибирский государственный технический университет, 630073, Россия, г. Новосибирск, пр. К. Маркса, 20, доктор физико-математических наук, тел. (383)330-60-46, e-mail: olga@nmsf.sscc.ru
Метод электромагнитного каротажа очень эффективен при изучении строения Земли. Основной целью этого метода является оценка электропроводности. При численном моделировании крупные неоднородные включения в среде учитываются в математической модели непосредственно с помощью граничных условий. Расчеты с учетом всех вариаций физических параметров требуют громадных вычислительных затрат. Чтобы решить эту проблему при численном моделировании многомасштабных неоднородных сред используют эффективные коэффициенты. В работе, используя метод подсеточного моделирования, получены эффективные коэффициенты в квазистационарных уравнениях Максвелла. Коэффициент электропроводности аппроксимируются мультипликативным каскадом Колмогорова с логарифмически устойчивым распределением вероятностей. Теоретические результаты проверяются прямым численным 3 D моделированием.
Ключевые слова: квазистационарные уравнения Максвелла, эффективные коэффициенты, мультипликативные каскады Колмогорова, устойчивые распределения вероятностей.
EFFECTIVE COEFFICIENTS FOR THE PROBLEM OF ELECTROMAGNETIC LOG IN MULTI-SCALE MEDIUM WITH LOG-STABLE CONDUCTIVITY
Olga N. Soboleva
Institute of Computational Mathematics and Mathematical Geophysics, 630090, Russia, Novosibirsk, 6 Akademik Lavrentiev Prospect; Novosibirsk State University - Baker Hughes Joint Laboratory of The Multi-Scale Geophysics and Mechanics, 630090, Russia, Novosibirsk, 2 Pirogova St.; Novosibirsk State Technical University, 630073, Russia, Novosibirsk, 20 K. Marks Prospect, D. Sc., tel. (383)330-60-46, e-mail: olga@nmsf.sscc.ru
Electromagnetic logging is an effective tool for studying Earth structure. The main aim is to estimate the medium conductivity as correctly as possible. The large-scale medium heterogeneities are taken into account in mathematical models with the help of some boundary conditions. The numerical solution of the problem with variations of parameters on all the scales requires high computer costs. The small-scale heterogeneities are taken into account in the effective parameters used. In this case, equations are found on the scales that can be numerically resolved. The effective coefficients in the quasi-steady Maxwell's equations are calculated for a multiscale isotropic medium by using a subgrid modeling approach. The conductivity is mathematically represented by a Kolmogo-rov multiplicative cascade with a log-stable probability distribution. The theoretical results obtained in the paper are compared with the results of a direct 3D numerical simulation.
Key words: quasi-steady Maxwell's equations, effective coefficients, Kolmogorov's multiplicative cascades, stable probability distribution.
Introduction
Electromagnetic logging is an effective tool for studying a medium structure. The main aim is to estimate the medium conductivity as correctly as possible. The large-scale medium heterogeneities are taken into account in mathematical models with the help of some boundary conditions. The numerical solution of the problem with variations of parameters on all the scales requires high computer costs. The small-scale heterogeneities are taken into account in the effective parameters used [1]. In this case, equations are found on the scales that can be numerically resolved. The method includes the two-scale homogenization is well known in the scientific community [2]. The authors consider physical problems arising in media with a periodic structure. Quite often the size of the period is small compared to the size of a sample of the medium, and, denoting their ratio by £ an asymptotic analysis, as ^ 0, is required: namely, starting from a microscopic description of a problem, the authors seek a macroscopic, or averaged, description. It has been shown that the irregularity of electrical conductivity, density, permeability, porosity, increases as the scale of measurements decreases for some natural media [1]. Many natural media are "scale regular" in the sense that they can be described by multifractals and hierarchical cascade models. In this paper, we apply the subgrid modeling method to hierarchical cascade models of media with log-stable distributions of electrical conductivity. The heterogeneities of the medium are represented by the spatial distribution of the local electrical conductivity that has essential variations of all scales from a finite interval at each spatial point. The width of the skin layer is assumed to be large as compared with the scale of heterogeneities of the medium and the parameter p characterizing the asymmetry of log-stable distribution is equal to unity, the parameter a satisfies the condition 1 < a < 2, hence, variance is infinite. The derived formulas for 3D media are verified by the direct numerical modeling.
Governing equations and the electrical conductivity model
According to [3], the quasi-steady approximation of Maxwell's equations for monochromatic fields E(x,t) = Re(E(x)e-iMt),H(x,t) = Re(H(x)e-iMt), in the absence of extraneous currents can be written as
rotH(x) = a(x)E(x), (1)
rotE(x) = iwH(x),
where E and H are the vectors of electric and magnetic field strengths, respectively; ^ is the magnetic permeability; a(x) is the electric conductivity; & is the cyclic frequency; and x is the vector of spatial coordinates. The magnetic permeability is assumed to be equal to the magnetic permeability of vacuum. We also assume that the electrical conductivity is constant outside of a finite volume V with a sufficiently smooth surface S. At the surface S, the tangent components of electric and magnetic field strengths are continuous.
Let the field of electrical conductivity be known. This means that the field is measured on a small scale l0 at each point x, g1q(x) = o(x). To pass to a coarser scale grid, it is not sufficient to smooth the field &lo(x) on the scale I, I > l0. The
field thus smoothed is not a physical parameter that can describe the physical process on the scales (I, L), where L is the maximum scale of heterogeneities. This is due to the fact that the fluctuations of electrical conductivity on the scale interval (l0,I) correlate with the fluctuations of electric field strength E induced by the electrical conductivity. To find an electrical conductivity that can describe a physical process on the scales (I, L) system (1) will be used in this paper. Following Kolmogorov [4] consider a dimensionless field which is equal to the ratio of two fields obtained by smoothing the field g1q(x) on two different scales 1,1'. Let al(x) denote parameter &io(x) smoothed on the scale l. Then ^ = al'(x)/ al(x). We obtain expanding the field ^ into a power series in I — I' and retaining the first order terms of series, at I' ^ I the following equation:
d-^) = y(x,H (2)
d\nl v '
where <p(x, I) = (d^(x, I, ly))ly=1. The solution of equation (2) is
°io(x) = °ooexp (ilLo M (3)
where a00 is constant. The field <p determines the statistical properties of the electrical conductivity. This approach is described in detail in [5]. According to the theorem of the sums of independent random variables if the variance of <p(x, I) is finite, the integral in (3) tends to a field with normal distribution as the ratio L/l0 increases. If the variance of <p(x, I) is infinite and there exists a non-degenerate limit of the integral in (3), the integral tends to a field with a stable distribution. In the present paper it is assumed that the field <p(x, I) is isotropic, statistically homogeneous with a stable distribution. We also assume that the fluctuations of <p(x, I) on different scales do not correlate. This assumption is standard in the scaling models [4]. This is due to the fact that the statistical dependence is small if the scales of fluctuations are different.
Subgrid modeling
The electrical conductivity function &lo(x) = a(x) is divided into two components with respect to the scale I. The large-scale (ongrid) component a(x, I) is obtained by statistical averaging over all <p(x, l1) with l0 < l1 < I, I — l0 = dl, where dl is small. A small-scale (subgrid) component is equal to a'(x) = a(x) — a(x,l). The large-scale (ongrid) components of electric and magnetic field strengths (x, I), H(x,l) are obtained by statistical averaging solutions to system (1) in which the large-scale component of conductivity a(x, I) is fixed and the small component a'(x) is a random variable. The subgrid components of the electric and of the magnetic field strengths are equal to H'(x) = H(x) — H(x, I), E'(x) = E(x) — E(x, I). Substituting relations for E(x), H(x) and a(x) in system (1) and averaging over the small-scale components we obtain
rotH(x, I) = a(x, l)E(x, I) + (a'E') (4)
rotE(x, I) = i<vH(x, I).
The subgrid term (a'E') in system (4) is unknown. This term cannot be neglected without preliminary estimation, since the correlation between the electrical conductivity and the electric field strength may be significant. The form of this term in (4) determines the subgrid model. The subgrid term is estimated using the perturbation theory for the fields, in which a small variation of the scale causes significant fluctuations of the field itself. This is a property of multiplicative cascades. The mul-tifractals can be obtained if a minimum scale l0 ^ 0 in formula (2). Therefore we suppose that a(x, I), E(x, I), H(x, I) and their first derivatives are believed to change slower than o',E',H' and their derivatives. Subtracting system (4) from system (1), and, taking into account only the first order terms, we obtain the subgrid equations:
rotH'(x) = a(x, l)E'(x) + a'E(x, I), (5)
rotE' = iwH'
If m^L2a(x, l) «1, using the solution of equation (5), we obtain the estimation of subgrid term. Substituting this estimation in (4), as dl ^ 0 we obtain the equation for new a0(l):
31no"o(i) , n 2(1-2a)+3 . . n , _
-nnr^0^"1-^-^, a0(l0)= ^00. (6)
2
Numerical testing
Equations (1) are solved in a unit cube with a00 = 1, H0 = 1 in dimensionless form. An alternating magnetic field with cyclic frequency w acts on a conducting medium. This field satisfies the condition H = On the boundary with
vacuum, z = 0, the component Hy(z) is assumed to be constant, Hy(0) = H0. Outside the cube and on the cube boundaries, the magnetic and electric fields strengths are specified by the formulas
Hy = exp(~kYz / yjl)exp[ikYz 1^2), Hx = Hz = 0,
Ex = exP (_kiz!
exp (iki z / 72 -n /4),
Ey = Ez = 0.
The conductivity field in the cube is simulated by formula (3) in which it is convenient to pass to a logarithm to base 2:
g(X) = 2£7=4-6<p(x,^,
where I = 2T, At is the t grid size. In our calculations At is taken to be one. A 256x256x256 grid is used for the space variables. The field v(x,t{) has the stable distribution with ft = 1. In the calculations we use: 0(0,1) = 0.2, = 0, L0 = 6hSkin, where hskin is the thickness of the skin-layer at a = 1. Figure 1 presents the average real parts of the component of electric and magnetic field strengths at a =
1.7 (log-stable distribution of a): line 1 is the result for (a) = 1, line 2 is the result with for aef, line 3 is the result numerical modeling for 30 realizations and line 4 is the numerical and theoretical result for a = 2 (lognormal distribution of a)
Figure 1. The average real parts of the component of electric and magnetic field
strengths
Acknowledgments
The work was supported by the RFBR N15-01-01458.
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3. Landau L. D. and Lifshitz E. M. Electrodynamics of Continuous Media. - New York : Pergamon Press, Oxford-Elmsford, 1984. - 423 p.
4. Kolmogorov A. N. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number // J. Fluid Mech. - 1962. -V.13. - P. 82-85.
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© O. Н. Соболева, 2017
