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ВЛИЯНИЕ КОРРЕЛЯЦИИ МЕЖДУ
ДИЭЛЕКТРИЧЕСКОЙ ПРОНИЦАЕМОСТЬЮ И ПРОВОДИМОСТЬЮ НА ЭФФЕКТИВНУЮ ОЦЕНКУ ЭНЕРГИИ В СКИН-СЛОЕ
Ольга Николаевна Соболева
Институт вычислительной математики и математической геофизики СО РАН, 630090, Россия, г. Новосибирск, пр. Академика Лаврентьева, 6; Новосибирский государственный университет - лаборатория многомасштабной физики и механики фирмы Бейкер Хьюз, 630090, Россия, 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 в неоднородной среде. 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, получены эффективные коэффициенты в уравнениях Максвелла. Коэффициенты электропроводности и диэлектрической проницаемости аппроксимируются мультипликативными каскадами Колмогорова с логарифмически нормальным распределением вероятностей. Показано влияние корреляций между параметрами на эффективную оценку энергии в скин-слое. Теоретические результаты проверяются прямым численным 3Б-моделированием.
Ключевые слова: уравнения Максвелла, эффективные коэффициенты, мультипликативные каскады Колмогорова, логарифмически нормальные распределения вероятностей.
INFLUENCE OF CORRELATIONS BETWEEN PERMITTIVITY AND CONDUCTIVITY ON EFFECTIVE ESTIMATE OF ENERGY FOR SKIN-LAYER
Olga N. Soboleva
Institute of the Computational Mathematics and Mathematical Geophysics SB RAS, 6, Prospect Аkademik Lavrentiev St., Novosibirsk, 630090, Russia; State University - Baker Hughes Joint Laboratory of The Multi-Scale Geophysics and Mechanics, 2, Pirogova St., Novosibirsk, 630090, Russia; Novosibirsk State Technical University, 20, Prospect K. Marx St., Novosibirsk, 630073, Russia, D. Sc., phone: (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 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. The effective coefficients in the Maxwell's equations are calculated for a multiscale isotropic medium. The conductivity and permittivity are mathematically represented by a Kolmogorov multiplicative cascade with a log-normal probability distribution. The theoretical results obtained in the paper are compared with the results of a direct 3D numerical simulation.
Key words: Maxwell's equations, effective coefficients, Kolmogorov multiplicative cascades, log-normal probability distribution.
The quasi-steady condition in a nomagnetic medium are wp0£< 0.1, where £ = £*£q, £q — 8.85 • 10~12F/m and £* <5 — 10 is the relative permittivity, ^ = 4n • 10~7H/m is the magnetic permeability, o = 1/Po, ° is the conductivity. For long probes of the well logging, the quasi-steady condition is satisfied with a high accuracy. However, in heterogeneous media, the quasi-steady condition may be violated, the permittivity affects a measured signal and £* may be equal to 40. In natural condition as a rule, the spatial geometry of small-scale heterogeneities is not exactly known, and the irregularity of electric conductivity and permittivity abruptly increases as the scale of measurements decreases. The numerical solution of the problem with variations of parameters at all the scales requires high computer costs. So, the small-scale heterogeneities are described by random fields with the joint probability distribution functions and they are taken into account with the help of the effective parameters. Many natural media are "scale regular" in the sense that they can be described by multifractals and hierarchical cascade models by [1], [2].
In this paper, by the method of subgrid modeling we obtain formulas of effective coefficients for Maxwell's equations in the frequency domain taking into account the first order terms of &>£(x)/<j(x). In this case, effective coefficients depend not only on means and variances of the parameters, but also on the correlation between the conductivity and permeability. The theoretical results are verified with the help of direct 3D numerical simulation.
Governing equations and the electrical conductivity and permittivity model
Maxwell's equations for monochromatic fields E(x, t) = Re(E(x)e~ia)t>),
in the absence of extraneous currents can be written as
rotH(x) = a(x)E(x) — io)£(x)E(x), rotE(x) = io)H(x), (1)
where E and H are the vectors of electric and magnetic field strengths, respectively; 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. The wavelength is assumed to be large as compared with the maximum scale of heterogeneities of the medium L.
For the approximation of the coefficients o"(x), e(x), we use the approach described in [3].Let the field of permittivity be known. This means that the fields are measured on a small scale l0 at each point x, 0"zo(x) = o"(x), £i0(x) = £(x). To pass to a coarser scale grid, it is not sufficient to smooth the fields <JiQ(x), £i0(x) on the scale I, l> l0. The fields thus smoothed are 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 and permittivity on the scale interval (l0,l) correlate with the fluctuations of electric field
strength E induced by the electrical conductivity and permittivity. To find an electrical conductivity that can describe a physical process on the scales (l,L) system (1) will be used a subgrid modeling. Following [3] we can write down
slo(x) = £00exp ¿1)^), (2)
^0(x) = Oooexp (j^ p(x, (3)
where ct00, £00 are constants. The fields x, V determine the statistical properties of the electrical conductivity and the permittivity. According to the theorem of the sums of independent random variables if the variances of 0, ^(x, Oare finite, the integrals in (2), (3) tends to a field with normal distribution as the ratio L/l0 increases. If the variances of j(x, 0, ^(x, I) are infinite and there exists a non-degenerate limit of the integrals in (2),(3), the integrals tend to a fields with a stable distribution. In the present paper it is assumed that the fields j(x, 0, ^(x, I) are isotropic, statistically homogeneous with a normal distributions. We also assume that the fluctuations of X(x, I), ^(x, I) on different scales do not correlate. This assumption is standard in the scaling models [3]:
<*(x, OKy, O) - (/(X, VHx(y, 0> = 4>**(|x-y|,Z)6(ln I - ln o <p(x, Z)p(y, 0> - 0)<p(y, 0> = OW(|x- y|,Z)5(ln I - ln I') (4) <p(x, ¿My, D> - <^(x, D)(x(y, 0> = Ocpx(|x — y|,/)5(ln I - ln I').
Here the angle brackets denote the ensemble averaging. . For a scale invariant medium, the following relation holds for any positive K
o(|x-y|,0 = o№-yU0
If for any / the equality (e(x)i) = £0 is valid (the conservative cascade), then it follows from (3), (4) that
= 2(x)
As the minimum scale l0 tends to zero, the permittivity and conductivity fields described in (2), (3) become multifractals, and we obtain an irregular fields on a Cantor-type set to be nonzero.
Subgrid modeling
The functions 0"zo(x) = tf(x) , £;0(x) = e(x) are divided into two components with respect to the scale /. The large-scale (ongrid) components o"(x, I), s(x, I) are obtained by statistical averaging over all ^(x, l1), j(x, l1) with l0 <l± <1,1 — l0 = dl, where dl is small. A small-scale (subgrid) components are equal to, &'(x) = &(x) — a(x, I),
£f(x) = s(x) — s(x, I). The large-scale (ongrid) components of electric and magnetic field strengths (x, I) , H(x, I) are obtained by statistical averaging solutions to system (1) in which the large-scale components of <r(x,l), s(x,l) are fixed and the small component o"'(x), £'(x) are a random variables. The subgrid components of the electric and of the magnetic field strengths are equal to H'(x) = H(x) — H(x,l), E'(x) = E(x) — E(x, I). Substituting relations for E(x), H(x) and ct(x) in system (1) and averaging over the small-scale components we obtain
rotH(x, I) = (&(x, I) — ia)s(x, Q)E(x,l) + ((a' -ia)£')E') (5)
rotE(x,l) = io)H(x,l).
The subgrid term ((ct' — ioae'^E') in system (5) 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 (5) 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. Therefore we suppose that o"(x, l),E(x, l),H(x, I) and their first derivatives are believed to change slower than a',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) = (<r(x, 0 -ia)£(x, 0)E'(x) + (a' -ia)s')E(x, I), (6)
rotE' =io)H'
If |<r(x,l) — io)£(x,l)ltetiL2 «1, using the solution of equation (6), we obtain the estimation of subgrid term. Substituting this estimation in (5), as dl ^ 0 we obtain the equation for new tf0(0, £o(0, neglecting the terms of second order of small-ness of ms(x, Q/<j(x, I):
-niir = ®(0,')-l-S0r-<<?>, °-o(io)= "oo (7)
2
Numerical testing
Equations (1) are solved in a unit cube with = 1, H0 = 1 in dimensionless form. An alternating magnetic field with cyclic frequency ^ 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^-k 1 exp^ik 1 zlj2Hx = Hz = 0, Ex = exp(-k 1 zl{P)exp(ik 1 zlJ2 - 74),
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:
where / = 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 ^(x, Tj) has the stable distribution with = 1. In the calculations we use: $(0,/) = 0.2, = 0, L0 = 6hSkin, where hskin is the thickness of the skin-layer at 0 = 1. Figure presents the average real parts of the component of electric and magnetic field strengths at a = 1.7 (log-stable distribution of ct): line 1 is the result for (a) = 1, line 2 is the result with for oef, 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").
The average real parts of the component of electric and magnetic field strengths
REFERENCES
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2. Bekele A., Hudnall H. W., Daigle J.J., Prudente A. and Wolcott M. (2005). Scale dependent variability of soil electrical conductivity by indirect measures of soil properties. Journal of Terramechanics, 42, P. 339-350.
3. Landau L. D. and Lifshitz E. M. Electrodynamics of Continuous Media. New York: Pergamon Press, Oxford-Elmsford; 1984.
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, pp.82-85.
5. Kuz'min G. A., Soboleva O. N. Subgrid modeling of filtration in porous self-similar media. J. Appl. Mech. Tech. Phys. 2002, V.43, pp.583-592.
© О. Н. Соболева, 2018
