Institute of Computational Mathematics and Mathematical Geophysics SB RAS

International Conference on Mathematical Methods in Geophysics «MMG-2008»

Akademgorodok, Novosibirsk, Russia, October 13-15, 2008

Abstracts

Mathematical Simulation in Geophysical Problems. Electromagnetic Methods

Mixing laws and two-scale homogenization of the Maxwell equations

Shelukhin V.V., Terentev S.A.

Lavrentyev Institute of Hydrodynamics of SB RAS (Novosibirsk),
Baker Hughes Russian Science Center (Novosibirsk)

We developed a mixing law theory for effective dielectric permittivity and effective electric conductivity by two-scale homogenization of the Maxwell equations. The approach is well justified for rocks with periodical structure, and it gives rise to a numerical algorithm which works well both for DC and AC frequencies. The code was tested successfully for the cubic array of nonintersecting spheres embedded in a matrix by means of comparing effective parameters obtained by the two-scale homogenization presented here and those computed by traditional mixture formulae such as the Hanai-Bruggeman formula.

As for real rock structures, calculations were performed for two rock models, with solid grains being intersecting spheres of the same radius. The first periodicity cell is formed of eight spheres centered at the unit cube vertices. The second cell of periodicity has one more sphere in the center of the cube. The Maxwell-Wagner dispersion effect is revealed to take place at rather high frequencies. Nevertheless a shift of the Maxwell-Wagner dispersion phenomena into the low frequency domain is possible when the rock cell is filled with plate grains. This suggestion is due to the analytic dispersion curve which we found for the cell with layered structure.

The homogenization method enables us to comment on the Archie formula. New inconsistencies of the Archie law were discovered. Particularly, we made it clear that porosity was not the only geometrical factor of importance in calculating effective conductivity; the percolation threshold should be taken into account as well, and the cementation factor depends significantly on the interval where porosity varies. Interestingly, according to our calculations, the value of the cementation factor is close to 1.5 for the best Archie percolation approximation of the homogenization conductivity/porosity curve.

Though in the present paper we considered rocks with simple geometrical structures, the method can be applied to rocks with complex structures as well, in contrast to other mixing rules. Moreover, the method allows us to take into account polarization of solid and fluid components.

Note. Abstracts are published in author's edition

Mail to Webmaster

[Home]
[Conferences]