Effective coefficients in the electromagnetic logging problem with log-normal distribution, multiscale conductivity and permittivity

The effective coefficients for Maxwell’s equations in the frequency domain for a multiscale isotropic medium by using a subgrid modeling approach are calculated. The correlated fields of conductivity and permittivity are approximated by the Kolmogorov multiplicative continuous cascades with a lognormal probability distribution. The wavelength is assumed to be large when compared with the scale of heterogeneities of the medium. The equations for effective coefficients are obtained in the first order terms of $$\omega \varepsilon (\mathbf {x})/\sigma (\mathbf {x})$$ ω ε ( x ) / σ ( x ) , where $$\varepsilon \left( \mathbf {x}\right) $$ ε x is the permittivity, $$\sigma \left( \mathbf {x}\right) $$ σ x is the electric conductivity and $$\omega $$ ω is the cyclic frequency. The obtained effective parameters are frequency-independent and therefore it follows that they are also the effective parameters in the time domain. The theoretical results are compared with the results from direct 3D numerical simulations. The permittivity under certain conditions can influence a measured signal in a quasi-steady case if the parameters $$\sigma $$ σ and $$\varepsilon $$ ε are weakly correlated.