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 $$\omega \varepsilon (\mathbf {x})/\sigma (\mathbf {x})$$ωε(x)/σ(x), where $$\varepsilon \left( \mathbf {x}\right) $$εx is the permittivity, $$\sigma \left( \mathbf {x}\right) $$σx 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.

中文翻译：

---

具有对数正态分布、多尺度电导率和介电常数的电磁测井问题中的有效系数

使用子网格建模方法计算了多尺度各向同性介质频域中麦克斯韦方程组的有效系数。电导率和介电常数的相关场由具有对数正态概率分布的 Kolmogorov 乘法连续级联近似。与介质的异质性尺度相比，假设波长很大。有效系数的方程在 $$\omega \varepsilon (\mathbf {x})/\sigma (\mathbf {x})$$ωε(x)/σ(x) 的一阶项中获得，其中 $ $\varepsilon \left( \mathbf {x}\right) $$εx 是介电常数，$$\sigma \left( \mathbf {x}\right) $$σx 是电导率，$$\omega $$ ω 是循环频率。获得的有效参数与频率无关，因此它们也是时域中的有效参数。将理论结果与直接 3D 数值模拟的结果进行比较。如果参数 $$\sigma $$σ 和 $$\varepsilon $$ε 弱相关，则在某些条件下的介电常数会影响准稳态情况下的测量信号。