## 2 Materials and Methods
### Theory: Electromagnetic Properties of Rock Capillarity
Points 3, 4, 5, and 6 mark the cross sections of the capillaries, where the total current density must be zero. Addition and subtraction are introduced intentionally for loop dividing. The original [PERSON]'s equations from Equation (3), where \(\sigma\) is the electric conductivity of the capillarity and \(\mu\) is the magnetic permeability of the media, can then be transformed into the complex form with the imaginary unit \(i\):  
\[ \begin{array}{l}rot\mathbf{H}=\sigma\mathbf{E}+i\omega\mathbf{D}=\mathbf{J}+ \mathbf{J}_{1}-\mathbf{J}_{1}+\mathbf{J}_{2}-\mathbf{J}_{2}=\mathbf{J}^{1}+ \mathbf{J}_{1}^{s}+\mathbf{J}_{2}^{s},\\ =rot\mathbf{J}^{1}+rot\mathbf{M}_{1}+rot\mathbf{M}_{2}\\ rot\mathbf{E}=-i\omega\mu\mathbf{H}.\end{array}  \tag{4}\]  
The model is now supplemented with a capillarity system oriented along the Y axis and not interacting with the original system (Figure 8) and a system of linear capillaries oriented along the vertical Z axis under conductivity, at which the conductivity of the rock will be the same in all directions. Ignoring the interaction between \(\Omega\) inclusions, we could apply coordinate averaging for electrical conductivity: \(\sigma^{e}=\frac{1}{\frac{1}{\Delta X}\int\frac{1}{\frac{1}{\Delta y}\frac{1}{ \sigma y}\sigma y}dx}\) (see Appendix A). Therefore, the material equations (Equation (2)) averaged over the rock volume in complex form are written as  
\[ \begin{array}{l}\mathbf{J}=\mathbf{i}J_{x}=\mathbf{i}\sigma^{e}E_{x}- \mathbf{i}i\omega\mu\alpha^{e}H_{y},\\ \mathbf{B}=\mathbf{j}\mu\beta^{e}E_{x}+\mu\mathbf{H},\end{array}  \tag{5}\]  
where \(\alpha^{e}=na\); \(\beta^{e}=n\beta\sigma^{e}\) are some parameters of pore and capillarity tortuosity; \(n\) is a quantity of inserts in a volume unit; \(\mathbf{i}\) and \(\mathbf{j}\) are capillarity units inside the volume; and \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\) are spatial coordinates. Thus, the effective parameters \(\xi\) and \(\zeta\) from Equation (2) become complex \(\xi=i\omega\mu\alpha^{e}\) and \(\xi=\mu\beta^{e}\). Accordingly, other spatial components are  
\[ \begin{array}{l}J_{x}=\sigma E_{x}-i\omega\mu A_{y};\ B_{x}=\mu H_{x}-\mu \beta E_{y};\\ J_{y}=\sigma E_{y}+i\omega\mu A_{x};\ B_{y}=\mu H_{y}+\mu\beta E_{x};\end{array} \]  
\(J_{z}=\sigma E_{z}\); \(B_{z}=\mu H_{z}\), because the model is horizontally layered. Figure 8: Representation of electric current densities in \(\Omega\) inclusions (in bianisotropic media) as a sum of three components: (**a**) upper part showing loop current density (\(\mathbf{J}_{1}^{s}\)); (**b**) central part showing loop current density (\(\mathbf{J}_{2}^{s}\)); (**c**) lower part showing linear current density (\(\mathbf{J}^{1}\)). The material equations in general view then take the complex vector form  
\[\mathbf{J} =\widehat{\sigma}\,\mathbf{E}-i\omega\eta\widehat{\alpha}\,\mathbf{H}, \tag{6}\] \[\mathbf{B} =\mu\,\widehat{\beta}\,\mathbf{E}+\,\widehat{\mu}\,\mathbf{H},\]  
where \(\widehat{\sigma}= \begin{pmatrix}\sigma&0&0\\ 0&\sigma&0\\ 0&0&\sigma\end{pmatrix} \); \(\widehat{\alpha}= \begin{pmatrix}0&\alpha&0\\ -\alpha&0&0\\ 0&0&0\end{pmatrix} \); \(\widehat{\beta}= \begin{pmatrix}0&-\beta&0\\ \beta&0&0\\ 0&0&0\end{pmatrix} \); \(\widehat{\mu}= \begin{pmatrix}\mu&0&0\\ 0&\mu&0\\ 0&0&\mu\end{pmatrix} \). The bianisotropic effective parameters \(\widehat{\alpha}\) and \(\widehat{\beta}\) are antisymmetric matrices, and their structure depends on the orientation of the \(\Omega\) inclusion relative to the straight, linear part of the capillarity. For instance, \(\widehat{\alpha}\) and \(\widehat{\beta}\) change signs (opposite signs) when the \(\Omega\) inclusion is rotated by \(180^{\circ}\) relative to the rectilinear segment (resulting in an L-shaped media). Other orientations lead to different structures for matrices \(\widehat{\alpha}\) and \(\widehat{\beta}\). For example, a \(90^{\circ}\) rotation gives rise to diagonal matrices known as Tellegen media [96].