## 2 Materials and Methods
### Theory: Electromagnetic Properties of Rock Capillarity
This type of media is composite. More complex models can be found in [70]. The problem of Maxwell homogenization is calculating the effective parameters for the suggested model of rock electric conductivity. To achieve this, we need to transform the complex system of electric current density \(\mathbf{J}\) into a simpler configuration. The idea behind this transformation is shown in Figure 6 and can be represented as the addition and subtraction of electric current densities at the bottom of each segment of the \(\Omega\) inclusion. In Figure 6, it can be observed that by adding and subtracting currents, whose sum is zero, the complex system of currents within the \(\Omega\) inclusion can be divided into current densities with simpler geometries: a straight current density, \(\mathbf{J}^{s}\), and two closed-loop current densities, \(\mathbf{J}^{s}_{1}\) and \(\mathbf{J}^{s}_{2}=-\mathbf{J}^{s}\). The closed electric currents form two coaxial magnetic dipoles. We can now apply homogenization to calculate the effective electromagnetic parameters of the capillarity system. Closed currents, similar to magnetic dipoles, are volume parameters, and the homogenization process reduces them to calculate the average magnetic moment  
Figure 5: An idealized model of a network of capillaries with a complex configuration and \(\Omega\) inclusions. Figure 6: Assumed diagram of the circulation of electric currents and their separation inside \(\Omega\)-inclusion capillarity, where \(\mathbf{J}^{s}\) is a straight current density and \(\mathbf{J}^{s}_{1}\) and \(\mathbf{J}^{s}_{2}=-\mathbf{J}^{s}\) are two closed-loop current densities. Addition and subtraction are introduced intentionally. for a given volume. Homogenization of a system of coaxial straight currents does not present significant difficulties. It is important to note that a similar transformation can be applied to more complex capillary flow geometries. The scheme for separating electric current densities, taking into account capillary size, is illustrated in Figures 7 and 8, along with the distribution of currents in the capillaries [94; 95]. The loop current density in the inset (Figure 8b) is divided by a dotted line into two components, \(\mathbf{J}^{s}=\mathbf{J}_{1}+\mathbf{J}_{2}\). The need for such a division will be clear from the following discussion. According to the procedure described above, the system of currents is added and subtracted from the current at the bottom of the \(\Omega\) inclusion (Figure 7), provided that  
\[ \begin{array}{l}\int\limits_{3}^{4}\left(\mathbf{J}+\mathbf{J}_{1}+\mathbf{ J}_{2}\right)_{z}dxdy=0,\\ \int\limits_{5}^{6}\left(\mathbf{J}_{1}+\mathbf{J}_{2}\right)_{x}dxdy=0,\end{array} \]  
where points 3, 4, 5, and 6 mark the cross sections of the capillaries where the total current density must be zero. The first condition eliminates currents that are normal to the surface of the capillarity, as shown by the dashed line in Figure 8a,b. The second condition relates the current density in the closed part of the capillarity to the current in its rectilinear part (Figure 8c). In other words, the introduced currents do not change [PERSON]'s equations, but the currents in the \(\Omega\) inclusion are presented as a sum of three components (Figure 8): a rectilinear current density (\(\mathbf{J}^{1}\)) and two closed current densities. The latter have the same direction and are written in the form \(\mathbf{J}_{1}^{s}=rot\mathbf{M}_{1}\) and \(\mathbf{J}_{2}^{s}=rot\mathbf{M}_{2}\) (Figure 8), where closed currents are the curls, which describe the rotation of a vector field enclosing charge \(\mathbf{M}_{1}\) in a bigger loop and \(\mathbf{M}_{2}\) in a smaller loop. In this case, the electric currents in the newly formed currents are continuous. Since the currents are relatively small, they will be ignored. Figure 7: Graphical solution for the separation of electric current densities in \(\Omega\) inclusions (in bianisotropic media).