## 2 Materials and Methods
### Theory: Electromagnetic Properties of Rock Capillarity
In a constant electromagnetic field, the effects of bianisotropic media disappear in an electric field but remain present in a magnetic field. One remarkable property of bianisotropic media is the violation of the principle of reciprocity, which can be illustrated in a simple manner (Figure 9). As a result, the electric current density becomes the sum of the conduction current (from the linear part of the capillarity) and the induction of the magnetic field due to the loop. When the electromagnetic field propagates from top to bottom (from the first source), the electric current in the \(\Omega\) inclusion is the sum of the electric current from the linear part of the capillarity and the electric current density induced by the magnetic field's induction in the loop (\(J_{x}=\sigma E_{x}+aH_{y}\)) (Figure 9). Conversely, when the electromagnetic field propagates from bottom to top (from the second source), the electric current in the \(\Omega\) inclusion is the sum of the electric current density from the linear part of the capillarity and another, which is induced by the magnetic field's induction in the loop but with a  
Figure 9: Demonstration of violation of electromagnetic reciprocity transfer (ERT) for capillarity in an \(\Omega\) inclusion. negative sign (\(J_{x}=\sigma E_{x}-aH_{y}\)). Therefore, the electric current density either increases or decreases depending on the direction of electromagnetic field propagation. This effect can be explained formally as follows:  
If we consider the system of [PERSON]'s equations (Equation (5)) in a homogeneous media with electromagnetic parameters, \(\sigma_{0}\) is media conductivity, \(\mu_{0}\) is the media permeability, and \(i\omega\mathbf{D}\) is equal to 0:  
\[rot\mathbf{H}=\sigma_{0}\mathbf{E},\]  
\[rot\mathbf{E}=-i\omega\mu_{0}\mathbf{H}. \tag{7}\]  
If we represent magnetic field \(\mathbf{\tilde{H}}=f\mathbf{H}\) and electric field \(\mathbf{\tilde{E}}=q\mathbf{E}\) as multiplexing by functions \(f\) and \(q\), which will be defined below, then [PERSON]'s equations for these vectors of the electromagnetic field, taking into account (7), take the form  
\[rot\mathbf{\tilde{H}}=rotf\mathbf{H}=rot\mathbf{H}+gradf\times \mathbf{H}=\sigma_{0}f\frac{q}{q}\mathbf{\tilde{E}}+\frac{gradf}{f}\times f \mathbf{H}=\sigma_{0}\frac{f}{q}\mathbf{\tilde{E}}+\frac{gradf}{f}\times \mathbf{\tilde{H}}\] \[rot\mathbf{\tilde{E}}=rotq\mathbf{E}=qrot\mathbf{E}+gradq\times \mathbf{E}=-i\omega\mu_{0}q\frac{f}{f}\mathbf{H}+\frac{gradq}{q}\times q \mathbf{E}=-i\omega\mu_{0}q\frac{q}{f}\mathbf{\tilde{H}}+\frac{gradq}{q}\times \mathbf{\tilde{E}}\]  
Considering \(f=e^{d}\) and \(q=e^{c}\) are exponential functions, where \(d\) and \(c\) are some functions of spatial coordinates, we obtain [94]  
\[rot\mathbf{\tilde{H}}=\sigma_{0}e^{d-c}\mathbf{\tilde{E}}+gradd \times\mathbf{\tilde{H}},\] \[rot\mathbf{\tilde{E}}=-i\omega\mu_{0}e^{c-d}\mathbf{\tilde{H}}+ gradc\times\mathbf{\tilde{E}}.\]  
Particularly, assuming \(d=c=\alpha_{23}x+\alpha_{13}y+\alpha_{12}z=g\), we obtain  
\[gradg\times= \begin{pmatrix}0&-\alpha_{12}&\alpha_{13}\\ \alpha_{12}&0&-\alpha_{23}\\ -\alpha_{13}&\alpha_{23}&0\end{pmatrix} =\alpha=\beta, \tag{8}\]  
from which we obtain the system of [PERSON]'s equations for bianisotropic media:  
\[rot\mathbf{\tilde{H}}=\sigma_{0}\mathbf{\tilde{E}}+\alpha\mathbf{\tilde{H}}, \tag{9}\] \[rot\mathbf{\tilde{E}}=-i\omega\mu_{0}\mathbf{\tilde{H}}+\beta \mathbf{\tilde{E}}.\]  
Thus, violation of the principle of reciprocity (ERT) in bianisotropic media leads to exponential growth of the electromagnetic field in one direction and an exponential decrease in the opposite direction (Equation (9) vs. Equation (6)). This property can be utilized to study the tortuosity of the capillarity system of electric conductivity in rocks. However, due to the challenges in solving forward and inverse problems for bianisotropic media, there is currently no scientifically justified technical solution for their use in electromagnetic monitoring.