## 2 Materials and Methods
### Theory: Electromagnetic Properties of Rock Capillarity  
An important problem in the detailed description of the physicoelectric parameters of rocks is the complex configuration of hydrodynamic capillarity systems. In the context of the theory of effective media approximation (EMA), capillarity is treated as a conductor, while the host rock is considered an insulator. This allows for the geological parameter known as the tortuosity of the capillarity system to be obtained. EMA helps us to avoid special theory for electrokinetic (due to capillarity fluid mineralization) and mechano-electrical effects (rock failure and piezoelectric effects). The study of rocks with a developed capillarity system leads us to the formulation of new material equations known as bianisotropic media theory [83; 84; 85; 86; 87; 88]. Bianisotropic media are characterized by linear electromagnetic properties described by general [PERSON]'s equations:  
\[\mathbf{J}=\sigma\mathbf{E}+\xi\mathbf{H},\quad\mathbf{B}=\zeta\mathbf{E}+ \mu\mathbf{H} \tag{2}\]  
\[\text{rot}\mathbf{H}=\sigma\mathbf{E}+\text{d}\mathbf{D}/\text{d}\text{t}, \text{rot}\mathbf{E}=-\text{d}\mathbf{B}/\text{d}\text{t}, \tag{3}\]  
where \(\mathbf{J}\) is the electric current density (A/m\({}^{2}\)), \(\mathbf{B}\) is the magnetic induction intensity (Wb/m\({}^{2}\)), \(\mathbf{E}\) is the electric field intensity vector (V/m), \(\mathbf{H}\) is the magnetic field intensity vector (A/m), \(\mathbf{D}=\epsilon\mathbf{E}\) is the electric displacement vector (C/m\({}^{2}\)), \(\epsilon\) is the dielectric constant, \(\sigma\) is the electric conductivity, and \(\mu\) is magnetic permeability [89; 68]. The items d\(\mathbf{D}/\text{d}\)t and d\(\mathbf{B}/\text{d}\)t are time derivatives. \(\xi\) and \(\zeta\) are called effective parameters and are new in the theory and application of electrical prospecting and have not yet received definite designations. Equation (3) is given in differential form. The effective parameters \(\xi\) and \(\zeta\) are necessary for an adequate description of the electromagnetic behavior of a rock with a complex system of capillaries. They take into account the tortuosity of capillaries filled with a conducting fluid. The physical properties depend on the connectivity of joint patterns [60]. Physical interpretation of these parameters is associated with the generation of electric currents through induced electromotive force (parameter \(\zeta\)) and the appearance of magnetic dipoles (parameter \(\zeta\)) due to the presence of closed conductors in the media. These conductors, or closed loop currents, arise due to the complex geometry of the capillarity system, which can theoretically be divided into simpler elements. The electromagnetic behavior and electrical resistivity features of bianisotropic media have been previously studied [88; 89; 90; 91; 92]. In previous works by the authors, it was demonstrated that the effective parameters \(\sigma\), \(\mu\), \(\xi\), and \(\zeta\) are \(3\times 3\) matrices because they correspond to three spatial components where anisotropy is observed [17; 91]. In this study, we will focus on obtaining the effective parameters for a rock with a complex conductivity system. We first consider a rock volume that contains a system of capillaries with a complex configuration and infinite length (closed porosity). In this case, the electric conductivity of the media is solely determined by the conductivity of fluids filling the thin, extended capillaries. The idealized model of a horizontally layered system with capillaries is shown in Figure 5, where \(x\), \(y\), and \(z\) are spatial coordinates. In this model, each individual capillary filled with fluid forms a rectilinear electric current that is galvanically connected to a loop-shaped capillary, referred to as the \(\Omega\) (omega shape) inclusion [93; 94]. The media in this case is considered dielectric, and the magnetic permeability remains constant along the entire length of the loop, equal to the magnetic permeability of a vacuum.