## 3 Theoretical Development
### Electrical Conductivity  
Adapting the reported approach for the electrical conductivity in porous media, e.g., [46; 83], the bulk conductivity within a single fracture is given by  
\[\sigma_{fb}=\sigma_{w}\frac{2 aw}{A_{REV\tau}}=\sigma_{w}\frac{2\beta w^{2}}{A _{REV\tau}}. \tag{17}\]  
The surface conductivity within a single fracture is given by  
\[\sigma_{fs}=\Sigma_{s}\frac{2w+4a}{A_{REV\tau}}=\Sigma_{s}\frac{2(1+2\beta)w} {A_{REV\tau}}, \tag{18}\]  
Figure 2: Fluid flow in a fracture and the coordinate along the aperture (\(a<<w<<L_{o}\)).  
recall that \(2 aw\) and \(2w+4a\) are the cross sectional area and the perimeter of the fracture (see Figure 1). Consequently, the total electrical conductivity in a fracture is given by  
\[\sigma_{f}(w)=\sigma_{fb}+\sigma_{fs}=\sigma_{w}\frac{2\beta w^{2}}{A_{REV\tau} }+\Sigma_{s}\frac{2(1+2\beta)w}{A_{REV\tau}}. \tag{19}\]  
The electrical conductivity of the REV \(\sigma\) under saturated conditions is obtained by integrating over all saturated capillaries as  
\[\sigma=\int_{w_{min}}^{w_{max}}\sigma_{f}(w)(-dN). \tag{20}\]  
Combining Equations (5), (19) and (20), the following is obtained  
\[\begin{split}\sigma=&\frac{1}{A_{REV\tau}}\Bigg{\{} 2\sigma_{w}\beta D_{f}w_{max}^{2}\frac{1-\alpha^{2-D_{f}}}{2-D_{f}}+2\Sigma_{ s}(1+2\beta)D_{f}w_{max}\frac{1-\alpha^{1-D_{f}}}{1-D_{f}}\Bigg{\}}.\\ &=\frac{2\beta D_{f}w_{max}^{2}(1-\alpha^{2-D_{f}})}{A_{REV}(2- D_{f})\tau}\Bigg{\{}\sigma_{w}+\frac{(1+2\beta)\Sigma_{s}}{\beta w_{max}}\frac{2-D_{ f}}{1-D_{f}}\frac{1-\alpha^{1-D_{f}}}{1-\alpha^{2-D_{f}}}\Bigg{\}}.\end{split} \tag{21}\]  
Substituting \(A_{REV}\) from Equation (8) into Equation (21), the electrical conductivity of porous media \(\sigma\) under saturated conditions is obtained as  
\[\sigma= \frac{\phi}{\tau^{2}}\Bigg{\{}\sigma_{w}+\frac{(1+2\beta)\Sigma_{s}}{ \beta w_{max}}\frac{2-D_{f}}{1-D_{f}}\frac{1-\alpha^{1-D_{f}}}{1-\alpha^{2-D_{ f}}}\Bigg{\}}. \tag{22}\]  
Equation (22) indicates that \(\sigma\) of fractured media is related to porosity, electrical conductivity of pore water, specific surface conductance and microstructural parameters of porous media (\(D_{f}\), \(\phi\), \(\alpha\), \(\beta\), \(w_{max}\)). It has a similar form to published models for porous media, e.g., [37; 84; 85; 86; 87; 88; 89]. From Equation (22), the formation factor \(F\) is deduced as [34]:  
\[F=\lim_{\Sigma_{s}\to 0}\Big{(}\frac{\sigma_{w}}{\sigma}\Big{)}=\frac{\tau^{2}}{ \phi}. \tag{23}\]  
[PERSON] et al. (2018) conducted a systematic numerical analysis for modeling electrical current flow in complex and heterogeneous fractured rocks [33]. For simple case of a simple fracture-matrix system with \(N_{f}\) horizontal fractures embedded in a matrix having domain size \(L\) (\(\tau=1\)), fracture aperture \(b_{f}\) and matrix porosity \(\phi_{b}\), the formation factor of such a domain is given by [33]  
\[F=\frac{L}{\Big{[}N_{f}b_{f}+(L-N_{f}b_{f})\phi_{b}^{m_{b}}\Big{]}}, \tag{24}\]  
where \(m_{b}\) is the cementation factor for the matrix. If a matrix is impervious to electrical current, that is, \(\phi_{b}=0\), for which the porosity only depends on the fractures as \(\phi=N_{f}b_{f}/L\), Equation (24) now becomes for fractured media \(F=\phi^{-1}\). This is expected from simple geometry principles, as for parallel straight and horizontal capillaries with no surface conductivity: \(\sigma=\phi\sigma_{w}\).  
Equation (22) can be rewritten as:  
\[\sigma= \frac{1}{F}\Bigg{\{}\sigma_{w}+\frac{(1+2\beta)\Sigma_{s}}{\beta w _{max}}\frac{2-D_{f}}{1-D_{f}}\frac{1-\alpha^{1-D_{f}}}{1-\alpha^{2-D_{f}}} \Bigg{\}}. \tag{25}\]