## 4 Results and Discussion  
Figure 4 shows the variation of the permeability of fractured media \(k\) with the fracture aspect ratio \(\beta\) and porosity \(\phi\) predicted from Equation (15) for three values of the ratio of minimum to maximum apertures \(\alpha\) (0.0001, 0.001 and 0.01) with representative values of \(w_{max}=200\times 10^{-6}\) m, \(\tau=1.2\): (a) \(\phi\) is fixed at 0.15 and (b) \(\beta\) is fixed at 0.001. Note that we determine \(D_{f}\) from \(\phi\) and \(\alpha\) using Equation (6). It is seen that the permeability increases with an increase of \(\phi\) as reported in literature for fractured rocks, e.g., [74; 98; 99; 100]. It can be explained by the fact that larger porosity means larger space occupied by fractures leading to larger permeability. Additionally, Figure 4 also shows that the permeability increases with an increase of \(\beta\) and \(\alpha\). The reason for the increase of \(k\) with an increase of \(\beta\) may be that larger the fracture aspect ratio leads to higher flow rate and therefore higher permeability. The increase of \(k\) with an increase of \(\alpha\) can be explained by the increase of the average fracture aperture with \(\alpha\) as shown by [74]. Figure 4: \(Cont.\)  
Figure 5 shows the variation of the electrical conductivity of fractured media \(\sigma\) with the fracture aspect ratio \(\beta\) and the fractal dimension of the fracture sizes \(D_{f}\) for three values of \(\alpha\) (0.0001, 0.001 and 0.01) with representative values of \(\sigma_{w}=10^{-2}\) S/m, \(\Sigma_{s}=1.5\times 10^{-9}\) S, \(F=20\) and \(w_{max}=200\times 10^{-6}\) m: (a) \(D_{f}\) is fixed at 1.7 and (b) \(\beta\) is fixed at 0.001. It is seen that the \(\sigma\) is very sensitive with \(\beta\), \(D_{f}\) and \(\alpha\). For given values of \(D_{f}\) and \(\alpha\), \(\sigma\) decreases with an increase of \(\beta\). It can be explained by the decrease of the surface conductivity when \(\beta\) increases. Additionally, \(\sigma\) increases with an increase of \(D_{f}\) and with a decrease of \(\alpha\). That can be explained by an increase of the total number of fractures in the REV and therefore, the increase of surface conductivity with the increase of \(D_{f}\) and with the decrease of \(\alpha\) (e.g., see [74] for more details). Figure 4: The variation of the permeability of fractured media \(k\) with the fracture aspect ratio \(\beta\) and porosity \(\phi\) predicted from Equation (15) for three values of \(\alpha\) (0.0001, 0.001 and 0.01) with representative values of \(w_{max}=200\times 10^{-6}\) m, \(\tau=1.2\): (a) \(\phi\) is fixed at 0.15; (b) \(\beta\) is fixed at 0.001. Figure 5: _Cont._  
Figure 6 predicts the variation of the SPCC with the fracture aspect ratio \(\beta\) and the maximum width \(w_{max}\) predicted from Equation (37) for three values of \(\alpha\) (0.0001, 0.001 and 0.01) with representative values of \(\sigma_{w}\) = 2 \(\times\) 10\({}^{-2}\) S/m, \(\Sigma_{s}\) = 1 \(\times\) 10\({}^{-9}\) S, \(\zeta\) = \(-\)30 mV, \(\phi\) = 0.15: (a) \(w_{max}\) is fixed at 200 \(\times\) 10\({}^{-6}\) m and (b) \(\beta\) is fixed at 0.001. Similarly to previously mentioned, the value of \(D_{f}\) is determined from Equation (6) with the knowledge of \(\phi\) and \(\alpha\). It is seen that the SPCC in magnitude increases with an increase of \(\beta\) for a given value of \(\alpha\). Additionally, one can see that when \(\alpha\) increases, the SPCC also increases. The reason is that \(\sigma\) decreases with an increase of \(\beta\) and with increase of \(\alpha\). As deduced from Equations (25) and (37), when \(\sigma\) decreases then the SPCC increases. The prediction also shows that the SPCC increases with an increase of \(w_{max}\). This may be attributed to the fact that a larger fracture size leads to a lower surface electrical conductivity and therefore an increase of the SPCC. When \(w_{max}\) exceeds a certain value, the SPCC becomes independent of fracture sizes.