## 4 Results and Discussion
The reason is that the surface electrical conductivity can be negligible at large fracture sizes as indicated by Equation (22). Figure 5: The variation of \(\sigma\) with the fracture aspect ratio \(\beta\) and the fractal dimension of the fracture sizes \(D_{f}\) predicted from Equation (25) for three values of \(\alpha\) (0.0001, 0.001 and 0.01) with representative values of \(\sigma_{w}\) = 5 \(\times\) 10\({}^{-2}\) S/m, \(\Sigma_{s}\) = 1 \(\times\) 10\({}^{-9}\) S, \(F\) = 20 and \(w_{max}\) = 200 \(\times\) 10\({}^{-6}\) m. (a) \(D_{f}\) is fixed at 1.7. (b) \(\beta\) is fixed at 0.001. Figure 6: \(Cont.\)  
Figure 7 shows the variation of the \(\widehat{Q}_{v}\) with the fracture aspect ratio \(\beta\) predicted from Equation (46) for three values of \(\alpha\) (0.0001, 0.001 and 0.01) with representative values of \(\sigma_{uv}=2\times 10^{-2}\) S/m, \(\Sigma_{u}=1\times 10^{-9}\) S, \(\zeta=-30\) mV, \(\phi=0.15\) and \(w_{max}=200\times 10^{-6}\) m. It is seen that the \(\widehat{Q}_{v}\) decreases with an increase of \(\beta\) and increase of \(\alpha\). As shown in Figure 4 the permeability \(k\) increases with increasing \(\beta\) or \(\alpha\). That is in good match with results reported in [19] or [101] in which the \(\widehat{Q}_{v}\) decreases with an increase of permeability \(k\) for porous media. Due to lack of experimental data for fractured media, the model result is only compared with the experimental data published by [102] for three fractured limestones from different depths (samples A, B and C). The permeability, porosity and maximum width of fractured samples A, B and C are reported to be \(1\times 10^{-17}\) m\({}^{2}\), 0.007, \(80\) um; \(4.69\times 10^{-16}\) m\({}^{2}\), 0.0107, \(200\) um and \(4.8\times 10^{-17}\) m\({}^{2}\), 0.006, \(150\) um, respectively. Figure 8 shows the comparison between the model result from Equation (16) with the measured data for samples A, B and C. For the comparison, the model parameters \(\beta\), \(\tau\) and \(D_{f}\) are determined by seeking a minimum root-mean-square error (RMSE) through the \"fminsearch\" function in the MATLAB. We found \(\beta=0.009\), \(\tau=1.5\) and \(D_{f}=1.8\) to give the minimum RMSE for fractured samples. Note that [PERSON] et al. (2019) found the values of \(\beta\) between 0.1 and 0.001 by fitting their model to experimental data of tensile fractures in the Krafla fissure swarm of northeast Iceland [76]. Besides the 1:1 line (the solid line), the 1:10 and the 10:1 lines are also shown in Figure 8 (the dashed lines). It is seen that the model predicts well the experimental data within less than 1 order of magnitude. Nevertheless,  
Figure 6: 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 \(\sigma_{uv}=2\times 10^{-2}\) S/m, \(\Sigma_{u}=1\times 10^{-9}\) S, \(\zeta=-30\) mV, \(\phi=0.15\): (a) \(w_{max}\) is fixed at \(200\times 10^{-6}\) m and (b) \(\beta\) is fixed at \(0.001\). Figure 7: The variation of the \(\widehat{Q}_{v}\) with the fracture aspect ratio \(\beta\) predicted from Equation (46) for three values of \(\alpha\) (0.0001, 0.001 and 0.01). additional experimental laboratory studies or well controlled field measurements should be conducted to test our model further.