# Laboratory Measurements of Zeta Potential in Fractured
## 3 Results and Discussion
### Effect of Confining Pressure
[66] reported the hydraulic aperture of order of 200 mm, which is three orders of magnitude larger than the values estimated for our sample using Equation (5). However, the values of hydraulic conductivity of 13 cm/s measured by [PERSON] et al. [66], which translates into the permeability of c. 13 \(\times\) 10\({}^{6}\) mD, is identical to that predicted by Equation (5), thus validating our approach and estimates. To assess the effect of the surface conductivity on the zeta potential, we used a model of the zeta potential of silica-NaCl systems [67], for which the expected surface conductivities for the two ionic strengths tested in our experiments were evaluated to be 3.0 \(\times\) 10\({}^{-9}\) S for 7.35 \(\times\) 10\({}^{-3}\) M, and 4.5 \(\times\) 10\({}^{-9}\) S for 0.7 M solution. To characterize the impact of the surface conductivity on the zeta potential, we used the dimensionless Dukhin number (\(Du\)), defined by \(Du=\frac{\sigma_{s}}{\varpi\varpi_{w}}\) for the fracture geometry described in Figure 5[7]:  
\[Du=\frac{\sigma_{s}}{w\varpi_{w}} \tag{12}\]  
From the estimates of the fracture aperture and surface conductivity, and measured bulk solution conductivities, \(Du\) translates to 7.9 \(\times\) 10\({}^{-2}\) for 7.35 \(\times\) 10\({}^{-3}\) M solutions at 4 MPa of confining pressure; 8.6 \(\times\) 10\({}^{-2}\) for 7.35 \(\times\) 10\({}^{-3}\) M solutions at 7 MPa; 1.9 \(\times\) 10\({}^{-3}\) for 0.7 M solutions at 4 MPa and 2.0 \(\times\) 10\({}^{-3}\) for 0.7 M solutions at 7 MPa. All estimated \(Du\) for our experimental conditions clearly satisfied the condition for a negligibly small contribution of the surface conductivity to the total effective conductivity of an aqueous solution, hence, a negligible effect on the zeta potential: \(Du\ll 1\). Therefore, for all tested solutions, the surface electrical conductivity could be neglected, and Equation (1) was appropriate for interpreting the zeta potential from the measured streaming potential coupling coefficient. To evaluate the aquifer depths at which surface conductivity becomes significant, we considered a study [66] that reported a near-linear correlation between the confining pressure and the hydraulic aperture of an artificially fractured granite sample. Thus, using estimates of the surface conductivity value for low-salinity solutions (Table 3), we were able to evaluate the confining pressure and the corresponding aquifer depth, at which the surface conductivity becomes significant, satisfying the condition of \(Du\approx 0.1\). The aperture that satisfies \(Du=0.1\) was calculated to be 310 nm using Equation (12), and the corresponding confining pressure was found to be 3.8 MPa, corresponding to an aquifer depth of 380 m, which is significantly deeper than the typical useful (from a resource point of view) depth of fractured gneiss aquifers, which varies between a few meters and a few hundred meters. At these depths, the impact of the surface conductivity on the zeta potential becomes substantial, hence, appropriate care should be taken to interpret the SP and ERT field data. Future experimental work will aim at acquiring additional zeta potential values in systems comprising other specimens of the Lewisian gneiss and confining pressures higher than 7 MPa. These experiments will complement the data reported here and expand the range of tested conditions not covered in this work, thus, further improving our understanding of the zeta potential at the gneiss-water interfaces under conditions relevant to naturally fractured systems. The results will also be used to inform future surface complexation and molecular dynamics simulation studies aimed at describing gneiss-water systems under realistic conditions.