# Laboratory Measurements of Zeta Potential in Fractured
## 3 Results and Discussion
### Effect of Confining Pressure  
In all of our experiments, we observed the zeta potentials to be independent of the confining pressure (Figure 4). The increasing confining pressure applied to the fractured rock sample resulted in a decrease of the apparent permeability from 51 mD (1 mD \(\cong\) 10\({}^{-15}\) m\({}^{2}\)) at 4 MPa to 43 mD at 7 MPa. The observed decrease in the permeability corresponds to a closure of the fracture, thus potentially leading to an increased effect of the surface electrical conductivity. To quantify the effect of the confining pressure on the measured streaming potential coupling coefficient, zeta potential and the fracture aperture, we considered the following geometry of the investigated rock sample (Figure 5). All our streaming potential experiments were conducted under steady-state fluid flow conditions and the induced flow through the fracture was consistent with the direction of the rock sample. Therefore, for the geometry described in Figure 5 and the experimental conditions, the Navier-Stokes equations simplify to 1-D flow in the Cartesian coordinate system as follows:  
\[\mu\frac{d^{2}v_{x}}{dy^{2}}=\frac{\Delta P}{L} \tag{2}\]  
where \(v_{x}(y)\) is the fluid velocity in \(x\)-direction inside the fracture. Applying the no-slip boundary conditions at both fracture walls, the fluid velocity profile is given by:  
\[V_{x}=\frac{1}{2\mu}\frac{\Delta P}{L}\Big{(}y^{2}-w^{2}\Big{)} \tag{3}\]  
Multiplying Equation (3) by the cross-section area to the flow, \(2 wH\), and integrating across the fracture aperture (between \(-w\) and \(+w\)) yields the well-known cubic law [64] for the total volumetric flow rate, \(Q\):  
\[Q=-\frac{2 Hw^{3}}{3\mu}\frac{\Delta P}{L} \tag{4}\]Now, equating the flow rate given by Equation (4) with that from the [PERSON]'s law, \(Q=-\frac{2 kHw}{\mu}\frac{P}{L}\), the effective permeability is expressed as:  
\[k=\frac{w^{2}}{3} \tag{5}\]  
Equation (5) is consistent with a recently published study [65] in the limit of a single fracture, and it implies that for a decrease of approximately 16% in sample's permeability (from 51 mD to 43 mD), the corresponding expected decrease in fracture aperture (\(2w\)) will be around 8% (from 0.782 mm at 4 MPa to 0.718 mm at 7 MPa). Decreasing fracture aperture implies a smaller distance of separation between the fracture walls, and as such, a higher confining pressure can potentially have an impact on the electrical properties of hydraulically active fractures via an increased impact of the surface electrical conductivity and the zeta potential. To investigate the impact of fracture aperture on the electrokinetic properties of the sample, we consider the streaming current under the steady state experimental conditions and for geometry described in Figure 5:  
\[I_{\mathrm{s}}=2\int\limits_{0}^{w}\Sigma(y)v_{x}(y)Hdy \tag{6}\]  
where \(\Sigma\) [C/m\({}^{3}\)] is the volumetric charge density. The flow of mobile excess counter-ions that contribute to the streaming current is confined to the diffuse part of the electrical double layer (EDL), which has a thickness of approximately 3 nm for 7.35 \(\times\) 10\({}^{-3}\) M and an even smaller thickness for higher salinity, so that the thickest expected EDL will still be two orders of magnitude smaller than the estimated fracture aperture. Therefore, we can use  
Figure 5: The tested fractured rock sample is assumed to have a single fracture of aperture \(2w\), height \(H\) and length \(L\).