# Laboratory Measurements of Zeta Potential in Fractured
## 3 Results and Discussion
### Effect of Confining Pressure
this condition to linearize the flow velocity that contributes to the streaming current in the region adjacent to the fracture walls, where \(y\approx w\):  
\[V_{x}=\frac{1}{2\mu}\frac{\Delta P}{L}\Big{(}y^{2}-w^{2}\Big{)}\approx\frac{1}{ 2\mu}\frac{\Delta P}{L}2w(y-w) \tag{7}\]  
Introducing a new variable \(\theta=y-w\), using Poisson equation for the charge density, \(\Sigma=-e\frac{d^{2}V}{dy^{2}}\) (where \(V\) is the electric potential and \(\varepsilon\) is the permittivity of the aqueous solution), substituting Equation (7) into Equation (6), and integrating by parts yields an expression for the streaming current through the fracture:  
\[I_{s}=-\frac{2 wH\varepsilon\zeta}{\mu}\frac{\Delta P}{L} \tag{8}\]  
where \(\zeta\) is the zeta potential at \(y=\pm w\). We now apply the boundary conditions of zero net current coming in or out of the fracture, and no charge separation along the fracture. In such case, the streaming current must be balanced by the conduction current of equal magnitude and opposite direction. The conduction current is given by [PERSON]'s law:  
\[I_{c}=2 wH\sigma_{w}\frac{\Delta V}{L}+2H\sigma_{s}\frac{\Delta V}{L}=2 wH\frac {\Delta V}{L}\Big{[}\sigma_{b}+\frac{\sigma_{s}}{w}\Big{]} \tag{9}\]  
where \(V\) is the electric potential difference (voltage) across the fracture, \(\sigma_{w}\) [S/m] is the electrical conductivity of the bulk aqueous solution and \(\sigma_{s}\) [S] is the surface electrical conductivity due to the presence of EDL. Using the condition of electro-neutrality, \(I_{s}=-I_{c}\), the streaming potential coupling coefficient can be expressed:  
\[V_{x}=\frac{1}{2\mu}\frac{\Delta P}{L}\Big{(}y^{2}-w^{2}\Big{)} \tag{3}\]0  
Note that Equation (10) explicitly expresses the effective electrical conductivity of brine with the contribution of the surface conductivity, but becomes equal to the Helmholtz-Smoluchowski equation (Equation (1)) when the surface conductivity is negligibly small. The zeta potential can now be expressed from the measured streaming potential coupling coefficient as:  
\[V_{x}=\frac{1}{2\mu}\frac{\Delta P}{L}\Big{(}y^{2}-w^{2}\Big{)} \tag{3}\]1  
For the two tested ionic strengths of 7.35 \(\times\) 10\({}^{-3}\) M and 0.7 M, the permittivity and viscosity of the NaCl solutions can be calculated using the approach described in [PERSON] et al. [28], while \(C_{SP}\) and the electric conductivity of the solutions were directly measured during the experiments. Rock, fluid and electrokinetic properties for all our NaCl experiments are summarized in Table 3 along with estimates of fracture aperture made using Equation (5). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**ID** & \(\mathbf{P}\)**, **MPa** & \(\mathbf{\mu}\)**, **Pa\(\cdot\)s** & \(\varepsilon\)**F\(\cdot\)m\({}^{-1}\)** & \(\mathbf{C_{SP}}\)**, **mV\(\cdot\)MPa\({}^{-1}\)** & \(\mathbf{\zeta}\)** & \(\mathbf{k}\)**, **mD** & **2\(\mathbf{w}\)**, **nm** \\ \hline LS & 4 & \(9.33\times 10^{-4}\) & \(7.00\times 10^{-10}\) & \(-86.4\) & \(-11.12\) & \(51\pm 1\) & 782 \\ \hline LS & 7 & \(9.33\times 10^{-4}\) & \(7.00\times 10^{-10}\) & \(-89.5\) & \(-11.51\) & \(43\pm 1\) & 718 \\ \hline HS & 4 & \(9.89\times 10^{-4}\) & \(6.97\times 10^{-10}\) & \(-1.21\) & \(-10.52\) & \(51\pm 1\) & 782 \\ \hline HS & 7 & \(9.89\times 10^{-4}\) & \(6.97\times 10^{-10}\) & \(-1.23\) & \(-10.69\) & \(43\pm 1\) & 718 \\ \hline \end{tabular}
\end{table} 
Table 3: Rock, fluid and electrokinetic properties of NaCl experiments at 7.35 \(\times\) 10\({}^{-3}\) M (LS) and 0.7 M (HS) ionic strengths. The confining pressure is denoted by \(P\) in the table and \(k\) is the measured absolute permeability. The fracture aperture (\(2w\)) is interpreted using Equation (5). The fracture aperture estimated from the sample permeability is consistent with the experimentally measured values reported for artificially fractured granite sample [66]. In their study, [PERSON] et al.