# Laboratory Measurements of Zeta Potential in Fractured
## 2 Materials and Methods
The experimental setup used in this study, including a pair of Ag/AgCl external electrodes used to measure the streaming potential, was the same as the one reported in previous studies by [PERSON] and co-workers (e.g., [33]). We used the paired-stabilized (PS) method [40] to measure the streaming potential coupling coefficient, from which the zeta potential was interpreted using the Helmholtz-Smoluchowski equation (e.g., [41]) and assuming negligible contribution of the surface electrical conductivity to the effective electrical conductivity of the aqueous solution occupying a mm-scale fracture (as discussed in subsequent sections):  
\[C_{SP}=\left.\frac{\Delta V}{\Delta P}\right|_{j=0}=\frac{\varepsilon\zeta}{ \mu\sigma_{w}} \tag{1}\]  
where \(C_{SP}\) [V/Pa] is the streaming potential coupling coefficient calculated from the measured voltage D\(V\) [V] and the pressure difference D\(P\) [Pa] across the sample when the total current density \(j\) [A/m\({}^{2}\)] is zero; \(\varepsilon\) [F/m] is the permittivity of water; \(\zeta\) [V] is the zeta potential; \(\mu\) [Pa-s] is the dynamic viscosity of water; and \(\sigma_{w}\) [S/m] is the electrical conductivity of the aqueous solution. Each measurement of the streaming potential coupling coefficient was carefully analyzed for potential error sources including voltage noise level, stability of voltage during the experiment and repeatability. The resulting cumulative \begin{table}
\begin{tabular}{c c c c c} \hline  & **AGW** & **LS** & **ASW** & **HS** \\ \hline
[Na\({}^{+}\)], M & \(1.522\times 10^{-3}\) & \(7.350\times 10^{-3}\) & 0.512 & 0.700 \\ \hline
[Ca\({}^{2+}\)], M & \(1.612\times 10^{-3}\) & - & 0.013 & - \\ \hline
[Mg\({}^{2+}\)], M & \(0.288\times 10^{-3}\) & - & 0.042 & - \\ \hline
[Cl\({}^{-}\)], M & \(5.072\times 10^{-3}\) & \(7.350\times 10^{-3}\) & 0.577 & 0.700 \\ \hline
[SO\({}_{4}\)\({}^{-}\)], M & \(0.125\times 10^{-3}\) & - & 0.022 & - \\ \hline pH & 6.0 \(\pm\) 0.1 & 6.5 \(\pm\) 0.1 & 6.2 \(\pm\) 0.1 & 6.9 \(\pm\) 0.1 \\ \hline \(\sigma_{w}\), S-m\({}^{-1}\) & 0.069 \(\pm\) 0.001 & 0.097 \(\pm\) 0.001 & 5.83 \(\pm\) 0.01 & 6.13 \(\pm\) 0.01 \\ \hline Ionic strength, M & \(7.350\times 10^{-3}\) & \(7.350\times 10^{-3}\) & 0.700 & 0.700 \\ \hline \end{tabular}
\end{table} 
Table 1: Chemical composition, equilibrium pH and electrical conductivity of tested solutions. The solution pH was measured using Mettler Toledo FiveGo pH meter, the electric conductivity of all solutions was measured using Jenway 4520 conductivity meter with the corresponding errors reflecting the instrument inaccuracy and measurement repeatability. AGW and LS solutions were prepared from high-salinity stock solutions of 0.735 M ionic strength, hence, errors in molar concentrations of all four tested solutions resulting from inaccuracies in weighing of the constituent salts are estimated at \(\pm\) 3% of the reported values. errors are presented by the individual error bars in Figures 2-4, as discussed below (all data accompanied by examples of SP experiment and error analysis are provided in S2 of the Supplementary Material). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sample; Mineralogy**} & \multirow{2}{*}{**Solution**} & \multicolumn{2}{c}{**Zeta Potential, mV**} & \multirow{2}{*}{**Source**} \\ \cline{3-3} \cline{5-5}  & & **LS** & **HS** & \\ \hline \multicolumn{5}{c}{**Grey shaded area**} \\ \hline Fontainebleau SS; \textgreater{}99\% quartz & NaCl & \(-\)52 & \(-\)20 & a, b \\ \hline \multicolumn{5}{c}{**Lochaline SS; \textgreater{}99\% quartz**} & NaCl & \(-\)77 & \(-\)25 & b \\ \hline \multicolumn{5}{c}{**Sainton SS; 90\% quartz, 5\% clays**} \\ \multicolumn{5}{c}{and feldspar} \\ \hline \end{tabular}
\end{table} 
Table 2: Summary of experimentally measured zeta potential in relevant samples saturated with NaCl and natural solutions.