## 3 Theoretical Development
### Effective Excess Charge Density  
Following the formalism presented by [52] or [54], we determine the effective excess charge density \(\widehat{Q}_{v}\) (C/m\({}^{3}\)) carried by the water flow in the REV. The effective excess charge density \(\widehat{Q}_{v}^{w}\) (C/m\({}^{3}\)) carried by the water flow in a single fracture with the width of \(w\) and aperture of \(2a\) is defined by, e.g., [50; 52; 54]:  
\[\widehat{Q}_{v}^{w}=\frac{1}{v_{av}(2 aw)}\int_{-a}^{+a}\widehat{Q}_{v}(y)v(y) (w.dy), \tag{39}\]  
recall that \(v(y)\), \(v_{av}\) and \(\widehat{Q}_{v}(y)\) is the velocity profile, the average velocity and the charge distribution in the fracture, respectively. Their expressions are presented in Equations (9), (10) and (27), respectively. Combining Equations (9), (10), (27) and (39), one obtains  
\[\widehat{Q}_{v}^{w}=-\frac{3\epsilon_{r}\epsilon_{o}\zeta}{\beta^{2}w^{2}} \left[1-\frac{\lambda}{a}\tanh(\frac{a}{\lambda})\right] \tag{40}\]  
Under a thin EDL condition, Equation (40) may be simplified as  
\[\widehat{Q}_{v}^{w}=-\frac{3\epsilon_{r}\epsilon_{o}\zeta}{\beta^{2}w^{2}}. \tag{41}\]  
The effective excess charge \(\widehat{Q}_{v}\) (C/m\({}^{3}\)) carried by the water flow in the REV is defined by, e.g., [50; 52]:  
\[\begin{split}\widehat{Q}_{v}=&\ \frac{1}{v_{D}A_{ REV}}\int_{w_{min}}^{w_{max}}\widehat{Q}_{v}^{w}v_{av}(2 aw)(-dN)\\ &=-\frac{1}{v_{D}A_{REV}}\frac{2\epsilon_{r}\epsilon_{o}\zeta \beta D_{f}w_{max}^{2}}{\eta\tau}\frac{(1-\alpha^{2-D_{f}})}{2-D_{f}}\frac{ \Delta P}{L_{o}},\end{split} \tag{42}\]  
where \(A_{REV}\) is the cross-section area of the REV perpendicular to the flow direction and \(v_{D}\) is the Darcy's velocity (m/s) that is given by  
\[v_{D}=\frac{k}{\eta}\frac{\Delta P}{L_{o}}, \tag{43}\]  
recall that \(k\) is the permeability of porous media.  
Combining Equations (42) and (43), the following is obtained:  
\[\widehat{Q}_{v}=-\frac{2\epsilon_{r}\epsilon_{o}\zeta\beta D_{f}w_{max}^{2}} {\tau kA_{REV}}\frac{(1-\alpha^{2-D_{f}})}{2-D_{f}}. \tag{44}\]  
Substituting \(A_{REV}\) from Equation (8) into Equation (44), \(\widehat{Q}_{v}\) in the REV is written as  
\[\widehat{Q}_{v}=-\epsilon_{r}\epsilon_{o}\zeta\frac{\phi}{\tau^{2}k}. \tag{45}\]  
Equation (45) shows the relationship between \(\widehat{Q}_{v}\), macroscopic hydraulic parameters of porous media (the permeability, porosity, tortuosity) and electrokinetic parameters (the zeta potential). It is seen that the fracture size distribution and geometrically shaped fracture structures (\(w_{max}\), \(a\) and \(\beta\)) do not directly appear in the closed-form equation for \(\widehat{Q}_{v}\) as reported in literature, e.g., [52; 94]. Note that Equation (45) shows a behavior that is completely consistent with the empirical relationship proposed by [19]. To obtain the dependence of \(\widehat{Q}_{v}\) on the parameters of \(w_{max}\), \(\alpha\) and \(\beta\), we combine Equations (15) and (45) to get the following:  
\[\widehat{Q}_{v}=-\epsilon_{r}e_{o}\zeta\frac{3(4-D_{f})(1-\alpha^{2-D_{f}})}{ \beta^{2}w_{max}^{2}(2-D_{f})(1-\alpha^{4-D_{f}})}. \tag{46}\]  
Comparing Equations (21), (37) and (44), one can find the SPCC from \(\widehat{Q}_{v}\) as  
\[C_{S}=-\frac{k\widehat{Q}_{v}}{\eta\sigma}. \tag{47}\]  
Equation (47) is the same as Equation (4) that has been developed for porous media using the volume averaging procedure, e.g., [43; 97]. However, Equation (47) is here explicitly developed from the assumptions made for fractured media. This confirms the findings of [47; 48; 94], the coupling coefficient dependence to the pore space geometry (pore size distribution and pore shapes) can be taken into account in the effective excess charge density \(\widehat{Q}_{v}\).