## 2 Methods
### Modelling Multi-Component Diffusion  
In contrast to [PERSON]'s laws, the transport of species in the MC diffusion approach is not based on a concentration gradient but on the electrochemical potential \(\mu\)[5; 6; 13; 37; 38]:  
\[\mu=\mu^{0}+RTIn\alpha+zF\psi \tag{1}\]  
with \(\mu^{0}\) the standard potential (J/mol), R the gas constant (8.314 J K\({}^{-1}\) mol\({}^{-1}\)), T the absolute temperature (K), \(\alpha\) the activity (-), z the charge number (-), F the Faraday constant (96,485 J V\({}^{-1}\) eq\({}^{-1}\)) and \(\psi\) the electrical potential of a charged surface (V). This enables calculation of the diffusive flux in uncharged (free porewater) and charged regions (DDL) of the clay minerals or the interlayer space between the clay layers. [PERSON] et al. [6] successfully applied the approach for the Opalinus Clay to model a diffusion experiment with neutral, cationic and anionic species. Furthermore, it was used to obtain the porewater composition in the Callovo-Oxfordian clay rock [37]. The diffusive flux of a species in solution \(J_{i}\) (mol m\({}^{-2}\) s\({}^{-1}\)) as result of the chemical and electrical potential gradients becomes [5; 6; 13; 38]:  
\[J_{i}=-\frac{u_{i}c_{i}}{|z_{i}|F}\frac{\partial\mu_{i}}{\partial x}-\frac{u_ {i}z_{i}c_{i}}{|z_{i}|}\frac{\partial\psi}{\partial x} \tag{2}\]  
with \(u_{i}\) the mobility in water (m\({}^{2}\) s\({}^{-1}\) V\({}^{-1}\)), \(c_{i}\) as species concentration in the accessible porewater (mol/m\({}^{3}\)) and x as spatial coordinate (m). The mobility in Equation (2) is related to the tracer diffusion coefficient in water \(D_{w,i}\) (m\({}^{2}\)/s):  
\[u_{i}=D_{w,i}\cdot\frac{|z_{i}|F}{RT} \tag{3}\]  
Due to the different transport velocities of ions, a charge with an associated potential is created and thus the gradient of the electrical potential (\(\partial\psi/\partial x\)) in Equation (2). The electrical potential in Equation (1) stems from a charged surface and is therefore fixed without inducing any electrical current. Accordingly, both potentials may differ from each other [5; 6; 13; 38]. If there is no electrical current, \(\sum z_{i}J_{i}=0\). This zero-charge flux condition allows to eliminate the electrical potential gradient (\(\partial\psi/\partial x\)) from Equation (2) by expressing it as a function of the other terms in Equation (2) [5; 6; 13; 38]. Any charge imbalance that may for instance exist in the DDL is maintained by the zero-charge flux condition and hence the equation is also valid, if the solution is not electrically neutral. Within PHREEQ, the diffusive flux for every species present in the system is calculated as the sum of the transport in the porewater as well as in the DDL for the MC option. Conceptually, the porous medium within a model element is subdivided along its length in paired cells for each mineral present (Figure 1). Accordingly, each pair of cells consists of a charge-balanced solution, the free porewater, and a charged surface with a DDL. In the case of negatively charged surfaces, cations are enriched in the DDL to compensate \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Shaly-76**} & \multicolumn{2}{c}{**Shaly-61**} & \multicolumn{2}{c}{**Sandy**} & \multicolumn{2}{c}{**Carb-Rich**} \\  & **Min** & **Max** & **Min** & **Max** & **Min** & **Max** & **Min** & **Max** \\ \hline Porosity e (-) & 0.191 & 0.141 & 0.191 & 0.141 & 0.177 & 0.130 & 0.171 & 0.155 \\ wet water content \(wc_{ext}\) (-) & 0.051 & 0.086 & 0.051 & 0.086 & 0.053 & 0.078 & 0.053 & 0.075 \\ dry bulk density \(\rho_{ab}\) (kg/m\({}^{3}\)) & 2230 & 2410 & 2230 & 2410 & 2260 & 2400 & 2280 & 2400 \\ wet bulk density \(\rho_{ab}\) (kg/m\({}^{3}\)) & 2420 & 2530 & 2420 & 2530 & 2440 & 2530 & 2450 & 2530 \\ \hline Rock per kg porewater (kg\({}_{rock}\)/kg\({}_{pw}\)) & 11.55 & 16.95 & 11.55 & 16.95 & 12.36 & 18.17 & 13.20 & 15.24 \\ \hline \end{tabular}
\end{table}