## 2 Methods
### Integrating Sorption Processes  
The varying mineralogy of the Opalinus Clay is implemented in the simulations with a bottom-up approach [27; 28]. The amount of sorption of contaminants in heterogeneous formations can be determined by the additive sorption on individual minerals. It is quantified with mechanistic surface complexation models (SCM) using the two-layer model of [PERSON] [29] and cation exchange [30]. The respective and proven SCM dataset [31] with all surface parameters and reactions is given in the Supplementary Materials of [PERSON] et al. [11]. We only consider sorption on the main clay mineralsillite, illite/smectite mixed layers, kaolinite and chlorite (Scenario MC\({}_{4 Clay}\)) as the contribution of the other minerals to the total sorption can be neglected (\(<\)1%) [11]. The illite/smectite mixed layers are handled as a combination of the pure clay minerals (illite and smectite) with a proportion of 1:1 [31]. Smectite is a generic term for a group of clay minerals and not further described for the Opalinus Clay. We decided to use montmorillonite as representative because it is the best investigated [31]. Cation exchange is considered for the main cations and uranyl-ions on illite and montmorillonite as exchange phases [11]. The respective reactions are given as well in the Supplementary Materials of [PERSON] et al. [11]. The distribution coefficient \(K_{d}\) (m\({}^{3}\)/kg) is defined by the ratio between the amount of a species adsorbed on the solid phase and the amount present in the liquid phase. The \(K_{d}\) is calculated in PHREEQC for each facies depending on the geochemical and mineralogical conditions following [PERSON] et al. [28]. In the literature, sorption of uranium on Opalinus Clay is commonly modeled using only the clay minerals illite and montmorillonite [32; 33; 34; 35]. We followed that approach and assessed the effect of various mineral compositions on the resulting diffusion length (Scenario MC\({}_{2 Clay}\)). Porosity, bulk density and water content vary for each facies [10]. These parameters influence the simulation results as they define the amount of porewater per volume unit Opalinus Clay and the amount of clay minerals per kilogram porewater. Therefore, we performed various scenarios of MC diffusion simulations for the minimum and maximum  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Unit** & **Shaly-76** \({}^{\text{a,b}}\) & **Shaly-61** \({}^{\text{c,d}}\) & **Sandy** \({}^{\text{e,f}}\) & **Carb-Rich** \({}^{\text{gh}}\) \\ \hline pH & - & 7.13 & 7.50 & 7.40 & 7.28 \\ \(N_{a}{}^{+}\) & mmol/L & 281 & 230 & 121 & 195 \\ \(K^{+}\) & mmol/L & 1.93 & 1.47 & 0.87 & 0.73 \\ \(Mg^{2+}\) & mmol/L & 21.97 & 16.79 & 5.91 & 15.89 \\ \(Ca^{2+}\) & mmol/L & 18.90 & 16.03 & 6.73 & 16.37 \\ \(Sr^{2+}\) & mmol/L & 0.46 & 0.47 & 0.35 & 0.44 \\ \(Fe_{total}\) & \(\mu\)mol/L & 29.62 & 8.37 \({}^{\text{d}}\) & 9.81 \({}^{\text{f}}\) & 9.81 \({}^{\text{f}}\) \\ \(U_{total}\) & nmol/L & 2.28 \({}^{\text{d}}\) & 2.28 \({}^{\text{d}}\) & 2.52 \({}^{\text{f}}\) & 2.52 \({}^{\text{f}}\) \\ \(Cl^{-}\) & mmol/L & 327 & 273 & 121 & 242 \\ \(SO_{4}^{2-}\) & mmol/L & 16.79 & 12.30 & 6.94 & 16.03 \\ \(PO_{4}^{3-}\) & \(\mu\)mol/L & 21.65 \({}^{\text{d}}\) & 21.65 \({}^{\text{d}}\) & 10.65 \({}^{\text{f}}\) & 10.65 \({}^{\text{f}}\) \\ Alkalinity & mmol/L & 3.85 & 1.24 & 2.61 & 2.05 \\ I \({}^{\text{i}}\) & mol/L & 0.38 & 0.32 & 0.16 & 0.28 \\ DDL \({}^{\text{j}}\) & nm & 0.49 & 0.54 & 0.76 & 0.57 \\ \hline Illite & wt.\% & 20 & 17 & 17 & 8 \\ IS mixed \({}^{\text{k}}\) & wt.\% & 16 & 12 & 8 & 6 \\ Kaolinite & wt.\% & 30 & 26 & 13 & 8 \\ Chlorite & wt.\% & 10 & 6 & 7 & 4 \\ \hline \(\sum\)Clay & wt.\% & 76 & 61 & 45 & 26 \\ \hline Calcite & wt.\% & 14 & 11 & 17 & 42 \\ Dolomite & wt.\% & n.a.