## 2 Materials and Methods
### Theory: Advective-Dispersion Equation (ADE)  
Solute transport and chemical reaction are primary components to affect the radionuclide activity in porous media. Three solute transport mechanisms describe the motion of mass through fluid in porous structure media: advection, diffusion, and dispersion. Chemical reactions are a series of complex activities that take place between radionuclides and media, including acid-base reactions, oxidation-reduction reactions, dissolution of precipitates, surface complex reactions, and ion exchange. Therefore, the study of transport behavior and chemical reaction in porous media requires complete information about the internal structures of the porous media, which at this time is not possible to obtain. Considering the solute transport of a conservative tracer in a flow field of porous media without chemical reactions, advection and hydrodynamic dispersion are the two mechanisms for the movement of solute particles. When the axis of the modeling domain aligns with the direction of the velocity vector, the dispersion coefficient is based on the dispersivity and molecular diffusion occurring in the principal direction only, as shown in [21]. By taking a two-dimensional case as an example, dispersion coefficients can be written for longitudinal and lateral dispersion as:  
\[D_{x}=\alpha_{L}\left(u_{x}/\left|\,u\,\right|\right)+\alpha_{T}\left(u_{y}/ \left|\,u\,\right|\right)+D_{m} \tag{1}\]  
\[D_{y}=\alpha_{L}\left(u_{y}/\left|\,u\,\right|\right)+\alpha_{T}\left(u/\left| \,u\,\right|\right)+D_{m} \tag{2}\]  
respectively, where \(D\) is the hydrodynamic dispersion, \(\alpha_{i}\) is the dispersivity, and \(u_{i}\) is the component of pore velocity \(u\) in the \(i\) direction. \(D_{m}\) is the molecular diffusion coefficient. In a one-dimensional isotropic medium, the dispersion coefficient can be further simplified as:  
\[D=\alpha u+D_{m}. \tag{3}\]  
The frequently used one-dimensional advective-dispersive equation for solute transport in the porous media can be expressed as:  
\[\frac{\partial C}{\partial t}=D\frac{\partial^{2}C}{\partial x^{2}}-u\frac{ \partial C}{\partial x} \tag{4}\]  
where \(C=\) solute concentration in the liquid (M/L\({}^{3}\)), \(D=\) dispersion coefficient (L\({}^{2}\)/T), \(t=\) time (T), \(u=\) pore water velocity (L/T), and x = distance (L). If the solute is subjected to linear equilibrium sorption, then Equation (4) can be written as:  
\[\frac{\partial C}{\partial t}=\frac{D}{R}\frac{\partial^{2}C}{\partial x^{2}}- \frac{u}{R}\frac{\partial C}{\partial x} \tag{5}\]  
The retardation factor, \(R\), can be given by:  
\[R=1+\left[(\rho_{b}*K_{d})/\theta\right] \tag{6}\]  
where \(\rho_{b}=\) bulk density (M/L\({}^{3}\)), \(K_{d}=\) distribution coefficient (L\({}^{3}\)/M), and \(\theta=\) porosity (-). The initial condition for the equation applied in this study is:  
\[C(\text{x},0)=0 \tag{7}\]  
and the boundary conditions for the upper and the lower boundary are written respectively as:  
\[C(0,t)=C_{0} \tag{8}\]  
\[\frac{\partial C(\infty,t)}{\partial x}=0 \tag{9}\]  
An alternative lower boundary condition could also be applied as:  
\[\frac{\partial C(L,t)}{\partial x}=0 \tag{10}\]  
where \(L\) is the column length and \(C_{0}\) is the inlet concentration. [PERSON] et al. [17; 22; 23; 24] have recommended using Equation (9) as the lower boundary condition for a column experiment since the derived analytical solution is simpler but still accurate in most situations.