## 2 Materials and Methods
### Transport Simulation: HTO, Se, and Cs  
The study of radionuclide transport behavior and chemical reaction in porous media requires complete information about the internal structures of the porous media, which, at the present time, cannot be obtained. Three mechanisms of solute transport describe the motion of mass through fluid in media with a porous structure: advection, diffusion, and dispersion. Chemical reactions are a series of complex equilibrium and non-equilibrium reactions between radionuclides and media, including acid-base reactions, oxidation-reduction reactions, precipitate dissolution, surface complex reactions, and ion exchange. Therefore, numerical simulations can be carried out using STANMOD (STudio of ANALyltical MODels), which is a package of programs used for predicting solute transport in saturated porous media using the analytical solutions for the classical advection-dispersion  
Figure 1: A reliable, precise ADE device with a PP (poly-propane) column apparatus used in this study. (**a**) schematic apparatus; (**b**) experimental set-up.  
equation [26]. Among those programs, CFITIM and CXTFIT can handle either the chemical or physical equilibrium and non-equilibrium transport issues. In this work, the equilibrium and the non-equilibrium model, so-called physical non-equilibrium simulations, were applied to simulate the experimental results of the radionuclide sorption kinetic mechanism for a solid/liquid system. Equation (12) can only be applied to linear equilibrium adsorption cases. For the non-equilibrium adsorption model, a similar but more complicated mathematical procedure is needed. Two-region (physical) and two-site (chemical) models were frequently applied for the possible non-equilibrium adsorption situations. The general dimensionless governing equations for the two non-equilibrium models are identical and are given as:  
\[\beta R\frac{\partial C_{1}}{\partial T}+(1-\beta)R\frac{\partial C_{2}}{ \partial T}=\frac{1}{P}\frac{\partial^{2}C_{1}}{\partial Z^{2}}-\frac{\partial C _{1}}{\partial Z} \tag{13}\]  
\[(1-\beta)R\frac{\partial C_{2}}{\partial T}=\omega(C_{1}-C_{2}) \tag{14}\]  
where (symbols with an identical definition as in the equilibrium model are not repeated here), for a two-region model, the dimensionless variables are given as:  
\(C_{1}=C_{m}/C_{0},C_{2}=C_{int}/C_{0}\),  
\(C_{m}=\) solute concentration in the mobile liquid phase (M/L\({}^{3}\)), \(C_{in}=\) solute concentration in the immobile liquid phase (M/L\({}^{3}\)),  
\(\beta=\left(\theta_{m}+f_{\rho_{v}}\times K_{d}\right)/(\theta+\rho b\times K _{d})\),  
\(\theta=\)_prosity_, \(\theta_{m}=\) porosity fraction of the mobile zone, \(f=\) sorbed mass fraction assigned to mobile zone,  
\(\omega=\alpha L/\theta_{m}\times v_{m}\),  
\(\alpha=\) mass transfer coefficient \((1/\mathrm{T})\),  
\(v_{m}=\) pore water velocity in the mobile zone \((\mathrm{L}/\mathrm{T})\).  
For a two-site model, the dimensionless variables are given as:  
\(\beta=\left(\theta+F_{\rho_{b}}\times K_{d}\right)/(\theta+\rho_{b}\times K_{ d})\),  
\(F=\) fraction of sites occupied by linear equilibrium adsorption,  
\(\omega=[\alpha(1-\beta)RL]/v\),  
\(C_{1}=C/C_{0}\), \(C_{2}=S_{2}/[(1-F)K_{d}C_{0}]\),  
\(S2=\) concentration subjected to kinetic non \(-\) equilibrium adsorption (M/M).  
The analytical solution of Equations (7)-(9), (13) and (14) is:  
\[C_{\epsilon}=G(T)\exp\left(-\frac{\omega T}{\beta R}\right)+\frac{\omega}{R} \int_{0}^{T}G(\tau)H(T,\tau)d\tau \tag{15}\]  
where  
\[H(T,\tau)=\exp(-a-b)[\frac{I_{0}(\xi)}{\beta}+\frac{\xi I_{1}(\xi)}{2b(1-\beta )}] \tag{16}\]  
\(a=\omega\tau/\beta R\),  
\(b=(\omega(T-\tau))/((1-\beta)R)\),  
\(\xi\)= \(2\ (ab)^{(1/2)}\),  
\(I_{0}\), \(I_{1}=\) modified Bessel functions of the first kind.  
\[\begin{array}{l}G(\tau)=\frac{1}{2}erfc[(\frac{p}{4 bR\tau})^{\frac{1}{2}}\ (\beta R-\tau)]+\\ \frac{1}{2}exp(P)erfc[(\frac{p}{4 bR\tau})^{\frac{1}{2}}\ (\beta R-\tau)]\end{array} \tag{17}\]