## 3 Comparison of Analytical Solution and Numerical Simulation for Radionuclide Transport  
This study analyzed the migration of key radionuclides affected by temperature in the engineering barrier and did not consider the migration of radionuclides in the geo-sphere or the dose assessment of radionuclides entering the biosphere. We used the analytical solution to verify the COMSOL model in the radionuclide transport simulation so we could confirm the performance of the model and the feasibility of the simulation results. The governing equation for radionuclide transport can be expressed as follows:  
\[(R_{i})\frac{\partial C}{\partial t}+\
abla\cdot(-D_{i}\
abla\cdot C_{i})=V \frac{\partial C}{\partial x}-\lambda_{i}C_{i} \tag{16}\]  
\[R_{i}=1+\frac{Kd\rho}{\epsilon} \tag{17}\]  
where \(R_{i}\) is the retardation factor, \(D\) is desperation or the diffusion coefficient, \(V\) is velocity, \(\lambda\) is a decay constant, \(Kd\) is the distribution coefficient, \(\rho\) is the density, and \(\epsilon\) is the porosity. We used the analytical solution of three different tracer tests (conservative, decaying, and adsorption) [45] to compare the simulation results of COMSOL in a one-dimensional transient transport. The model domain was set as 10 m and the water flow velocity was constant. At the inflow boundary (\(x\) = 0), the concentrations of all three tracers were main-tained at 1 mol/m\({}^{3}\) at 0-15,000 s and 0 afterward. Table 4 lists the model parameter values for the simulation of the tracer test. The COMSOL simulated concentration in the model domain was compared with the advection-dispersion analytical solution of the three dif-ferent tracer tests (conservative, decaying, and adsorption) at 20,000 s (see Figures 5-7). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value**[45] & **Units** \\ \hline Velocity & \(10^{-4}\) & m/s \\ Dispersion coefficient & \(10^{-4}\) & m\({}^{2}\)/s \\ Porosity & 0.4 & - \\ Decay constant & \(5\times 10^{-5}\) & 1/s \\ Distribution coefficient & \(6.8\times 10^{-4}\) & mol/kg \\ Liquid density & 1000 & kg/m\({}^{3}\) \\ Solid gran density & 2000 & kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter values for simulation in transient advection and dispersion. Figure 5: Comparison between COMSOL 1D transport model and analytical solution in the case of the transient-conservative tracer. Figure 6: Comparison between COMSOL 1D transport model and analytical solution in the case of the transient decay tracer. Figure 7: Comparison between COMSOL 1D transport model and analytical solution in the case of the transient-adsorbing tracer. Analytical solution of the transient-conservative tracer case is the following equation:  
\[\frac{c0}{2}\cdot\left( \begin{array}{c}\left(\exp((q\cdot x)/(\varepsilon\cdot D ))\cdot erfc\Big{(}(x+(q/phi)\cdot t)/\Big{(}2\cdot\sqrt{D\cdot t}\Big{)} \Big{)}\right)\\ +erfc\Big{(}(x-(q/\varepsilon)\cdot t)/\Big{(}2\cdot\sqrt{D\cdot t}\Big{)} \end{array} \right) \tag{18}\]  
Analytical solution of the transient-decaying tracer case can be expressed as follows:  
\[\frac{c0}{2}\cdot\left(\left( \begin{array}{c}\exp(x\cdot\Big{(}A+\sqrt{B} \Big{)})\cdot erfc\Big{(}\Big{(}x+2\cdot t\cdot\sqrt{(B\cdot D^{2})}\Big{)}/ \Big{(}2\cdot\sqrt{D\cdot t}\Big{)}\Big{)}\\ +\exp\Big{(}x\cdot\Big{(}A-\sqrt{B}\Big{)}\cdot erfc\Big{(}\bigg{(}x-2\cdot t \cdot\sqrt{\Big{(}B\cdot D^{2}\Big{)}}\Big{)}/\Big{(}2\cdot\sqrt{(D\cdot t)} \Big{)}\Big{)}\end{array} \right)\end{array}\right) \tag{19}\]  
\[A=q/(2\cdot\varepsilon\cdot D) \tag{20}\]  
\[B=\log(2)/(D\cdot T)+A^{2} \tag{21}\]  
Analytical solution of the transient-adsorbing tracer case is presented in (22):  
\[\frac{c0}{2}\cdot\left( \begin{array}{c}\left(\exp((q\cdot x)/(\varepsilon \cdot D))\cdot erfc\Big{(}(R\cdot x+(q/phi)\cdot t)/\Big{(}2\cdot\sqrt{D \cdot R\cdot t}\Big{)}\Big{)}\right)\\ +erfc\Big{(}\big{(}x-(q/\varepsilon)\cdot t\big{)}/\Big{(}2\cdot\sqrt{D\cdot R \cdot t}\Big{)}\Big{)}\end{array}