## 4 Inversion of the Penetration Parameters of a High Core Rockfill Dam
### Verification of the Permeability Coefficient  
Add the permeability coefficient into the finite element method to calculate the pressure water head of the observation point to verify the accuracy. The absolute error and relative error are as Figure 10:  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**K (cm/s)** & **K (cm/s)** & **Observe 1** & **Observe 2** & **Observe 3** & **Observe 4** & **Observe 5** & **Observe 10** & **Observe 14** \\ \hline \(3.00\times 10^{-6}\) & 0.1 & 3.75 & 9.68 & 12.13 & 28.53 & 38.91 & 156.86 & 243.22 \\ \(1.10\times 10^{-6}\) & 0.14 & 6.56 & 22.3 & 33.97 & 51.98 & 67.73 & 165.41 & 244.31 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The maximum and minimum values of the pressure water head of computational combinations.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Pressure Water Head (m)** & **Observe 1** & **Observe 2** & **Observe 3** & **Observe 4** & **Observe 5** & **Observe 10** & **Observe 14** \\ \hline rock a = 0.02 n = 2 & 6.41 & 18.81 & 28.84 & 45.76 & 59.81 & 163.85 & 244.09 \\ core a = 0.02 n = 2 & 6.6 & 21 & 31.55 & 47.23 & 58.78 & 162.07 & 243.87 \\ \hline rock a = 0.03 n = 2 & 9.9 & 21.21 & 30.11 & 47.28 & 61.76 & 164.11 & 244.15 \\ \hline rock a = 0.03 n = 2 & 9.87 & 21.1 & 28.52 & 45.01 & 58.1 & 162.35 & 243.95 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Effects caused by changes in shape materials.  
It can be seen that the calculation errors of the observation points at the lower part of the core wall are relative to each other. In other areas, the three inversion methods have their own advantages at different computing nodes. The average of the seven-point absolute errors and relative errors is as Table 5:  
It can be seen that the BP method's average of the absolute error is the minimum, though the absolute error of the MPA-BP method is smaller than that of the COMSOL-MATLAB method, the second's relative error is smaller, and the law of variance is also similar to the law of error. So, the COMSOL-MATLAB method may cause a large deviation at a certain code, and the BP method shows the best accuracy in terms of prediction.  
There exists another storage plan that involves adding the calculated permeability coefficient into the second calculated conditions and changing the medium-term storage speed to 1 m/d; the result is as Table 6:  
The objective function calculated by the permeability coefficient obtained by COMSOL-MATLAB is the minimum, which is 0.253. The distribution of the pore water pressure in the dam body is as Figure 11:  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Pressure Water Head (m)** & **K (cm/s)** & **K (cm/s)** & **1** & **2** & **3** & **4** & **5** & **10** & **14** \\ \hline observed value & & 5.9 & 17.9 & 30.4 & 46.4 & 58.1 & 152 & 247.9 \\ MPA-BP & 4.02 \(\times\) 10\({}^{-6}\) & 0.0698 & 5.64 & 13.73 & 19.84 & 36.64 & 49.87 & 160.98 & 243.66 \\ BP & 6.93 \(\times\) 10\({}^{-6}\) & 0.442 & 6.23 & 19.52 & 28.98 & 46.68 & 61.46 & 164.56 & 244.46 \\ COMSOL–MATLAB & 6.89 \(\times\) 10\({}^{-6}\) & 0.949 & 5.64 & 19.05 & 29.47 & 45.84 & 60.64 & 164.56 & 244.49 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Result of the second calculation condition.  
Figure 10: Absolute error and relative error of each inversion method. (**a**) Absolute error of each observed point; (**b**) relative error of each observed point.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Average of** & **Average of** & **Variance in** & **Variance in** \\  & **Absolute Error (m)** & **Relative Error** & **Absolute Error** & **Relative Error** \\ \hline MPA-BP & 3.52 & 0.105 & 3.15 & 0.109 \\ BP & 2.21 & 0.058 & 1.43 & 0.035 \\ COMSOL–MATLAB & 3.68 & 0.083 & 3.78 & 0.068 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The error’s average and variance.  
Compared with Figure 9, the wetting line in the core wall is closer to the downstream direction, since the extension of the water storage time causes the materials in the dam body to become more saturated.