## 5 Results  
### Effect of Dataset Size on the Outcome of Bayesian Updating  
Using the surrogate model and the experimental data, the iterative Bayesian updating was done until the simulated value was close to the observed value. Bayesian updating is an iterative process. First, it sets the prior distribution of SWCC parameters and uses the MCMC algorithm to generate a large number of samples that obey the prior distribution. Then it performs iterative calculations, using surrogate models and experimental water cut monitoring data. Bayesian updating is performed until the simulated value is close to the observed value. For each iteration, a set of samples of SWCC parameters are generated. All the data of the iteration are presented into a plotmatrix, as shown in Figure 10a. In this figure, the top-left figure is the frequency diagram of parameter \(a\), the bottom-right figure is the frequency diagram of parameter \(n\), and the bottom-left and top-right figures represent the correlation. Figure 10b shows the distribution of SWCC parameters obtained by Bayesian updating, converged after 5 iterations (\(j=5\)). This is the plotmatrix of the SWCC parameters after the fifth iteration.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Samples** & **MSE** & **R** \\ \hline Training & 700 & 1.43430 \(\times\)\(10^{-7}\) & 9.99991 \(\times\)\(10^{-1}\) \\ Validations & 150 & 1.67754 \(\times\)\(10^{-7}\) & 9.99989 \(\times\)\(10^{-1}\) \\ Testing & 150 & 5.56267 \(\times\)\(10^{-7}\) & 9.99968 \(\times\)\(10^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The results of the prediction with the artificial neural networks (ANN).  
Figure 9: Neural network training regression: (**a**) The regression diagram of the 70% group of the training data; (**b**) The regression diagram of the 15% group of the validation data; (**c**) The regression diagram of the 15% group of test data; (**d**) The regression diagram of all data.  
In order to study the changes in the posterior distribution when monitoring the increase in the amount of data, we used 200, 500, 1000, 2000, and 5000 sets of monitoring data to calculate the posterior distribution and analyze the results. As the amount of data increased, the trends of Bayesian updating results are shown in Figure 11, and the values of Bayesian updating results are shown in Table 2. In that figure, the red line represents the median, the upper and lower lines of the blue rectangle represent the upper and lower quartiles, the short red lines represent outliers, and the short black lines represent the minimum and maximum values. The trend of the change of parameter \(a\) is shown in Figure 11a. It can be seen that the median value of parameter \(a\) fluctuates around 1.46 and has a tendency to increase slightly. Outliers are distributed below the minimum value, indicating that the posterior distribution is left skewed. The trend of the change of parameter \(n\) is shown in Figure 11b. It can be seen that the median value of the parameter \(n\) fluctuated around 3.98. Outliers are distributed over the maximum value, indicating that the posterior distribution skewed right. When the number of data was 1000, the data of the posterior distribution were relatively stable. Therefore, in the following analysis, 1000 sets of observation data were used as samples to study the influence of the distribution type on the posterior distribution.  
Figure 11: Impact of data volume on update results: (**a**) Parameter \(a\); (**b**) parameter \(n\).  
Figure 10: Results of using 1000 sets of water content monitoring data to update: (**a**) prior distribution; (**b**) posterior distribution.