# Spatial Variability of Grouting Layer of Shield Tunnel and Its Effect on Ground Settlement
## 4 Results and Analysis  
The maximum surface settlements in the random models were extracted and the hypothesis test was conducted. Figure 7 presents the empirical cumulative distribution function (CDF) of the maximum surface settlement. The Anderson-Darling test results indicate that the overall settlements of all the cases can be successfully fitted by the normal distribution, passing the test at the 5% significance level. After confirming that the maximum surface settlement follows the normal distribution, the probability density function (PDF) of the maximum settlements was plotted. The probability density function describes the distribution of a variable. The larger the peak point of the PDF curve, the larger the mean of the maximum surface settlement. The flatter the PDF curve, the stronger the variability of the maximum surface settlement. Figure 8 shows the effect of the COVs of soil on the maximum surface settlement. It can be seen that the COV of the soil strength parameters, friction angle and cohesion, has little effect on the PDF of the maximum settlement. While the COV of the soil Young's modulus has a great influence on the maximum settlement. The maximum settlement PDF becomes taller and thinner with the decrease of the COV of the soil Young's modulus, indicating the variability of Young's modulus is the main geological reason for the fluctuation of the surface settlement. Figure 9 presents the effect of the SOFs of soil on the maximum surface settlement. It shows that when the horizontal SOF (\(s_{h}\)) increases from 10 m to 30 m, the mean value of the maximum surface settlement increases slightly. The vertical SOF (\(s_{v}\)) nearly does not influence the PDF of surface settlement. This is because the vertical SOF (\(s_{v}\)) is relatively smaller than the soil cover and the surface settlement is a result of stress and strain propagation in the global space. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Remarks**} & \multirow{2}{*}{**COV (\(c\))**} & \multirow{2}{*}{**COV (\(\varphi\))**} & \multirow{2}{*}{**COV (\(E\))**} & \multicolumn{2}{c}{**SOF (soil)/m**} & \multirow{2}{*}{**\(t\) (mm)**} & \multirow{2}{*}{**\(\delta\) (\(mm\))**} & \multirow{2}{*}{**COV (\(t\))**} \\ \cline{5-5}  & & & & & \(s_{v}\) & & & & \\ \hline Reference case & 0.2 & 0.1 & 0.15 & 2.0 & 10 & 180 & 20 & 0.2 \\ \hline \multirow{2}{*}{Effect of COV of soil cohesion} & 0.1 * & 0.1 & 0.15 & 2.0 & 10 & 180 & 20 & 0.2 \\  & 0.2 * & 0.1 & 0.15 & 2.0 & 10 & 180 & 20 & 0.2 \\  & 0.3 * & 0.1 & 0.15 & 2.0 & 10 & 180 & 20 & 0.2 \\ \hline \multirow{2}{*}{Effect of COV of soil friction angel} & 0.2 & 0.1 * & 0.15 & 2.0 & 10 & 180 & 20 & 0.2 \\  & 0.2 & 0.2 * & 0.15 & 2.0 & 10 & 180 & 20 & 0.2 \\ \hline \multirow{2}{*}{Effect of COV of soil Young’s modulus} & 0.2 & 0.1 & 0.05 * & 2.0 & 10 & 180 & 20 & 0.2 \\  & 0.2 & 0.1 & 0.15 * & 2.0 & 10 & 180 & 20 & 0.2 \\ \hline \multirow{2}{*}{Effect of soil SOF} & 0.2 & 0.1 & 0.15 & 2.0 * & 10 & 180 & 20 & 0.2 \\  & 0.2 & 0.1 & 0.15 & 2.0 * & 10 & 180 & 20 & 0.2 \\  & 0.2 & 0.1 & 0.15 & 3.0 * & 10 & 180 & 20 & 0.2 \\  & 0.2 & 0.1 & 0.15 & 2.0 & 10 * & 180 & 20 & 0.2 \\Figure 10a shows that when the gap parameter \(\delta\) increases from 10 mm to 30 mm (corresponding to the grouting-layer thickness \(t\) from 190 mm to 170 mm), the mean value and standard deviation of the maximum ground-surface settlement both increase appreciably. The apparent increase in the mean is well understood, which is mainly because the larger the radial gap, the greater the amplitude of the ground movement. It is remarkable that the standard deviation also rises observably, indicating  
Figure 8: Effect of the soil COVs on settlement. (**a**) COV(\(c\)); (**b**) COV(\(\varphi\)); (**c**) COV(\(E\)). Figure 7: Anderson–Darling goodness-of-fit test of normal distribution. that the increase of the gap parameter \(\delta\) can also increase the variability of the ground settlement. This failure risk induced by the variability of the settlement should be fully evaluated.