## 4 Parametric Analysis and Parametric-Sensitivity Analysis
### Parametric-Sensitivity Analyses  
During the analysis of the aforementioned parameters, we discovered that the change in each parameter has a different amplitude of influence on the settlement. The sensitivity coefficient (SC) was used to analyse and compare the influence of each parameter on the settlement [31]. SC denotes the rate at which the model output value changes due to a slight change in the parameter value. In mathematics, SC is the partial derivative of the dependent variable for each independent variable, and it is therefore expressed in this study as a first-order partial derivative of the settlement function, \(w(x,y,t)\), with respect to \(\alpha\), \(G_{1}\), \(G_{2}\) and \(\eta\):  
Figure 6: Creep curves under different \(\eta\). \[SC_{\alpha}=\frac{\partial w(x,y,t)}{\partial x}=-\frac{F(x,y)}{4\pi} \bigg{(}\frac{1}{G_{2}}\underset{1}{\overset{\infty}{\Sigma}}\Psi_{1}+\Phi_{1} \underset{1}{\overset{\infty}{\Sigma}}\Psi_{2}\bigg{)} \tag{17}\] \[SC_{G_{1}}=\frac{\partial w(x,y,t)}{\partial G_{1}}=\frac{F(x,y)} {4\pi}\bigg{\{}-\frac{1}{G_{1}^{2}}-\frac{3}{(3K+G_{1})^{2}}+\Phi_{2}\Big{(}1- E_{\alpha}\Big{(}-\frac{t^{\alpha}}{\Omega_{2}}\Big{)}\Big{)}\] \[-\Phi_{1}\underset{1}{\overset{\infty}{\Sigma}}\Psi_{3}\bigg{\}}\] \[SC_{G_{2}}=\frac{\partial w(x,y,t)}{\partial G_{2}}=\frac{F(x,y) }{4\pi}\bigg{\{}-\frac{1}{G_{2}^{2}}\Big{[}1-E_{\alpha}\Big{(}-\frac{t^{\alpha }}{\Gamma_{1}}\Big{)}\Big{]}-\frac{\alpha}{G_{2}^{2}}\underset{1}{\overset{ \infty}{\Sigma}}\Psi_{4}\] \[-\Phi_{3}\Big{[}1-E_{\alpha}\Big{(}-\frac{t^{\alpha}}{\Omega_{2} }\Big{)}\Big{]}-\Phi_{1}\underset{1}{\overset{\infty}{\Sigma}}\Psi_{5}\bigg{\}}\] \[SC_{\eta}=\frac{\partial w(x,y,t)}{\partial\eta}=\frac{F(x,y)}{4 \pi}\bigg{(}-\frac{1}{G_{2}}\underset{1}{\overset{\infty}{\Sigma}}\Psi_{6}- \Phi_{1}\underset{1}{\overset{\infty}{\Sigma}}\Psi_{7}\bigg{)}\]  
where \(\Psi_{1}=\frac{n\big{(}-\frac{t^{\alpha}}{\Gamma_{1}}\big{)}^{n}ln\big{(} \frac{G_{2}}{\Gamma_{2}}\big{)}}{\Gamma(1+an)}-\frac{n\big{(}-\frac{t^{\alpha} }{\Gamma_{1}}\big{)}^{n}\int_{0}^{\infty}t^{\alpha n}e^{-t}\ln t}{[\Gamma(1+ an)]^{2}}\), \(\Psi_{2}=n\Big{(}-\frac{t^{\alpha}}{\Omega_{2}}\Big{)}^{n}\frac{ln\big{(}G_{2}^{ \alpha}\big{)}}{\Gamma(1+an)}-n\Big{(}-\frac{t^{\alpha}}{\Omega_{2}}\Big{)}^{n}\), \(\frac{\int_{0}^{\infty}t^{\alpha n}e^{-t}\ln t}{[\Gamma(1+an)]^{2}}\), \(\Psi_{3}=\frac{\big{(}-\frac{t}{\Omega}\big{)}^{n}}{(3K+G_{1})(3 KG_{1}+3 KG_{2}+ G_{1}G_{2})}\), \(\Psi_{4}=\frac{n\big{(}-\frac{t^{\alpha}}{\Omega}\big{)}^{n}}{\Gamma(1+an)}\), \(\Psi_{5}=\frac{n\big{(}-\frac{t^{\alpha}}{\Omega}\big{)}^{n}}{\Gamma(1+an)}\), \(\frac{(\alpha-1)3 KG_{1}+3 KG_{2}+G_{1}G_{2}\)}{(3 KG_{1}+3 KG_{2}+G_{1}G_{2})G_{2 }}\), \(\Psi_{6}=\frac{\big{(}-\frac{t^{\alpha}}{\Omega}\big{)}^{n}\big{(}-\frac{t^{ \alpha}}{\Omega}\big{)}}{\Gamma(1+an)}\), \(\Psi_{7}=\frac{\big{(}-\frac{t^{\alpha}}{\Omega}\big{)}^{n}\big{(}-\frac{t^{ \alpha}}{\Omega}\big{)}}{\Gamma(1+an)}\), \(\Phi_{1}=\frac{3G_{1}^{2}}{(3K+G_{1})(3 KG_{1}+3 KG_{2}+G_{1}G_{2})}\), \(\Phi_{2}=\frac{18 KG_{1}^{2}G_{2}+27K_{1}^{2}+54K^{2}G_{1}G_{2}}{[(3K+G_{1})(3 KG _{1}+3 KG_{2}+G_{1}G_{2})]^{2}}\), \(\Phi_{3}=\frac{3G_{1}^{2}}{(3 KG_{1}+3 KG_{2}+G_{1}G_{2})^{2}}\). The SCs of the parameters solved by the first-order partial derivatives produce large errors after 100 \(d\) due to the influence of computational errors. Therefore, the SCs are calculated using this method for the first 100 \(d\), and thereafter they are approximated using the differential method as follows:  
\[SC_{j}(t)=\frac{w\big{(}t+\Delta X_{j}\big{)}-w(t)}{\Delta X_{j}} \tag{18}\]  
where \(w(t)\) is the creep displacement corresponding to the basic parameter; \(\Delta X_{j}\) is the \(j\)th mechanical parameter change in the base parameter; \(w\big{(}t+\Delta X_{j}\big{)}\) is the creep displacement corresponding to the change in the \(j\)th mechanical parameter. Equation (18) is used to compute the four mechanical parameters after increasing them by 1%. Figure 7 shows the results of the SCs for each parameter.