## 2 Methodology
### Theoretical Calculation Methodology  
#### 2.3.1 Penetration Depth Prediction  
Given the time and cost constraints associated with model testing and numerical simulation, ultimate bearing capacity theory is widely adopted for the rapid estimation of spudcan penetration resistance in engineering practice. The theoretical models governing spudcan mechanical responses during penetration through heterogeneous strata, based on soil stratification characteristics and distinct failure mechanisms, are illustrated in Figure 10. For a square spudcan, the ultimate bearing capacity \(q_{u}\) (kPa) in three geotechnical scenarios is calculated as follows [22; 23; 24]:  
\[q_{u}=\left\{\begin{array}{c}c_{u}N_{c}s_{c}d_{c}+p^{\prime}_{0},\;clay\\ 0.5\gamma^{\prime}BN_{\gamma}s_{\gamma}d_{\gamma}+N_{q}s_{q}d_{p}p^{\prime}_{0 },\;sand\\ 0.5\gamma^{\prime}BN_{\gamma}s_{\gamma}i_{\gamma}+N_{q}s_{q}i_{q}p^{\prime}_{0 }+c_{u}N_{c}s_{c}i_{c},\;layers\end{array}\right. \tag{4}\]  
Here,  
\[N_{c}=\left\{\begin{array}{c}5.14,\;clay\\ \left(N_{q}-1\right)/tamp,\;layers\end{array}\right. \tag{4b}\]  
\[d_{c}=\left\{\begin{array}{c}1-0.4(D/B),\;D/B\leq 1\\ 1+0.4 arctan(D/B),\;D/B>1\end{array}\right. \tag{4c}\]  
\[d_{q}=\left\{\begin{array}{c}1+2 tan\varphi(1-sin\varphi)^{2}(D/B),\;D/B\leq 1 \\ 1+2 tan\varphi(1-sin\varphi)^{2}arctan(D/B),\;D/B>1\end{array}\right. \tag{4d}\]  
where \(q_{u}\) denotes the ultimate bearing capacity at the spudcan base (kPa). \(c_{u}\) represents the undrained shear strength of clay (kPa), calculated as the depth-averaged value within B/2 below the spudcan-soil interface [20]. \(N_{c}\) stands for the bearing capacity factor for clay. \(s_{c}\) and \(d_{c}\) indicate the shape and depth correction factors for clay, respectively. \(B\) and \(L\) correspond to the spudcan width and length (m). \(\varphi\) signifies the soil internal friction angle (\({}^{\circ}\)). \(D\) is the penetration depth from the seabed to the spudcan base (m). \(p^{\prime}_{0}\) denotes the effective overburden pressure at the spudcan base (kN). \(\gamma^{\prime}\) represents the effective unit weight of the soil above the spudcan (kN/m\({}^{3}\)). \(N_{q}\) stands for the bearing capacity factor for sand. \(s_{\gamma}\) and \(d_{\gamma}\) indicate the shape and depth correction factors for the overlying soil, respectively. \(s_{q}\) and\(d_{\delta}\) correspond to the shape and depth correction factors for sand, respectively. \(i_{\gamma}\), \(i_{q}\), and \(i_{c}\) signify the inclination factors (default = 1 for vertical loading).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Parameter** & **Physical** & **Mesh Refinement Area** & **Grid Refinement Size** & **Penetration Rate (m/s)** & **Underined Shear Strength \(s_{u}\) (kPa)** & **Internal Friction Angle \(\varphi\) (kPa)** & **Elastic Modulus \(E\)** & **Poisson’s Ratio \(\mu\)** \\ \hline Malaysian & As given & & M1 = 0.025 L; & \(vp1=0.1\); & & \\ Malaysian & As given & & M2 = 0.003 L; & \(vp2=0.2\); & & \\ kaolin clay & in Table 2 & & M3 = 0.006 L; & \(vp3=0.4\); & & \\  & & & M4 = 0.009 L; & \(vp4=0.8\) & & \\ Fujian & & & M1 = 0.025 L; & \(vp1=0.1\); & & \\ standard & As shown & & M2 = 0.003 L; & \(vp2=0.2\); & & \\ sand & in Table 3 & & M3 = 0.006 L; & \(vp3=0.4\); & & \\  & & & M4 = 0.009 L & \(vp4=0.8\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Values of parameters used in numerical simulations.