## 2 Derivation of \(e-log\sigma^{\prime}{}_{v}\) Equations for Laboratory Test Results
### Equation for the Virgin Compression Line (VCL)  
As previously mentioned, the equations for the virgin compression line (VCL) found in the literature include linear functions (LF) [26; 27; 28], curved functions (CF) [33; 34; 35; 36; 37; 38; 39], and combined linear-curved formulations, such as the equation proposed by [PERSON] and [PERSON] (2023) [40], referred to here as the arc joint via optimum parameters (AJOP). These LF, CF, and AJOP equations describe the relationship between the void ratio (\(e\)) and the mean effective stress (\(p^{\prime}\)), where \(p^{\prime}\) is defined as \((\sigma^{\prime}{}_{v}+2\sigma^{\prime}{}_{h})/3\) where \(\sigma^{\prime}{}_{v}\) and \(\sigma^{\prime}{}_{h}\) represent the effective vertical and horizontal stresses, respectively. The mathematical expressions for the LF, CF, and AJOP models are presented in Equations (3)-(5), and a comparison of their corresponding graphs is shown in Figure 1.  
\[e=\Gamma-\lambda\cdot ln{p^{\prime}} \tag{3}\]  
\[e=\Gamma-\lambda\cdot\left(\frac{p^{\prime}}{p^{\prime}{}_{a}}\right)^{\xi} \tag{4}\]  
\[e=\Gamma-\alpha\cdot ln\left(\frac{p^{\prime}}{p^{\prime}{}_{ref}}\right)- \sqrt{\theta+\left(\alpha\cdot ln\left(\frac{p^{\prime}}{p^{\prime}{}_{ref}} \right)\right)^{2}} \tag{5}\]Based on the graphical comparison shown in Figure 1 between LF, CF, and AJOP equations, the AJOP equation provides the most accurate representation of the e-Inp relationship across both low and high stress levels. In this study, the AJOP equation was adopted, and it was revised such that the independent variable (the variable on the horizontal axis of the graph) was made compatible with the consolidation test results. This aimed to simplify and streamline the calculation process. Thus, \(p\) was changed to \(\sigma^{\prime}v\), the reference pressure \(p^{\prime}_{ref}\) was changed to a reference positive value \(R\), and the _log_ was changed to the natural logarithm _ln_. \[e=\Gamma-\alpha\cdot log\left(\frac{\sigma^{\prime}v}{R}\right)-\sqrt{\theta+ \left(\alpha\cdot log\left(\frac{\sigma^{\prime}v}{R}\right)\right)^{2}} \tag{6}\]  
\[\overline{C}_{c}=-\frac{\Delta e}{\Delta log\sigma^{\prime}v}=\alpha+\left\{ \sqrt{\theta+\left(\alpha log\left(\frac{\sigma^{\prime}v2}{R}\right)\right)^ {2}}-\sqrt{\theta+\left(\alpha log\left(\frac{\sigma^{\prime}v1}{R}\right) \right)^{2}}\right\}/log\left(\frac{\sigma^{\prime}v2}{\sigma^{\prime}v1}\right) \tag{7}\]  
Equation (6) presents the revised form of the AJOP function. The new fitting parameters of AJOP equation become \(\Gamma\), \(R\), \(\alpha\), and \(\theta\). According to the explanation in [40], \(\Gamma\) and \(R\) control the vertical shift and the horizontal shift of the graph, respectively. While \(\alpha\) controls the slope of the linear portion, \(\theta\) controls the curvature of the curved portion. Here is a short guideline for determining parameters \(\Gamma\), \(R\), \(\alpha\), and \(\theta\). All four parameters \(\Gamma\), \(R\), \(\alpha\), and \(\theta\) are strictly positive. The parameter \(\Gamma\) acts as the upper bound of the void ratio and should be slightly greater than the maximum void ratio observed in the experimental data. The parameter R controls the horizontal position of the curve and can be initially set to atmospheric pressure (e.g., 101 kPa), with further adjustment of the remaining parameters as needed. However, all parameters can be freely adjusted to better fit the experimental results. Parameter \(\alpha\) is typically close to half the slope of the linear function (LF) method. It is important to note that \(\alpha\) equals exactly half the LF slope only when \(\theta\) is set to zero. As for parameter \(\theta\), initially it should be assumed as zero and gradually increased depending on the curvature of the dataset. A smaller curvature radius (i.e., sharper curve) requires a smaller \(\theta\). A value of \(\theta\) equal to zero corresponds to a sharp corner with no curvature.