Numerous attempts have been made to create various one-dimensional consolidation models that take into account the nonlinear variations in permeability and compressibility [4; 5; 6; 7; 8; 9; 10; 4; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Recently, [PERSON] et al. (2021) [21] successfully developed the analytical solution for \(U\) of multilayered soil under time-dependent loading. Based on this information, the calculation of \(U\) therefore has high accuracy. On the other hand, in the literature, the calculation of \(\rho_{t}\) still relies on an approximation method. This method is based on a linear relationship between the void ratio (\(e\)) and effective vertical stress (\(log\sigma^{\prime}{}_{v}\)) derived from the one-dimensional consolidation tests conducted in the laboratory. In soil mechanics, two types of clay are explained via \(e-log\sigma^{\prime}{}_{v}\): normally consolidated clay (NC), which is clay where the current stress is the highest it has ever been, and overconsolidated clay (OC), which is clay that has experienced higher stress in the past than the current stress (e.g., [22; 23; 24]). The virgin compression line (VCL) is defined as the portion of the graph that represents NC. Additionally, in the experiment, there will be parts of the graph showing unloading and reloading portions (referred as the hysteresis); both of them represent OC. Although the test results are nonlinear, the calculation method assumes linear approximation for both the VCL and the hysteresis for simplicity. The equations used to calculate consolidation settlement based on this approximation, which applies to the difference in void ratio (\(\Delta e\)) from the experimental result graphs, are presented in Equations (1) and (2) (e.g., [25]). This approximation contributes to the ongoing inaccuracy in calculating consolidation settlement \(\rho_{t}\) and \(\rho\). Normally consolidated clay:  
\[\rho=H\bigg{[}\frac{\Delta e}{1+e_{0}}\bigg{]}=\frac{H}{1+e_{0}}\bigg{[}C_{c} log\bigg{(}\frac{\sigma^{\prime}{}_{v0}+\Delta\sigma}{\sigma^{\prime}{}_{v0}} \bigg{)}\bigg{]} \tag{1}\]  
Overconsolidated clay:  
\[\rho=H\bigg{[}\frac{\Delta e}{1+e_{0}}\bigg{]}=\frac{H}{1+e_{0}}\bigg{[}C_{r} log\bigg{(}\frac{\sigma^{\prime}{}_{p}}{\sigma^{\prime}{}_{v0}}\bigg{)}+C_{c} log\bigg{(}\frac{\sigma^{\prime}{}_{v0}+\Delta\sigma}{\sigma^{\prime}{}_{p}} \bigg{)}\bigg{]} \tag{2}\]  
In Equations (1) and (2), \(H\) is the soil layer's thickness, \(e_{0}\) is the initial void ratio, \(\sigma^{\prime}{}_{v0}\) is the effective overburden stress, \(\sigma^{\prime}{}_{p}\) is the maximum past pressure, and \(\Delta\sigma\) is the stress increment due to surface loading. To address the inaccuracies in calculating consolidation settlement, it is essential to use appropriate equations for the \(e-log\sigma^{\prime}{}_{v}\) curve rather than relying on linear approximations. In literature, the relationship between \(e\) and \(log\ (p)\), where \(p\) represents the mean effective stress, from one-dimensional consolidation tests is frequently utilized for modeling soils, including clay (e.g., [26; 27; 28; 29; 30; 31; 32]) and sand (e.g., [33; 34; 35; 36; 37; 38; 39]). A comprehensive review by [PERSON] and [PERSON] (2023) [40] found that linear equations are accurate for high stress levels but tend to be quite inaccurate at lower stress levels due to the curved nature of experimental results. In contrast, curved equations in the form of power functions, which are commonly used for granular soils, are accurate at lower stress levels because the experimental results follow a similar tendency. However, power functions can be significantly inaccurate at high stress levels as they cannot accurately represent a straight line. Additionally, [PERSON] and [PERSON] (2023) [40] were the first to introduce an equation that combines a curve with a straight line on a semi-logarithmic scale, using only four fitting parameters. This approach can significantly reduce errors in calculating the changes in void ratio (\(\Delta e\)) for normally consolidated clay (NC).