## 2 Richards Model  
[PERSON] (1959) proposed the general differential form of the Bertalanffy equation as follows:  
\[\frac{dV}{dt}=\eta V^{m}-\gamma V \tag{1}\]  
The authors of [26; 27] gave the general solution to Equation (1) under the conditions of initial values of \(t=t_{0}\) and \(v=v_{0}\) as follows:  
\[V_{t}=A(1-Be^{-Kt})^{\frac{1}{t-m}} \tag{2}\]  
Here, \(A\), \(B\), \(K\), and \(m\) are the parameters to be estimated, where \(A\) is the limit value of biological growth, \(B\) is a parameter related to the initial conditions of biological growth, \(K=\gamma\) (\(1-m\)) is a parameter describing the growth rate, and \(m\) is the shape parameter of the curve. Because of the introduction of the shape parameter \(m\), the Richards equation has strong plasticity and a strong description ability for S curves with various deformations. According to Equation (2), we can find the different forms of the Richards equation under different \(m\) values, as shown in Table 1.  
In this study, the cumulative deformation of a fill foundation at time \(t\) is defined as \(S(t)\), the average deformation rate is \(v(t)\), and the acceleration is \(a(t)\). When the Richards \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Value** & **Equation** & **Compatibility Equation** \\ \hline \(m=0\) & \(V_{t}=A(1-B\cdot\mathrm{e}^{-Kt})\) & Mitscherlich Brody equation \\ \(m=0\), \(B=1\) & \(V_{t}=A(1-\mathrm{e}^{-Kt})\) & Knothe equation \\ \(m=2/3\) & \(V_{t}=A(1-B\cdot\mathrm{e}^{-Kt})^{3}\) & [PERSON] equation \\ \(m\to 1\) & \(V_{t}=A\cdot exp[(1-\mathrm{e}^{-Kt})]\) & Gompertz equation \\ \(m=2\) & \(V_{t}=A/(1+B\cdot\mathrm{e}^{-Kt})\) & Logistic equation \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Compatibility of Richards equation. model is used to describe the deformation of a fill foundation, Equation (2) can be rewritten as follows:  
\[S(t)=\frac{a}{\left[1+e^{(b-ct)}\right]^{\frac{1}{2}}} \tag{3}\]  
We calculate the first derivative and the second derivative of \(S(t)\) with respect to time \(t\) to obtain the expressions of the deformation rate and acceleration, as shown in Equations (4) and (5), respectively:  
\[v(t)=\frac{\mathrm{d}S(t)}{\mathrm{d}t}=\frac{ac\cdot e^{(b-ct)}}{d\cdot\left[ 1+e^{(b-ct)}\right]^{\left(\frac{1+d}{d}\right)}} \tag{4}\]  
\[a(t)=\frac{\mathrm{d}^{2}S(t)}{\mathrm{d}t^{2}}=\frac{ac^{2}}{d^{2}}e^{(b-ct)} \frac{e^{(b-ct)}-d}{\left[1+e^{(b-ct)}\right]^{\left(2+1/d\right)}} \tag{5}\]  
In combination with the characteristics of the Richards equation and foundation deformation, the possibility of using this model to describe the deformation of a fill foundation is explained below. (1) Monotonic increasing characteristic  
The Richards model curve is an asymmetric S-shaped curve. It can be seen from Equation (4) that \(v(t)=\mathrm{d}S(t)/\mathrm{d}t>0\), indicating that \(S(t)\) is a monotone increasing function about \(t\); that is, under the action of additional load, the settlement deformation of a foundation continuously increases with time, and its settlement deformation rate is always positive. (2) Asymptote  
It can be seen from Equations (3)-(5) that \(S(\infty)=\mathrm{a}\), \(v(0)=0\), and \(a(\infty)=0\). This shows that with the gradual completion of the consolidation of the fill foundation, the final settlement of the foundation converges to a certain value, and the settlement deformation rate and acceleration converge to zero. (3) Inflection point  
From Equation (5), let \(a(t)=0\) to find the inflection point of the [PERSON] model curve (\(\tau=\frac{b-\mathrm{ln}d}{c}\), \(S(\tau)=\frac{a}{\left(1+d\right)^{1/d}}\)), where \(v(t)\) reaches the maximum value \(v(\tau)=v_{\mathrm{max}}\), \(a(\tau)=0\). When \(t\in\)(0, \(\tau\)), \(v(t)\): \(v(0)\)\(\rightarrow\)\(v_{\mathrm{max}}\), \(a(t)\): \(a(0)\rightarrow+a_{\mathrm{max}}\to 0\), and when \(t\in\)(\(\tau\),\(\infty\)), \(v(t)\):\(v_{\mathrm{max}}\rightarrow\
u(\infty)=0\), and \(a(t)\):\(0\rightarrow-a_{\mathrm{max}}\to 0\).