## 4 SWCC Model Considering Fibre Reinforcement
The addition of fibres increases the contact area between soil and water molecules, influencing soil parameters, such as pore-structure size and compaction state. The large surface area of soil particles requires more water to satisfy the high matric suction. The energy requirement for the interaction with pore water  
Figure 11: Comparison of four models on five sand samples. affects the water-retention characteristic and improves the water-retention capacity of the soil. Therefore, based on the VG model, a modified model considering fibre (VG-CFR) is derived and verified. The VG model [39] expressed by Equation (5) describes a relatively simple water content-hydraulic pressure head relationship:  
\[ \begin{cases}\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}}=\left[1+\left|ah \right|^{n}\right]^{-m},h<0\\ \theta(h)=\theta_{s},h\geq 0\end{cases}  \tag{5}\]  
where \(\theta\) is the water content; \(\theta_{s}\) is the saturated water content; \(\theta_{r}\) is the residual water content; \(h\) is the pressure head; \(\alpha\) is the parameter associated with the inlet value; and \(m\) and \(n\) are the curve shape parameters (\(m=1-n\) or \(m=1-2n\)) based on the VG model. The model of [PERSON] [57] is used to predict the hydraulic conductivity, \(K_{r}\), as follows:  
\[K_{r}=\Theta^{1/2}\left[\int_{0}^{\Theta}\frac{1}{h(x)}dx/\int_{0}^{1}\frac{1} {h(x)}dx\right]^{2} \tag{6}\]  
The dimensionless water content of fibre-reinforced soil is expressed by Equation (7):  
\[\Theta=\frac{\theta-\theta_{r}}{\theta_{s}-\theta_{r}} \tag{7}\]  
The dimensionless expression of the pressure head is given by Equation (8):  
\[\Theta=\left[\frac{1}{1+\left(ah\right)^{n}}\right]^{m} \tag{8}\]  
where \(a\), \(n\), and \(m\) are three unknown parameters. To simplify the equation in the VG model, parameters \(m\) and \(n\) in Equation (8) are limited; \(m=1\) has been verified [58]. A simple closed-form expression for \(K_{r}(\Theta)\) is obtained. Solve \(h=h(\Theta)\), and substitute the resulting expression into Equation (6):  
\[K_{r}(\Theta)=\Theta^{1/2}\left[\frac{f(\Theta)}{f(1)}\right]^{2} \tag{9}\]  
\[f(\Theta)=\int_{0}^{\Theta}\left[\frac{x^{1/m}}{1-x^{1/m}}\right]^{1/n}dx \tag{10}\]  
Substitute \(x=y^{m}\) into Equation (10) as follows:  
\[f(\Theta)=\int_{0}^{\Theta^{1/m}}y^{m-1+1/n}(1-y)^{-1/n}dy \tag{11}\]  
In Equation (11), it is easily shown that, for all integer values of \(k=m-1+1/n\), the integration can be carried out without difficulties. In a specific case, \(k=0\) and \(m=1-1/n\). Accordingly, the closed-form expression can be obtained as  
\[f(\Theta)=1-\left(1-\Theta^{1/m}\right)^{m},m=1-1/n \tag{12}\]  
Because fibre content has a more significant effect on the hydraulic characteristics of fibre-reinforced soil, the change in the water-retention capacity of the soil is only related to the change in fibre content, and the influence of fibre length is ignored. Let \(k=ac\) (where \(a\) is the parameter of the change in water-retention property caused by an increase in fibre content, and \(c\) is the fibre content). In the VG model, \(m\) is related to the symmetry of the SWCC. After adding the fibres, the symmetry of the SWCC changed. Note that if \(k=ac\)then \(ac=m-1+1/n\). Parameter \(m\) in the formula is related to \(n\) and the fibre content \(c\).