2.4 Model for predicting soil loss in Hole Erosion Test (HET)
In the HETs, the temporal variation of erosion can be understood explicitly under equation [21]. The accumulated soil loss is used for estimation of the radius expanded. The radius in the specimen is considered to be eroding uniformly, which is calculated by the following equations:
\begin{equation} m=\rho_{s}\bullet V\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [26]\nonumber \\ \end{equation}\begin{equation} m=\rho_{s}\bullet R\bullet 2\pi R\bullet L\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [27]\nonumber \\ \end{equation}\begin{equation} dm=2\pi RL\rho_{s}\bullet dR\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [28]\nonumber \\ \end{equation}\begin{equation} \int\text{dm}=\int{2\pi RL\rho_{s}\bullet dR\ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [29]\nonumber \\ \end{equation}\begin{equation} m=\pi L\rho_{s}(R^{2}-R_{0}^{2})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [30]\nonumber \\ \end{equation}\begin{equation} R=\sqrt{\frac{m}{\text{πL}\rho_{s}}+R_{0}^{2}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [31]\nonumber \\ \end{equation}
As suggested in equation [21], erosion coefficient can be interpreted by the curve of the temporal variation of erosion with the parameter (\(\lambda\)) in the HET. In order to simplify the calculation procedure to determine \(\lambda\), equation [21] is transformed into,
\begin{equation} R^{4}=\lambda^{4}\left(t+C\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [32]\nonumber \\ \end{equation}
Thus, a linear relationship is found between \(R^{4}\) and \(t\). The slope in the \(R^{4}\)-\(t\) curve reflects \(\lambda^{4}\). Based on the known \(R(t)\), other physical quantities can be determined.
\begin{equation} \dot{}=\ \rho_{s}\frac{\lambda}{4}\left(t+C\right)^{-\frac{3}{4}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [33]\nonumber \\ \end{equation}\begin{equation} \tau=\frac{4\nu Q\rho_{w}}{\pi\lambda^{3}}\left(t+C\right)^{-\frac{3}{4}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [34]\nonumber \\ \end{equation}\begin{equation} \Delta P=\frac{8\nu Q\rho_{w}L}{\pi\lambda^{4}(t+C)}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [35]\nonumber \\ \end{equation}\begin{equation} m=\text{πL}\rho_{s}[\lambda^{2}\left(t+C\right)^{\frac{1}{2}}\ \ \ \ -R_{0}^{2}]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [36]\nonumber \\ \end{equation}
It should be noted that the applicability of these equations is under three assumptions as physical boundary conditions: 1) approximate homogeneity and continuity of soil material is adopted; 2) erosion is visualized as uniform at the fluid-soil interface; 3) the eroded radius has a physical limit due to the weakness of hydraulic shear stress in the erosion process (i.e., \(\operatorname{}{R^{{}^{\prime}}(t)=0}\)).
In the understanding of erosion process every concerned parameter, including erosion rate, hydraulic shear stress, and pressure drop, is linked with the determination of the radius. Based on equation [21], the radius is interpreted from the HET and other quantities could be further determined.