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.