4.1 Determination of erosion characteristics based on newly developed model
As illustrated in Fig. 2(b), the soil specimens were eroded approximately axisymmetric during a HET. Although irregular eroded hole in the HETs was observed in the previous studies (Benaissa et al. 2012; Říha and Jandora 2015; Lachouette et al. 2008; Xie et al. 2018), it was still reasonable to utilize a linear erosion law and homogeneity assumption to predict erosion enlargement (Benaissa et al. 2012). Hence, the results were presumptively reasonable to conduct further analyses.
Figure. 3 presented the observed erosion rate and extended radius in the erosion hole under six flow rates. The soil sample was subjected to erosion with a fixed flow rate. It is expected that erosion rate achieved is highest in the beginning. Erosion rate decreased gradually with time during the tests. The radius of the soil hole was continually increased until erosion rate became negligible. The results showed that eroded hole enlargement was nonlinear with time, which is in agreement with the previous studies of Sterpi (2003), Cividini and Gioda (2004), Wilson (2009), and Jiang and Soga (2019). The developed equation [21] was applied for explanation of the temporal variation of erosion in the tests. The data of radius was successfully fitted by equation [21]. In general, the fitted equation indicated that\(R(t)\) did follow the function of time, with an exponent of 0.25. As explained in equation [14], erosion rate could be reviewed to obey the first derivative equation of \(R(t)\). Thus, equation [33] was used to fit the data of erosion rate over time. Although both equations are describing the same erosion process, it should be noted that the goodness-of-fit of equation [21] is higher than equation [33].
As indicated in Fig. 4, proposed equations (equation [21] and [36]) fitted considerably with the temporal variation of erosion. Eroded depth was found to reduce with time. It can be also visualized that rate of change of hole size (as indicated in Fig. 2(b)) gradually becomes negligible at the end of test. It was found from Fig. 4 that eroded depth and cumulative soil loss increased with flow rate. Based on the corresponding deformation of the soil pipe, the mechanical characteristics (i.e. hydraulic shear stress and water pressure) were obtained and interpreted in the soil pipe, as documented in Fig. 5. The variation of hydraulic shear stress and pressure drop showed a similar trend with erosion rate over time. The decline of hydraulic shear stress led to the decrease of erosion rate and eroded depth as also observed from erosion law (Wall and Fell 2004). As observed from equation [12], hydraulic shear stress decreased with an enlargement of hole over time. A higher flow rate resulted into higher hydraulic shear stress and pressure drop. Interestingly, at the latter half of tests, hydraulic shear stresses of different flow rates became constant, which is defined as equilibrium shear stress in the current study. The equilibrium shear stress (\(\tau_{e}\)) will be further discussed in the next section. The recorded profile of pressure drop is consistent with that of Benahmed and Bonelli (2012). The pressure drop was balanced in the final phase of the experiment, which was referred as the semi-equilibrium condition in the study of Wilson (2009). This result supports the opinion that the generated pore water pressure from a high hydraulic gradient would be dissipated attributed to the enlargement of the soil pipe (Hicher 2013; Ouyang and Takahashi 2016). Hydraulic non-equilibrium within soil pipes would be exhausted by soil erosion and the hydraulic potential energy would be diminished during the process of sediment detachment and transport (Wilson et al. 2009; Sang et al. 2015; Nguyen and Indraratna 2020).
To interpret erosion characteristics, the \(R^{4}-t\) curves were plotted based on equation [32]. As indicated in Fig. 6 (a), proportional relationship between \(R^{4}\) and time was found in each experiment for different flow rates. Table 2 summarized the interpretation of erosion characteristics. Equation [32] was used to fit the data of HET. The slopes of the best-fit lines reflected the values of \(\lambda\). Equation [25] was used to determine erosion coefficient based on the results from equation [32]. Fig. 6 (b) proves that \(\lambda^{4}\) is proportional to flow rate for a given soil as indicated in Equation [25]. This theoretically suggests that the erosion coefficient is likely to remain constant under different experimental conditions (Indraratna et al. 2008).