1. Introduction
Internal erosion progressively erodes geotechnical infrastructures, such as earthen dam, road embankment, and landfill cover through preferential flow or rise in water level (Nieber et al. 2019). Internal erosion driven by seepage force would lead to the change of soil properties (i.e., soil stiffness, porosity, permeability), and further trigger the occurrence of piping, settlement, sinkhole, landslide, and even collapse (Cividini and Gioda 2004; Elkholy et al. 2015; Sato and Kuwano 2018; Indiketiya et al. 2019). The damage of geotechnical infrastructures induced by internal erosion is a worldwide issue and has caused substantial socioeconomic loss (Foster et al. 2000; Zhang et al. 2020). Therefore, it is essential to study the influence of internal erosion process on the susceptibility, vulnerability, instability, and even failure of hydraulic earth structures (Foster et al. 2000; Cividini et al. 2009). The interpretation of erosion process is significant for the safety surveillance of the earthen structures to reveal failure mechanism and provide advance warning of internal erosion (Foster et al. 2002; Fell et al. 2003; Chang and Zhang 2013; Haghighi et al. 2013).
Erosion characteristics including erosion coefficient and critical shear stress are generally based on mechanical responses of the soils to water flow (Wall and Fell 2004a). Erosion characteristics have been extensively studied as the governing factors in the process of internal erosion (Arulanandan and Perry 1983). The Hole Erosion Test (HET) is an effective and widely adopted approach to study the mechanical responses of the soils under given hydraulic gradient (Wan and Fell 2004b; ASTM 2006; Fattahi et al. 2017). The HET approach was originated from the critical hydraulic gradient method, which was commonly applied to evaluate the safety of earthen dams in the field (Nadal-Romero et al. 2011; Chang and Zhang 2013; Haghighi et al. 2013). Existing studies about HETs found that erosion characteristics were mainly influenced by soil basic properties such as Atterberg limits, clay percentage, soil density, water content, and grain size distribution (Arulanandan and Perry 1983; Wall and Fell 2004b; Indraratna et al. 2008; Benahmed and Bonelli 2012; Haghighi et al. 2013; Pereyra et al. 2019). HETs have been also utilized to qualitatively understand the potential of dam failures (Fell et al. 2003; Wan and Fell 2004a).
Generally, the mechanism of internal erosion is idealized as a soil pipe model (Nieber et al. 2019). The moving fluid flow acts on the soil boundaries causing the deformation (i.e. erosion) of the soil pipe while the enlargement of the soil pipe in turn affects the flow motion (Onate et al. 2011). This highly coupled mechanism of internal erosion induced by hydraulic shear stress is recognized as a fluid–structure interaction problem (Onate et al. 2011). Several studies have been carried out to explore the process of internal erosion in a soil pipe experimentally and theoretically. Chang et al. (2011), Fattahi et al. (2017), and Zhang et al. (2020) conducted experimental investigations on the internal erosion and suggested empirical equations of nonlinear incremental radial erosion propagation. Dumberry et al. (2017) and Xie et al. (2018) applied X-ray microcomputed tomography and invented a visual HET apparatus to observe the temporal variation of internal erosion. Besides, theoretical formulations were successively developed to model the erosion process governed by Bernoulli principles. Wan and Fell (2004a) proposed the erosion constitutive law expressing the linear relationship between erosion rate and hydraulic shear stress to capture the process of sediment detachment in a soil pipe (Haghighi et al. 2013). Bonelli and Brivois (2008) found that the radial erosion propagation under the given pressure drop obeyed a scale exponential law. Sang et al. (2015) developed semi-physically models for predicting the enlargement of the hole in the HET. The mechanism of progressive internal erosion leading to instability of the structure was interpreted and modelled by Onate et al. (2011) and Hicher (2013). Benaissa et al. (2012) and Říha and Jandora (2015) investigated the spatial distribution of hydraulic pressure conditions in the soil pipe. Nguyen and Indraratna (2020) developed the energy transformation model describing the fluid-solid interaction in the dynamic erosion process. Many studies (Cividini and Gioda 2004, Bonelli et al. 2006, Benaissa et al. 2012, Hicher 2013, Parron Vera et al. 2014, Yang et al. 2020, and Zhang et al. 2020) performed the numerical simulation based on the mass conservation equations for predicting sediment transport. However, these numerical approaches using the computational fluid dynamics or finite element method had limitation in determining erosion characteristics of soils in the HET (Lachouette et al. 2008). Limited studies considered erosion characteristics (especially erosion coefficient) in the theoretical deductions of numerical simulation, in which soil erosion were mostly considered as soil dispersity or diffusion instead of shearing movement of soil particles (Onate et al. 2011; Hicher 2013; Nieber et al. 2019). Bernatek-Jakiel and Poesen (2018) and Wilson et al. (2018) pointed out that the mechanistic mathematical formulations explaining sediment detachment and transport within soil pipes have not been fully established yet.
The main objective of this study is to formulate a theoretical model for interpretation of the erosion process (sediment detachment as well as transport) in the HETs. Bernoulli’s equation and erosion constitutive law has been adopted for formulation of the model. The constitutive relationship between erosion rate and hydraulic shear stress was developed into a second-order nonlinear ordinary differential function. An analytical solution was deduced to determine the realistic erosion propagation with an assumption of homogeneous radial erosion along the length of the hole. The analytical solution was substituted into the model to deduce the expression of temporal variations of pressure conditions. Furthermore, a new equation generated from the model was proposed to determine erosion coefficient from the realistic variation of the measured sediment in the soil pipe. Finally, the validity of the proposed model has been examined by performing the HETs with pre-formed soil pipes on a sandy lean clay subjected to different hydraulic conditions. The advantage of the proposed model is that the formulation accounts for the change in radius of the hole during erosion with given erosion coefficient.