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.