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Modeling coupled erosion and filtration of fine particles in granular media

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Abstract

One of the major causes of instability in geotechnical structures such as dikes or earth dams is the phenomenon of suffusion including detachment, transport and filtration of fine particles by water flow. Current methods fail to capture all these aspects. This paper suggests a new modeling approach under the framework of the porous continuous medium theory. The detachment and transport of the fine particles are described by a mass exchange model between the solid and the fluid phases. The filtration is incorporated to simulate the filling of the inter-grain voids created by the migration of the fluidized fine particles with the seepage flow, and thus, the self-filtration is coupled with the erosion process. The model is solved numerically using a finite difference method restricted to one-dimensional (1-D) flows normal to the free surface. The applicability of the model to capture the main features of both erosion and filtration during the suffusion process has been validated by simulating 1-D internal erosion tests and by comparing the numerical with the experimental results. Furthermore, the influence of the coupling between erosion and filtration has been highlighted, including the development of material heterogeneity induced by the combination of erosion and filtration.

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Acknowledgement

The financial supports provided by the French National Institute for Industrial Environment and Risks (INERIS) and the National Natural Science Foundation of China (51579179) are gratefully acknowledged.

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Correspondence to Zhen-Yu Yin or Farid Laouafa.

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Appendix A. Finite difference solution for 1D suffusion process

Appendix A. Finite difference solution for 1D suffusion process

Defining \(r_{1} = {{\Delta t} \mathord{\left/ {\vphantom {{\Delta t} {\left( {\Delta x} \right)^{2} }}} \right. \kern-0pt} {\left( {\Delta x} \right)^{2} }}\) and \(r_{2} = {{\Delta t} \mathord{\left/ {\vphantom {{\Delta t} {\Delta x}}} \right. \kern-0pt} {\Delta x}}\) allows Eqs. (31)–(33) to be rewritten

$$- r_{1} \left[ {A_{{p_{w} }} } \right]_{j - 1/2}^{k} p_{wj - 1}^{k + 1} + \left\{ {1 + r_{1} \left( {\left[ {A_{{p_{w} }} } \right]_{j - 1/2}^{k} + \left[ {A_{{p_{w} }} } \right]_{j + 1/2}^{k} } \right)} \right\}p_{wj}^{k + 1} - r_{1} \left[ {A_{{p_{w} }} } \right]_{j + 1/2}^{k} p_{wj + 1}^{k + 1} = p_{wj}^{k}$$
(36)

with\(\begin{aligned} \left[ {A_{{p_{w} }} } \right]_{j - 1/2}^{k} & = \left( {\frac{0.5}{{\left[ {A_{{p_{w} }} } \right]_{j - 1}^{k} }} + \frac{0.5}{{\left[ {A_{{p_{w} }} } \right]_{j}^{k} }}} \right)^{ - 1} ,\quad \left[ {A_{{p_{w} }} } \right]_{j + 1/2}^{k} = \left( {\frac{0.5}{{\left[ {A_{{p_{w} }} } \right]_{j}^{k} }} + \frac{0.5}{{\left[ {A_{{p_{w} }} } \right]_{j + 1}^{k} }}} \right)^{ - 1} ,\quad \left[ {A_{{p_{w} }} } \right]_{j}^{k} = \left[ {\frac{{Ek\left( {f_{c} ,\phi } \right)}}{{\eta \bar{\rho }\left( c \right)}}} \right]_{j}^{k} \end{aligned}\)

$$\begin{aligned} - r_{2} A_{\phi } \phi_{j - 1}^{k + 1} + \left( {1 + r_{2} A_{\phi } } \right)\phi_{j}^{k + 1} = \left( {1 - \Delta tB_{\phi } } \right)\phi_{j}^{k} - \Delta tC_{\phi } \\ \end{aligned}$$
(37)

with\(\begin{aligned} A_{\phi } & = \frac{{u_{j}^{k + 1} - u_{j}^{k} }}{\Delta t},\quad B_{\phi } = - \frac{{\varepsilon_{vj}^{k + 1} - \varepsilon_{vj}^{k} }}{\Delta t},\quad C_{\phi } = \frac{{\varepsilon_{vj}^{k + 1} - \varepsilon_{vj}^{k} }}{\Delta t} + \left[ {\left( { - \lambda_{e} \left( {1 - \phi } \right)\left( {f_{c} - f_{c\infty } } \right) + \lambda_{f} \frac{{\phi - \phi_{\hbox{min} } }}{{\phi^{\beta } }}c} \right)\left| {q_{w} } \right|} \right]_{j}^{k} \end{aligned}\)

$$\begin{aligned} - r_{2} A_{c} c_{j - 1}^{k + 1} + \left( {1 + r_{2} A_{c} } \right)c_{j}^{k + 1} = \left( {1 - \Delta tB_{c} } \right)c_{j}^{k} - \Delta tC_{c} \\ \end{aligned}$$
(38)

with

$$A_{c} = \left( {\left[ {\frac{{q_{w} }}{\phi }} \right]_{j}^{k} + \frac{{u_{j}^{k + 1} - u_{j}^{k} }}{\Delta t}} \right),\quad B_{c} = \frac{1}{{\phi_{j}^{k} }}\left( {\frac{{\phi_{j}^{k + 1} - \phi_{j}^{k} }}{\Delta t} + \left[ {{\text{div}}\left( {q_{w} } \right)} \right]_{j}^{k} + \frac{{\phi_{j + 1}^{k} - \phi_{j - 1}^{k} }}{2\Delta x}\frac{{u_{j}^{k + 1} - u_{j}^{k} }}{\Delta t} - \phi_{j}^{k} \frac{{\varepsilon_{vj}^{k + 1} - \varepsilon_{vj}^{k} }}{\Delta t}} \right),$$
$$C_{c} = \left[ {{{\left( { - \lambda_{e} \left( {1 - \phi } \right)\left( {f_{c} - f_{c\infty } } \right) + \lambda_{f} \frac{{\phi - \phi_{\hbox{min} } }}{{\phi^{\beta } }}c} \right)\left| {q_{w} } \right|} \mathord{\left/ {\vphantom {{\left( { - \lambda_{e} \left( {1 - \phi } \right)\left( {f_{c} - f_{c\infty } } \right) + \lambda_{f} \frac{{\phi - \phi_{\hbox{min} } }}{{\phi^{\beta } }}c} \right)\left| {q_{w} } \right|} \phi }} \right. \kern-0pt} \phi }} \right]_{j}^{k}$$

where \(\left( {j{\kern 1pt} {\kern 1pt} = 1,2,3 \ldots ,N - 1;{\kern 1pt} \;k = 1,2,3 \ldots ,M - 1} \right)\).

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Yang, J., Yin, ZY., Laouafa, F. et al. Modeling coupled erosion and filtration of fine particles in granular media. Acta Geotech. 14, 1615–1627 (2019). https://doi.org/10.1007/s11440-019-00808-8

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