# Modeling Water-Induced Base Particle Migration in Loaded Granular Filters Using Discrete Element Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model for Base Soil Erosion

#### Model and Simulation Scheme

_{d}) does not apply to the DEM assemblies [17]; therefore, the base-filter systems were modeled at a target porosity of 50% to indicate a medium dense and stable assembly for which R

_{d}≈ 50% could be assumed. Prior geometrical assessments using retention criteria of Terzaghi [18] ($\left({D}_{15}/{d}_{85}\right)\le 4$), NRCS [19] (${D}_{15}\le 4\times {d}_{85R}$), and Indraratna et al. [20] ($\left({D}_{c35}/{d}_{85SA}\right)<1$) have been made. Here, ${D}_{15}$ and ${D}_{c35}$ represent the particle size at 15% finer by mass and controlling the constriction size at 35% finer by surface area for the filter soil, respectively. ${d}_{85}$, ${d}_{85R}$, and ${d}_{85SA}$ represent base soil particle sizes at 85% finer by mass, regraded base particle size at 85% finer by mass, and base particle size at 85% finer by surface area, respectively. It revealed that both the filters NF1 and NF2 were ineffective in retaining the base soil-NB (potential for ineffectiveness; NF2 > NF1), but they were individually internally stable [21,22]. The base particles were assigned different colors according to their group sizes to visualize their positions during the simulations. Similarly, the z-positions of the eroding particles were monitored to determine the particle infiltration depths and hence the retention capacity of the filters.

## 3. Results and Discussion

_{c}distributions for specimen NF1-NB when subjected to ${\sigma}_{vt}^{\prime}$ = 10–50 kPa. The initial contact force, f

_{c}, and hence the contact stress ${\sigma}_{c}^{\prime}$ distribution within the filter layer is almost uniform for the given ${\sigma}_{vt}^{\prime}$, but it is reduced significantly inside the base soil layer, although there too it remained uniform. This substantial ${\sigma}_{c}^{\prime}$ reduction during transfer from the filter to the base soil clearly indicated that the magnitudes of stress reduction factor ($\beta ={\sigma}_{base}^{\prime}/{\sigma}_{filter}^{\prime}$) in Appendix B for the current base-filter systems were less than unity. Similarly, the magnitudes of ${\sigma}_{c}^{\prime}$ inside both the base and the filter layers markedly increased with the corresponding increase in ${\sigma}_{vt}^{\prime}$. The final f

_{c}distributions (at t = 0.2 s) in the filter remained almost the same compared to the base, wherein the seepage stresses markedly reduced the base particle contact stresses.

## 4. Validation of Proposed Model

#### Theoretical Envelope

## 5. Conclusions

_{a}and ${\sigma}_{vt}^{\prime}$) and the geometrical and physical characteristics of filter media (e.g., PSD, R

_{d}, and ∅’ etc.) was presented.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Basic Coupled CFD-DEM Modeling Principles

## Appendix B. Equivalent Hydraulic Gradient for the Mechanical Constraint

## References

- Fourie, A.B.; Copeland, A.M.; Barrett, A.J. Optimization of the as-placed properties of hydraulic backfill. J. S. Afr. Inst. Min. Met.
**1994**, 94, 199–210. [Google Scholar] - Rice, J.D.; Duncan, J.M. Deformation and cracking of seepage barriers in dams due to changes in the pore pressure regime. J. Geotech. Geoenviron. Eng.
**2010**, 136, 16–25. [Google Scholar] [CrossRef] - Aziz, M. Using grain size to predict engineering properties of natural sands in Pakistan. Geomech. Eng.
**2020**, 22, 165–171. [Google Scholar] - Chang, D.; Zhang, L. Critical Hydraulic Gradients of Internal Erosion under Complex Stress States. J. Geotech. Geoenviron. Eng.
**2013**, 139, 1454–1467. [Google Scholar] [CrossRef] - Xiao, M.; Shwiyhat, N. Experimental investigation of the effects of suffusion on physical and geomechanic characteristics of sandy soils. Geotech. Test. J.
**2012**, 35, 890–900. [Google Scholar] [CrossRef] [Green Version] - Terzaghi, K. Failure of dam foundations by piping and means for preventing it. Wasserkr. Z. Gesamte Wasserwirtsch.
**1922**, 17, 445–449. (In German) [Google Scholar] - Dirkx, W.; Beek, R.V.; Bierkens, M. Piping the Influence of Grain Size Distribution on the Hydraulic Gradient for Initiating Backward Erosion. Water
**2020**, 12, 2644. [Google Scholar] [CrossRef] - Skempton, A.W.; Brogan, J.M. Experiments on piping in sandy gravels. Geotechnique
**1994**, 44, 449–460. [Google Scholar] [CrossRef] - Irfan, M.; Uchimura, T. Modified triaxial apparatus for determination of elastic wave velocities during infiltration tests on unsaturated soils. KSCE J. Civ. Eng.
**2016**, 20, 197–207. [Google Scholar] [CrossRef] - Indraratna, B.; Radampola, S. Analysis of critical hydraulic gradient for particle movement in filtration. J. Geotech. Geoenviron. Eng.
**2002**, 128, 347–350. [Google Scholar] [CrossRef] - Li, M.; Fannin, R.J. Comparison of two criteria for internal stability of granular soil. Can. Geotech. J.
**2008**, 45, 1303–1309. [Google Scholar] [CrossRef] - Indraratna, B.; Israr, J.; Rujikiatkamjorn, C. Geometrical method for evaluating the internal instability of granular filters based on constriction size distribution. J. Geotech. Geoenviron. Eng.
**2015**, 141. [Google Scholar] [CrossRef] [Green Version] - Feng, Q.; Ho, H.; Man, T.; Wen, J.; Jie, Y.; Fu, X. Suffusion Internal Stability Evaluation of Soils. Water
**2019**, 11, 1439. [Google Scholar] [CrossRef] [Green Version] - Trani, L.D.O.; Indraratna, B. Assessment of Subballast Filtration under Cyclic Loading. J. Geotech. Geoenviron. Eng.
**2010**, 136, 1519–1528. [Google Scholar] [CrossRef] - Cundall, P.; Strack, O. A discrete numerical model for granular assemblies. Geotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - Tsuji, Y.; Kawaguchi, T.; Tanaka, T. Discrete particle simulation of two-dimensional fluidized bed. Powder Technol.
**1993**, 77, 79–87. [Google Scholar] [CrossRef] - Shire, T.; O’Sullivan, C. Experiments on piping in sandy gravels. Acta Geotech.
**2013**, 8, 81–90. [Google Scholar] [CrossRef] [Green Version] - Terzaghi, K. Soil mechanics—A new chapter in engineering science. J. Inst. Civ. Eng.
**1939**, 12, 106–142. [Google Scholar] [CrossRef] - Natural Resources Conservation Services (NRCS). Gradation Design of Sand and Gravel Filters. In National Engineering Handbook; USDA: Washington, DC, USA, 1994; Chapter 26, Part 633. [Google Scholar]
- Indraratna, B.; Raut, A.K.; Khabbaz, H. Constriction-based retention criterion for granular filter design. J. Geotech. Geoenviron. Eng.
**2007**, 133, 266–276. [Google Scholar] [CrossRef] [Green Version] - Kenney, T.C.; Lau, D. Internal stability of granular filters. Can. Geotech. J.
**1985**, 22, 215–225. [Google Scholar] [CrossRef] - Raut, A.K.; Indraratna, B. Further advancement in filtration criteria through constriction-based techniques. J. Geotech. Geoenviron. Eng.
**2008**, 134, 883–887. [Google Scholar] [CrossRef] - Zou, Y.; Chen, Q.; Chen, X.; Cui, P. Discrete numerical modelling of particle transport in granular filters. Comput. Geotech.
**2013**, 32, 340–357. [Google Scholar] - Qing-Fu, H.; Mei-Li, Z.; Jin-Chang, S.; Yu-Long, L.; Bao-Yu, S. Investigation of fluid flow induced particle migration in granular filters using a DEM-CFD method. J. Hydrodyn.
**2014**, 26, 406–415. [Google Scholar] - Langroudi, M.F.; Soroush, A.; Tabatabaie, S.P.; Shafipour, R. Stress transmission in internally unstable gap-graded soils using discrete element modelling. Powder Technol.
**2013**, 247, 161–171. [Google Scholar] [CrossRef] - Shamy, U.E.; Zeghal, M. Coupled continuum-discrete model for saturated granular soils. J. Geotech. Geoenviron. Eng.
**2004**, 131, 413–426. [Google Scholar] [CrossRef] - Itasca. Particle Flow Code, PFC3D, Release 4.0.; Itasca Consulting Group Inc.: Minneapolis, MN, USA, 2004. [Google Scholar]
- Christie, D. Bulli field trial: Vertical and lateral pressure measurement. In Proceedings of the Rail CRC Seminar; University of Wollongong: Wollongong, NSW, Australia.

**Figure 2.**Illustration of (

**a**) simulation cell, (

**b**) numerical model specimen, and (

**c**) the Hertz–Mindlin contact model.

**Figure 3.**Effective stress distributions in specimens; (

**a**) NF1-NB at t = 0 s, (

**b**) NF1-NB at t = 0.2 s, (

**c**) NF1-NB at t = 0.3 s, (

**d**) NF2-NB at t = 0 s, (

**e**) NF2-NB at t = 0.2 s, and (

**f**) NF2-NB at t = 0.3 s.

**Figure 4.**Initial contact force distribution at ${\sigma}_{vt}^{\prime}$= (

**a**) 10 kPa, (

**b**) 20 kPa, (

**c**) 30 kPa, (

**d**) 40 kPa, and (

**e**) 50 kPa for specimen NF1-NB (Note: time t = 0 s and t = 0.2 s represent conditions before and after the coupled simulations, respectively).

**Figure 5.**(

**a**) Observed relationship between erosion ratio R

_{e}and effective stress magnitude in base soil, (

**b**) spatial distributions of particles eroded and captured at various depths within the filter medium at t = 0.2 s, and (

**c**) spatial distributions of particles eroded and captured at various depths within the filter medium at t = 0.3 s. (

**Note:**Red lines in Figure 5b,c represent ineffective filters).

**Figure 6.**(

**a**) Plot of hydraulic gradient for unit displacement of base particle ${i}_{c0}$ observed [10] and that estimated by Equation (7) versus mean constriction size ${D}_{cm}$ values, and (

**b**) comparison between experimentally observed and theoretically estimated ${i}_{cr}$ data for the complete erosion of base soil through filter.

**Figure 7.**Illustration of base particle erosion and capture inside the filter NF2 under ${\sigma}_{vt}^{\prime}$=; (

**a**) 0 kPa, (

**b**) 10 kPa, (

**c**) 30 kPa, and (

**d**) 50 kPa.

**Figure 8.**The comparison between observed and estimated ${i}_{cr}$ values by the proposed model for numerical modeling data in Table 2.

**Figure 10.**Validation through coupled CFD-DEM simulation results (test series numbers in Table 2 for data).

Parameters | Balls | Walls | Fluid | Others |
---|---|---|---|---|

Number: | ||||

NF1-NB | 16,735 | 6 | - | - |

NF2-NB | 16,295 | 6 | - | - |

Density $\rho $ (kg/m^{3}) | 2700 | - | 1000 | - |

Friction coefficient | 0.5 | 0.5 | - | - |

Porosity ($n$) | 0.5 | 0.5 | - | - |

Stiffness (N/m): | ||||

Normal (${k}_{n}$) | $1\times {10}^{6}$ | $1\times {10}^{6}$ | - | - |

Shear (${k}_{s}$) | $1\times {10}^{6}$ | $1\times {10}^{6}$ | - | - |

Time step (s): | ||||

DEM | - | - | - | $1\times {10}^{-6}$ |

CFD | - | - | - | $1\times {10}^{-3}$ |

Viscous coefficient μ (N-s/m^{2}) | - | - | 0.001 | - |

Test Series Number | Sample ID | ${\mathit{n}}_{\mathit{f}}$ $\left(\mathit{\%}\right)$ | ${\mathit{i}}_{\mathit{c}\mathit{t}}$ | ${\mathit{i}}_{\mathit{a},\mathit{m}\mathit{a}\mathit{x}}$ | Filter Effectiveness | References | |
---|---|---|---|---|---|---|---|

Observed | Current Model | ||||||

N1 | B-F1 | 50 | 99.7 | 87.5 | Yes | Yes | Qing-fu et al. [23] (Upward Flow) |

N2 | B-F2 | 50 | 31 | 187.5 | Yes | No | |

N3 | B-F3 | 50 | 6.1 | 87.5 | No | No | |

N4 | B-F4 | 50 | 3.8 | 87.5 | No | No | |

N5 | Base-F1 | 50 | 99.7 | 6 | Yes | Yes | Zou et al. [24] (Horizontal Flow) |

N6 | Base-F1 | 50 | 99.7 | 30 | Yes | Yes | |

N7 | Base-F1 | 50 | 99.7 | 83 | Yes | Yes | |

N8 | Base-F2 | 50 | 31 | 6 | Yes | Yes | |

N9 | Base-F2 | 50 | 31 | 30 | Yes | Yes | |

N10 | Base-F2 | 50 | 31 | 83 | Yes | No | |

N11 | Base-F3 | 50 | 6.1 | 6 | No | Yes | |

N12 | Base-F3 | 50 | 6.1 | 30 | No | No | |

N13 | Base-F3 | 50 | 6.1 | 83 | No | No | |

N14 | Base-F4 | 50 | 3.8 | 6 | No | No | |

N15 | Base-F4 | 50 | 3.8 | 30 | No | No | |

N16 | Base-F4 | 50 | 3.8 | 83 | No | No | |

N17 | NF1-NB-0 | 50 | 9.1 | 35 | No | No | Current Study (Upward Flow) |

N18 | NF1-NB-10 | 50 | 19.2 | 35 | No | No | |

N19 | NF1-NB-20 | 50 | 29.3 | 35 | No | No | |

N20 | NF1-NB-30 | 50 | 39.4 | 35 | Yes | Yes | |

N21 | NF1-NB-40 | 50 | 49.5 | 35 | Yes | Yes | |

N22 | NF1-NB-50 | 50 | 59.6 | 35 | Yes | Yes | |

N23 | NF2-NB-0 | 50 | 6.1 | 20 | No | No | |

N24 | NF2-NB-10 | 50 | 13.5 | 20 | No | No | |

N25 | NF2-NB-20 | 50 | 20.9 | 20 | No | Yes | |

N26 | NF2-NB-30 | 50 | 28.3 | 20 | Yes | Yes | |

N27 | NF2-NB-40 | 50 | 35.7 | 20 | Yes | Yes | |

N28 | NF2-NB-50 | 50 | 43.1 | 20 | Yes | Yes | |

N29 | NF1-NB-0 | 50 | 9.1 | 20 | No | No | |

N30 | NF1-NB-10 | 50 | 19.2 | 20 | No | No | |

N31 | NF1-NB-20 | 50 | 29.3 | 20 | Yes | Yes | |

N32 | NF1-NB-30 | 50 | 39.4 | 20 | Yes | Yes | |

N33 | NF1-NB-40 | 50 | 49.5 | 20 | Yes | Yes | |

N34 | NF1-NB-50 | 50 | 59.6 | 20 | Yes | Yes | |

N35 | NF2-NB-0 | 50 | 6.1 | 35 | No | No | |

N36 | NF2-NB-10 | 50 | 13.5 | 35 | No | No | |

N37 | NF2-NB-20 | 50 | 20.9 | 35 | No | No | |

N38 | NF2-NB-30 | 50 | 28.3 | 35 | No | No | |

N39 | NF2-NB-40 | 50 | 35.7 | 35 | No | Yes | |

N40 | NF2-NB-50 | 50 | 43.1 | 35 | Yes | Yes |

**Note:**Here ${n}_{f}$, ${i}_{a,max}$, and ${i}_{ct}$ define filter porosity, maximum applied hydraulic gradient, and hydraulic gradient governing particle migration estimated from Equation (8), respectively.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, G.; Israr, J.; Ma, W.; Wang, H.
Modeling Water-Induced Base Particle Migration in Loaded Granular Filters Using Discrete Element Method. *Water* **2021**, *13*, 1976.
https://doi.org/10.3390/w13141976

**AMA Style**

Zhang G, Israr J, Ma W, Wang H.
Modeling Water-Induced Base Particle Migration in Loaded Granular Filters Using Discrete Element Method. *Water*. 2021; 13(14):1976.
https://doi.org/10.3390/w13141976

**Chicago/Turabian Style**

Zhang, Gang, Jahanzaib Israr, Wenguo Ma, and Hongyu Wang.
2021. "Modeling Water-Induced Base Particle Migration in Loaded Granular Filters Using Discrete Element Method" *Water* 13, no. 14: 1976.
https://doi.org/10.3390/w13141976