Numerical modelling of suffusion by discrete element method: a new internal stability criterion based on mechanical behaviour of eroded soil

Non-cohesive soils subjected to a flow may have a behavior in which fine particles migrate through the interstices of the solid skeleton formed by the large particles. This phenomenon is termed internal instability, internal erosion or suffusion, and can occur both in natural soil deposits and also in geotechnical structures such as dams, dikes or barrages. Internal instability of a granular material is its inability to prevent the loss of its fine particles under flow effect. It is geometrically possible if the fine particles can migrate through the pores of the coarse soil matrix and results in a change in its mechanical properties. In this work, we uses the three-dimensional Particle Flow Code (PFC3D/DEM) to study the stability/instability of granular materials and their mechanical behavior. Kenney and Lau criterion sets a safe boundary for engineering design. However, it tends to identify stable soils as unstable ones. The effects of instability and erosion, simulated by clipping fine particles from the grading distribution, on the mechanical behaviour of glass ball samples were analysed. The mechanical properties of eroded samples, in which erosion is simulated and gives a new approach for internal stability. A proposal for a new internal stability criterion is established, it is deduced from the analysis of relations between the mechanical behaviour and internal stability, including material contractance.


Introduction
The construction of earth structures such as dams, levees and road embankments often requires the use of granular soils.These cohesionless granular media are often composed of two parts: a solid (coarse) skeleton that can support and transmit the external stresses, and fine particles that can move through the constrictions.These granular soils can be internally stable or unstable, and more or less resistant to the phenomena of internal erosion and suffusion.These phenomena caused by seepage of water through the material affect its mechanical behaviour and may lead to its collapse.In granular soils, suffusion may occur only when these three following conditions are satisfied: the fine particles are smaller than the size of constrictions formed by the coarse particles; fine particles do not fill the constrictions and are free to move; the hydraulic gradient is high enough.Kenney and Lau [1][2] proposed a criterion to assess the internal stability of granular media.This criterion is based on the assumption that each class of grain, characterized by its diameter d must be prevented to move by the grains whose diameter is comprised between d and 4d.This criterion consists in plotting the PSD curve of a soil in a diagram with the mass fraction F of particles whose diameter is smaller than d on the X-axis and the mass fraction H of the particles whose diameter is comprised between d and 4d on the Y-axis.The soil is considered as stable if, for F < 20 or 30%, the curve obtained is located below the line H = F. Otherwise, if some portion of this curve passes above the line H = F, the soil is considered as unstable.According to Kenney and Lau [1][2], F values of 20% apply to the soils with widely graded primary fabric (Cu > 3) and the value of 30% to the soils with narrowly graded primary fabric (Cu < 3).The mechanism of internal stability has also been widely studied by other authors [3][4][5][6][7][8][9], mainly in order to define criteria for soils likely to develop internal erosion.Kézdi [4] proposed a criterion to assess the internal stability of soil based on the division of the grain size distribution curve into its coarser and finer components.Each component must satisfy the criterion D'15 / D''85 = 4 [5], where D'15 is the particle diameter of the filter corresponding to 15% of passing and D''85 is the particle diameter of the soil to be protected, corresponding to 85% of passing.A unified approach combining the methods of Kézdi [4] and Kenney and Lau [1][2] was established by Li and Fannin [6].The common feature of both methods is the examination of the slope of the gradation curve over a discrete interval of its length.This approach yielded a more reliable estimation of soil stability [7].Internal instability has also been assessed by Andrianatrehina et al. [8].The stability of nine samples with different grain size distribution curves, derived from direct grain size measurements carried out after triaxial and vibration tests was studied.The results were compared with the predictions of several methods [1,4,5,9,10].In the literature, many works have focused on the evaluation of the internal stability of materials, but few of them studied the change in the mechanical behaviour resulting from grain size distribution changes after flow.The objective of this work is to establish a relationship between the internal stability parameters and the mechanical behaviour of materials.To achieve this, a Discrete Elements Method (DEM) is used to model soil sample submitted to a triaxial paths.A new criterion of internal stability taking into account the mechanical behaviour of soils is proposed.

Numerical modelling
This numerical study was performed with PFC 3D that uses the discrete element method (DEM) originally developed by Cundall and Strack [11,12].The DEM method is very suitable for numerical modelling of granular systems behaviour (granular soils; rocks...).Materials are discretized in elements of simple shape (spheres) that interact with neighbouring contact elements.A material is generated with a target granular distribution and porosity and then subjected to a triaxial test.Three stages were employed for the numerical specimen tests: (1) specimen preparation; (2) consolidation and (3) specimen shearing.

Numerical sample generation
The generation procedure of granular sample with a given grain size distribution curve and a given porosity developed by Abdoulaye Hama et al [13,14] is used.

Triaxial modelling
The triaxial test in PFC3D is performed by confining a cylindrical sample within three walls.The top and the bottom walls simulate loading platens and the lateral one simulates the confining pressure experienced by the sample sides.The void volume can be calculated using the total volume of the sample minus the total volume of the solid particles.The axial strain can be calculated by the following equation (which can be obtained by a combination of the Eulerian and Lagrangian strains): Where H0 is the initial height of the sample and H1 is the current height during the test.
The tridimensional particle flow determines the wall stress and adjusts the wall velocity to reduce the difference between the current wall stress and the target wall stress σ ୵ , σ ୲ .The wall velocity u̇୵ must satisfy: Where G is the "gain" parameter that is estimated by: ‫ܩ‬ = ೢ ∆௧ [3]

Results and discussion
When it occurs in a soil sample the phenomenon of suffusion, its PSD changes and this affects its internal stability.To study the impacts of suffusion on the mechanical behaviour of granular soils, five PSD identified by S1; S2; S3; S4 and S5 were defined from an alluvial deposit from the Rhône River in France [14].Table 1 summarises the Characteristics of these samples.S stands for "Stable" and U for "Unstable"

Triaxial results
Each sample was subjected to a consolidated and drained triaxial test simulation, at same initial void ratio of e=0.66 and constant confining stress V 3 =200 kPa.
Results are presented in the deviatoric (H 1 ; q/p) and the volume change (H 1 , H v ) planes (Fig 1).They are used to calculate the stress ratio (q/p) max and the maximum contractancy (H vmax ) corresponding at projection of the characteristic point in the (H 1 ; q/p) plane.3 showed a comparison between mechanical parameters (q/p) max and H vmax ) and internal stability ones ((H/F) ୫୧୬; (D ଵହ / d85) max.).H vmax (%) 0.5 0.02 0 0.01 0.9 It is noted that the parameters of internal stability influence the mechanical behaviour of soil samples (Table 3).Samples with higher deviatoric stress and lower contractancy present a minimum ratio of stability higher than their maximum coefficient of retention: (H/F) ୫୧୬ > (D ଵହ /d ଼ହ ) ୫ୟ୶ .

Proposition of a new stability criterion
Internal instability of a granular material is its inability to prevent the loss of its fine particles under flow effect.It is geometrically possible if the fine particles can migrate through the pores of the coarse soil matrix and results in a change in its mechanical properties.

From
Fig 2: Proposed criterion 4 Validation of proposed criterion

Fig 3 :,
Fig 3: Validation of proposed criterion with literature results.'Hollow triangle' means experimentally unstable 'Full triangle' means experimentally stable

Table 1 :
Characteristics of different samples

Table 2 .
One can note that the two criteria diverge on samples S1 and S5.

Table 2 :
Samples classification according to the Kenney and Lau and Kezdi criteria

Table 3 :
Comparison of mechanical and internal stability parameters