**Abstract:** The suffusion susceptibility of the soil samples is evaluated through an erosion resistance index. Thanks to existing statistical analyses, the erosion resistance index is estimated from several soil parameters. In actual exploitation, the soil properties with the input parameters related to the grain distribution of the soil… vary greatly from the original design value due to the influence of many factors. One of the factors is the inherent variability. Inherent soil variability is modeled as a random field. The usual problems used to assess the suffusion susceptibility may be not give accurate results or fully evaluate the actual working ability of the ground in each case. This is one of the reasons why dams are still eroded when they are put into use. The paper aims assess the suffusion susceptibility of the earth dam body using two-dimensional (2D) Stochastics random field, modelling the initial problem, considering the variability spatial of soil properties, using the assumption of a Normal random field of soil characteristics parameters. An illustration of a numerical simulation of homogeneous earth dam body is presented in this paper. The paper shows the predicted results of the two-dimensional spatial variability of erosion resistance index and probability of suffusion susceptibility of the homogeneous earth dam body.

**Keywords:** Internal erosion, suffusion susceptibility, numerical simulation, random field, forecasting

1. Introduction

Internal erosion is one of the main causes of instabilities within hydraulic earth structures such as dams, dikes, or levees in [1]. According to reference [2], there are four types of internal erosion: concentrated leak erosion, backward erosion, contact erosion and suffusion. Concentrated leak erosion may occur through a crack or hydraulic fracture. Backward erosion mobilizes all the grains in regressive way (i.e., from the downstream part of earth structure to the upstream part) and includes backward erosion piping and global backward erosion. Contact erosion occurs where a coarse soil is in contact with a fine soil. The phenomenon of suffusion corresponds to the process of detachment and then transport of the finest particles within the porous network under seepage flow. The finer fraction eroded and leaving the coarse matrix of the soil will further modify the hydraulic conductivity and mechanical parameters of the soil. This suffusion process may result in an increase of hydraulic conductivity, seepage velocities and hydraulic gradients, possibly accelerating the rate of suffusion in [3]. The development of suffusion may cause the incidents of dam including piping and sinkholes.

In the literature, some researchers assume that suffusion is best represented by its initiation. Reference [4] take into account the main initiation conditions for suffusion include three components: material susceptibility, critical hydraulic load and critical stress condition. Several methods have been proposed to characterize the initiation of suffusion confronting material susceptibility criteria and hydraulic criteria in [5].

In the literature, the suffusion susceptibility characterization was mainly researched through grain size based on criteria for the initiation of process. Several criteria based on the study of grain size distribution have been proposed in literature in [6–7]. Reference [8] concluded that the most widely used methods based on particle size distribution are conservative. In the case, the geometrical conditions allow particle movements, the hydraulic conditions must be studied in [9]. The hydraulic loading on the grains is often described by three distinct parameters characterizing the hydraulic loading: the hydraulic gradient in [10], the hydraulic shear stress in [11] and the pore velocity in [12]. The critical values of these three quantities can then be used to characterize the suffusion initiation in [10, 13, 12]. However, suffusion tests carried out with permeameters of different sizes indicate that scale effects exist when measuring critical hydraulic criteria in [14].

Reference [14] showed the critical hydraulic gradient concept depends on the length of the seepage path. Moreover, the value of critical hydraulic gradient is affected significantly by the hydraulic loading history in [15]. Therefore, the suffusion susceptibility of dam scales cannot be evaluated by these approaches. Besides, Reference [16] focused on the estimation of whole suffusion process. Reference [17] proposed a new analysis based on the energy expended by the seepage flow which is a function of both the flow rate and the pressure gradient. Reference [18] performed many the suffusion tests to “final state”. This ‘final state’ is obtained towards the end of each test when the hydraulic conductivity is constant while the rate of erosion decreases. The expended energy E_{flow} is the time integration of the instantaneous power dissipated by the water seepage for the test duration. For the same duration the cumulative eroded dry mass is determined, the erosion resistance index is expressed by:

Depending on the values of Iα index, Reference [19] proposed six categories of suffusion susceptibility from highly erodible to highly resistant (corresponding susceptibility categories: highly erodible for Iα < 2; erodible for 2 ≤ Iα < 3; moderately erodible for 3 ≤ Iα < 4; moderately resistant for 4 ≤ Iα < 5; resistant for 5 ≤ Iα < 6; and highly resistant for Iα ≥ 6). Since the erosion resistance index Iα has been proven to be intrinsic, i.e., independent of the sample size in [20] and of the loading path in [15], at least at the laboratory scale, it may be applied to the structure scale of a dam. Reference [21] gave a method to assess the suffusion susceptibility of low permeability core soil in compacted dams based on construction data. They showed the one-dimensional (1D) spatial variability of all material parameters, in particular the hydraulic conductivity, the dry unit weight and the grain size distribution which affect the erosion resistance index. However, the suffusion susceptibility of earth dam body through the erosion resistance index needs to be assess the two-dimensional spatial variability. A two-dimensional contour map of the erosion resistance index would provide additional valuable information.

Reference [22] showed the disparate sources of uncertainties. One of the primary sources of geotechnical uncertainties is inherent soil variability. When we repeat the experiment many times at the same location, or at different locations, we always don't get the same result. To suppress or eliminate the influence of this source, we often use a very large number of samples. However, in practice, this implementation is not feasible because the experimental conditions do not allow, or the cost is too great. So, in the current calculation, there is always this random source. The objectives of the paper are to assess the suffusion susceptibility of earth dam considering variability spatial of soil properties. To tackle this objective, the contour map of 2D spatial variability of erosion resistance index of earth dam body is presented. This approach is based on two-dimensional Stochastics random field.

2. Description

*2.1 Assessment of soil suffusion susceptibility*

Reference [18] performed many suffusion tests on 32 different soils to measure the value of erosion resistance index. For each test, the erosion resistance index Iα was measured at the ‘final state’ in [15]. From the study of several grain size-based criteria, a statistical analysis was developed to predict the erosion resistance index Iα. Reference [23] described the key influence of the grain size distribution on the suffusion process and they distinguished three main gradation curves: linear distribution, discontinuous distribution and upwardly concave distribution. The concave distribution consists of a poorly graded coarse fraction associated to a highly graded fine fraction. The soils that are likely to suffer from suffusion are, according to reference [24] “internally unstable”, i.e., their grain-size distribution curve is either discontinuous or upwardly concave. Based on this information, several criteria have been proposed in literature, and recently reference [25] proposed three categories of soil erodibility from the comparison of criteria of Istonima, Kézdi and Kenney and Lau. They defined P as the mass fraction of particles finer than 0.063mm. For gap-graded soil, Chang and Zhang defined the gap ratio as: Gr = d_{max}/d_{min} (d_{max} and d_{min}: maximal and minimal particle sizes characterizing the gap in the grading curve). For P less than 10%, the authors assumed that the stability is correctly assessed using the criterion G_{r} < 3. For P higher than 35%, the gap graded soil is reputed stable, and with P in the range 10% to 35% the soil is stable if G_{r} < 0.3P. According to Chang and Zhang, their method is only applicable to low plasticity soils.

By using a triaxial erodimeter, reference [14] determined the suffusion susceptibility of three mixtures of kaolin and aggregates. The results indicate that suffusion process depends on the grain angularity of coarse fraction. With a same grain size distribution, angularity of coarse fraction grains contributes to increase the suffusion resistance. Thus, the parameter of shape of grains plays an important role on suffusion susceptibility.

The physicochemical characteristics of the fluid and solid phases are also crucial, particularly in the case of cohesive soils. For the same type of clay, several tests performed in clayey sands have shown that suffusion decreases with the increase of the clay content in [26].

In addition, with material susceptibility, reference [4] consider two others main initiation conditions for suffusion: the stress condition and the hydraulic load. For the same material susceptibility, the modification of the effective stress can induce grain rearrangements. Several tests performed in oedometric conditions on unstable soils showed that a rise in the effective stress causes an increase of the soils’ resistance to suffusion in [14]. In the same manner, when tests were carried out under isotropic confinement in [26], the increase in the confinement pressure, and then the increase of soil density allowed a decrease in the suffusion rate.

Reference [18] showed the correlation equation between physical parameters and erosion resistance index Iα for all soils

Iα = –13.57+0.43g_{d} + 0.18j – 0.02Finer KL +0.49V_{BS} + 189.70k_{i}+3.82 min(H/F)

+0.18P +0.28G_{r} +19.51d_{5} +1.06d_{15} - 0.84d_{20}+0.81d_{50} -0.98 d_{60} -0.10d_{90} (1)

Where: dry unit weight g_{d}, blue methylene value V_{BS}, internal friction angle j, initial hydraulic conductivity k_{i}, minimum value of ratio H/F, percentage of finer fraction (based on Kenney and Lau’s criteria) Finer KL, gap ratio G_{r}, d_{5}, d_{15}, d_{20}, d_{50}, d_{60}, d_{90} (diameters of the 5%, 15%, 20%, 50%, 60%, 90% mass passing, respectively) and P (percentage of finer than 0.063mm)

For widely graded soils, the correlation of physical parameters with the erosion resistance index: (N=10, R^{2} =0.99)

Iα = -26.34+0.43g_{d} + 0.66 j – 0.16Finer KL + 1.15V_{BS} +0.37P +6.82d_{5} -1.26d_{60}

(2)

For gap-graded soils, the correlation of physical parameters with the erosion resistance index: (N=21, R^{2}=0.90)

Iα = -37.62+0.67 g_{d} + 0.64 j + 0.09Finer KL - 0.03V_{BS} -1.43P + 0.63G_{r} + 0.76d_{5} -0.97d_{60} +0.61d_{90} (3)

The distinction between widely graded or gap-graded is based on the gap-ratio in [24] and in [18]). Gap-graded soils are defined by G_{r} > 1. If this distinction is not obvious, the smallest value of Iα should be taken from the two values calculated by equations (2) and (3) to ensure a conservative estimation. The equation (2) may be viewed as a resilient tool that can be adapted to the available parameters of a given construction site.

*2.2 Assessment of the relative suffusion potential*

The erosion resistance index (Iα) is just a material parameter that characterizes the susceptibility of a given soil to suffusion. Hence, it cannot be interpreted as a ‘security factor’ to distinguish between ‘probable occurrence of erosion’ and ‘no erosion’ in [21], This distinction requires additionally the estimation of the hydraulic loading. The erosion resistance index is estimated from several soil parameters using 2D Stochastics random field. Therefore, the relative suffusion potential of the earth dam body may be characterized by the 2D contour map of the erosion resistance index Iα. A contour map shows the suffusion susceptibility at locations in the homogeneous earth dam body through the erosion resistance index value Iα. Cross-section of the earth dam will be pointed out with the spatial variability of Iα, may be low of high resistance to suffusion. Two other maps show the 2D spatial variability of density, internal friction angle.

3. Numerical simulation

*3.1 A numerical example of homogeneous earth dam body*

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

The earth dam has the structure homogeneous with assumed parameters: the height of dam crest: +21.1 m, the width of dam crest is 6.0 m. The dam slope in upstream: m = 3.5 ¸ 3.0 and in downstream: m = 2.5 ¸ 3.0. The data of dam include full grain size distributions with widely graded soil, dry unit weight, internal friction angle, initial hydraulic conductivity, and other parameters with the following assumed average values: dry unit weight g_{d} =17 kN/m^{3}; internal friction angle j=27^{0}; percentage of fines (%) (based on Kenny &Lau, 1985 criterion) Finer KL=20%; the percentage finer than 0.063 mm P=24%; d_{5} =0.1mm; d_{60} =1mm. The suffusion susceptibility will be estimated through erosion resistance index.

*3.2 Simulation methodology*

In this paper, the soil characteristic parameters are modeled as a random field. These parameters are inputted in the model using the two-dimensional (2D) Stochastics random field which is researched in [27]. In a random finite element method, the spatial variability g, j, Finer KL, P, d_{5}, d_{60} are simulated by a random field with assumed coefficient of variance (cov) cov = 0.05 and mapped onto the finite element mesh. This estimation is based on equation (2) since all soil samples are widely graded. Among the seven parameters of equation (2), the blue methylene value (V_{BS}) was considered constantly V_{BS}=0.5g/100g in the dam. The forecasting result of spatial variability of erosion resistance index with the contour map 2D is showed.

*3.3 Numerical results*

3.3.1 Forecasting the erosion resistance index and the probability of suffusion susceptibility

The result of erosion resistance index with contour map 2D is showed in figure 2. Several locations in the map pointed out with a low resistance to suffusion (blue color). Some zones with yellow color represent the high resistance to suffusion. The predicted values of Iα lies within the range from 2 to more than 9. The result of map shows the spatial variability of erosion resistance index. These results may be explained by soil spatial variability. According to reference [21], they show the one-dimensional spatial variability of erosion resistance index which erosion resistance index is estimated equally for one layer in the dam core. These results may be not given accurate results at different locations. Based on two- dimensional random field model, the two- dimensional spatial variability of erosion resistance index is predicted in the whole dam.

.4. Conclusions

The result of paper assesses the suffusion susceptibility of the dam body using two-dimensional random field considering soil spatial variability. With illustration of a numerical simulation, the predicted result of spatial variability of erosion resistance index is showed in a contour map 2D. Furthermore, the probability of suffusion susceptibility is also forecasted correspond to classification of suffusion susceptibility. This result demonstrates that the actual state of practice would be to account for the two-dimensional spatial variability.

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Published on 30/09/22

Submitted on 22/09/22

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