Olivier Cuisinier
email: olivier.cuisinier@ensg.inpl-nancy.fr

Akbar A Javadi

Alireza Ahangarasr

Farimah Masrouri

IDENTIFICATION OF COUPLING PARAMETERS BETWEEN SHEAR STRENGTH BEHAVIOUR OF COMPACTED SOILS AND CHEMICAL'S EFFECTS WITH AN EVOLUTIONARY-BASED DATA MINING TECHNIQUE

Keywords: Nuclear waste repository, Chemo-mechanical couplings, Artificial Intelligence, Data Mining, soil modelling, Compacted soil

When subjected to a very high-pH, most of the soil minerals undergo physico-chemical transformation. This could induce strong modifications of the shear strength behaviour of the soil. This issue is of high interest in the framework of the design of deep nuclear wastes repositories, since the degradation of the concrete lining of deep galleries after thousands of years will generate an alkaline solute (pH > 12) that would circulate through the backfill, and alter its hydromechanical characteristics. A study was undertaken to assess the impact of high-pH fluid circulation on the shear strength behaviour of a backfill material. Because of the complexity of the existing constitutive theories, a new approach was used, based on evolutionary polynomial regression (EPR), for modelling of these processes. EPR is an evolutionary data mining technique that generates a transparent and structured representation of the behaviour of a system directly from data. An EPR model was developed and validated using results from a comprehensive set of triaxial tests. Through a sensitivity analysis, the EPR model permitted to identify the specific surface, and to a lesser extent the micropore void ratio, as coupling parameters between hydromechanical behaviour alteration during alkaline fluid circulation and a physical process.

INTRODUCTION

In the French concept of nuclear waste storage, the stability of the galleries will be ensured with a concrete lining and this repository will be backfilled after use. Among other functions, this backfill will limit the convergence of the galleries after the concrete lining breaks (i.e. after thousands of years); it should also constrain the bentonite seals intended to limit water circulation along the galleries during their hydration. One key point is the degradation of the concrete lining of the galleries that will generate alkali-rich and high-pH solutes (e.g. [START_REF] Anderson | Chemical composition of cement pore solutions[END_REF] ) that will then diffuse into the backfill. This will give rise to a phenomenon called the hyperalkaline plume. Under extremely alkaline conditions, i.e. pH > 12, most of the usual soil minerals undergo extensive physicochemical transformations (e.g. [START_REF] Huertas | Effects of cement on clay barrier performance -Ecoclay project[END_REF] [START_REF] Boardman | Development of stabilisation and solidification in lime-clay mixes[END_REF] ). Very high pH water causes the dissolution of the soil primary minerals accompanied by the formation of secondary minerals like calcium silicate hydrates (CSH) and calcium aluminate hydrates (CAH) (pozzolanic reactions) (e.g. [START_REF] Bauer | Kaolinite and smectite dissolution rate in high molar KOH solutions at 35 ºC and 80 ºC[END_REF] [START_REF] Bauer | Smectite transformation in high molar KOH solutions[END_REF] [6] [START_REF] Ramírez | Hydrothermal alteration of «La Serrata » bentonite (Almería, Spain) by alkaline solutions[END_REF] , [START_REF] Claret | Experimental investigation of the interaction of clays with high-ph solutions: a case study from the callovo-oxfordian formation, Meuse-Haute Marne underground laboratory (France)[END_REF] ). Many geochemical modelling studies have also been performed (e.g. [START_REF] Gaucher | Modeling diffusion of an alkaline plume in a clay barrier[END_REF] [START_REF] Savage | Modelling the interaction of bentonite with hyperalkaline fluids[END_REF] ). However, only very few studies ( [START_REF] Rodwell | Work package 6: summary and performance assessment[END_REF] [12] [START_REF] Robinet | Effects of an alkaline plume on the hydraulic and hydromechanical properties of the bentonite MX-80[END_REF] have been carried out on these processes at the scale of compacted clay samples to characterise potential chemo-mechanical couplings between such high-pH water circulation and its impact on the geomechanical behaviour (shear strength, compressibility, permeability, etc.). In this context, Cuisinier et al. [START_REF] Cuisinier | Microstructure of a compacted soil submitted to an alkaline plume[END_REF] [START_REF] Cuisinier | Shear strength behaviour of compacted clayey soils submitted to an alkaline plume[END_REF] have carried out a study to depict the influence of the circulation of very high-pH water on the hydromechanical behaviour of compacted argillite, pure or mixed with an additive (sand, bentonite or quicklime), that are the candidate materials to be used for backfilling. The geomechanical behaviour and the microstructure of the considered materials were followed over a period of alkaline water circulation of 12 months. The impact of the alkaline fluid on the geomechanical properties of the materials appeared to be a direct function of the nature of the additive. The geomechanical behaviour of the sand-argillite mixture remained almost stable over a period of 12 months of alkaline water circulation while, over the same period, dramatic modification of the lime-argillite mixture was observed.

A subsequent step of this research would be to identify the key coupling parameter(s) between the modifications induced by the alkaline water circulation and the alteration of the hydromechanical behaviour of the tested mixtures. However, the large amount of tested configurations, the four different mixtures, and experimental conditions rendered difficult the determination of such coupling parameter(s).

In this context, a new approach is presented, based on evolutionary polynomial regression (EPR) to model the complex hydromechanical behaviour of the soil during alkaline fluid circulation. EPR is a data mining technique, based on evolutionary computing, aimed to search for polynomial structures representing a system. The key idea behind the EPR is to use evolutionary search for exponents of polynomial expressions by means of a genetic algorithm (GA) engine [START_REF] Giustolisi | A Symbolic Data-driven Technique Based on Evolutionary Polynomial Regression[END_REF] . The main advantage of EPR over other data mining techniques such as neural networks (which is widely used in modelling of engineering systems) is that it provides a transparent and structured representation of the behaviour of the system in the form of a clear mathematical expression that is readily accessible to the user. EPR has shown its potential in application in modelling of complex behaviour of soils (e.g. [18] [19] ).

The main objective of the paper is to depict the different steps followed to achieve the identification of coupling parameters between modifications induced by the alkaline water circulation and the magnitude of the resultant alteration of the shear strength. First, the main trends of the impact on the shear strength of the different mixtures are presented. An emphasis is placed on the identification of candidate coupling parameters. In the second part of the paper, the steps followed to elaborate the EPR model are presented. An EPR model is developed and evaluated based on results from test data involving various additives circulation times (i.e. 0, 3, 6, and 12 month). Four cases of data, not employed for the training phase, were chosen to be used in the testing stage to evaluate the generalisation capabilities of the developed model. In the last section, a parametric study is carried out where all parameters were set to their mean values except the one which was changed from its minimum to the maximum value in the training and testing data sets. The results of this analysis are discussed in the last section of the paper to identify the most adequate coupling parameter(s) between chemical effects and shear strength behaviour alteration.

EXPERIMENTAL CHARACTERISATION OF THE IMPACT OF THE

CIRCULATION OF ALKALINE WATER

In this section, details about the soil used and the experimental set up are first given.

Then, an overview of the results, both from microscopic and geotechnical points of views are presented.

Materials properties

An outcrop of the Callovo-Oxfordian argillite provided the material used in this study.

The Manois Argillite (MA) was carefully homogenised and crushed into a very fine grain powder. Chemical and XRD analyses indicated that the MA contains 26 to 32 % calcite, 22 to 27 % quartz, and 41 to 49 % clays (fraction lower than 2 µm). The clays are mainly illite, kaolinite and an interstratified illite-smectite. The specific surface determined is 40.4 ± 1 m 2 .g -1 .

For the study of the backfilling of deep galleries, three different additives, intended to improve the properties of the argillite, were considered. The first mixture was made of a 50 % sand and 50 % MA, on a dry-weight basis. The addition of sand increases the dry density and the frictional characteristics of the compacted argillite (e.g. [20] [21] ). The second additive was MX-80 bentonite, which was selected to enhance the sealing properties of the backfill through improved swelling (e.g. [START_REF] Dixon | A study of the compaction properties of potential clay-sand buffer mixtures for use in nuclear waste disposal[END_REF] ). The mixture used was 20 % MX-80 and 80 % MA on a dry-weight basis. The third additive used was quicklime, which improves both the cohesion and the friction angle of clayey materials (e.g. [START_REF] Fossberg | Some fundamentals engineering properties of a lime-stabilized clay[END_REF] [23] [START_REF] Little | Stabilization of pavement subgrades and base courses with lime[END_REF] [25] [START_REF] Le Runigo | Impact of the initial state on fabric and permeability of a lime treated silt under long term leaching[END_REF] ). A lime content of 4 % with the remainder MA on a dry-weight basis, was selected. The characteristics of the materials are given in Table 1.

Aging procedure and samples preparation

Special cells were developed to impose a circulation of alkaline fluid through compacted samples of the different considered mixtures under accelerated conditions over a long period of time. Details about the development of these cells are available in [START_REF] Cuisinier | Shear strength behaviour of compacted clayey soils submitted to an alkaline plume[END_REF] . The fluid was injected through the base of the cells and circulation was maintained up to 12 months. A portlandite-saturated water with a pH of about 12.4 was used to simulate the cement-pore water. That composition was selected for ease of preparation and to insure chemical stability of the cement water. The experiments were conducted at an elevated temperature in order to speed up the kinetics of the chemical reactions. The dissolution kinetics of clay particles in high pH media and pozzolanic reactions increase sharply at high temperatures ( [START_REF] Little | Stabilization of pavement subgrades and base courses with lime[END_REF] [2] ). The circulation of the selected cement water was performed under a high hydraulic gradient (50 m/m) and a high temperature (60°C), up to a circulation duration of 12 months. These conditions were much more severe than those expected in situ where a progressive saturation of the backfill by the alkaline water by diffusion is likely to occur.

The samples were prepared by static compaction in seven layers inside the circulation cells with a static compaction pressure was about 470 kPa. This pressure was used to prepare all tested samples. All mixtures in the study were prepared at their respective optimum water content determined from the standard Proctor test. The optimum water contents for each mixture statically compacted under a pressure of 470 kPa are given in Table 2 together with the initial dry density of each mixture when static compaction was used.

After the compaction of the samples, the circulation cell was closed and transferred to the oven set to 60°C. The alkaline fluid circulation began immediately, with an injection pressure of 40 kPa. If the sample pore volume is used as the volume unit, about 18 pore volumes flowed through a sample of compacted MA each month, considering an initial permeability of the Manois argillite of about 5.10 -9 m.s -1 for an initial porosity of 40 %.

During the experiment, the fluid pH was periodically measured at the outlet of the cells. A few days after the fluid circulation began, it was observed that the pH stabilized at a value between 12 and 13. That pH value remained nearly constant throughout the experiment.

Alteration of shear strength parameters by alkaline water circulation

At the end of the circulation period, each cell was carefully dismantled. Triaxial core samples were then trimmed to the desired dimensions (diameter of 3.5 cm and height of 7 cm) and used to perform triaxial consolidated undrained tests with a measurement of the internal pore water pressure. All the mechanical tests were performed at room temperature (about 20°C). The failure criterion was based on the maximum deviator stress for this study. The stress paths followed during the tests are given in Figure 1 and in Figure 2 where t corresponds to the deviatoric stress (' 1 -' 3 )/2, and s' to the mean stress (' 1 + ' 3 )/2. The shear strength envelope parameters determined during the tests are given in Table 3.

In the case of the pure MA argillite after its compaction, the initial effective cohesion c' was 25.6 kPa and the effective friction angle ' was 18.6°. It is representative of the behaviour of a compacted soil that is slightly overconsolidated. The alkaline fluid circulation for 12 months led to a slight modification of the shear strength behaviour. The effective friction angle increased to 22.2°, and this was accompanied by a reduction of the effective cohesion to 9.5 kPa (Table 3). For the Sand / MA argillite mixture, the effect of the alkaline fluid circulation was relatively limited. The alkaline fluid circulation over 12 months did not significantly alter the mechanical behaviour of the sand / MA mixture. As it can be seen from Table 3, the circulation duration had no significant effect on the shear strength envelope parameters. Before any alkaline fluid circulation, it can be seen that the addition of 20 % of bentonite tends to lower the cohesion of the MA (Table 3). The effect of the MX-80 addition on the friction angle was not significant. After the alkaline fluid circulation period of 12 months at 60°C, it can be seen in Table 3that the effective friction angle decreased to 13.5°, while the effective cohesion increased to about 30 kPa.

The shear strength behaviour of the 4 % lime treated material was tested after 90 days of curing at constant water content at 20°C. If the maximum shear stress failure criterion is used, the effective cohesion is 50.3 kPa and the effective friction angle is 39.5°. The shear strength behaviour of the lime-treated material was determined after 3, 6 and 12 months of cement water circulation. Under these conditions, the effective cohesion is 282.5 kPa and the mean effective friction angle is 53.7°. The greatest change in the shear strength behaviour of the lime-treated MA occurred during the first three months of water circulation. Longer circulation periods slightly lowered the peak shear strength of the lime-treated MA.

Alteration of microstructural parameters by alkaline water circulation

In parallel to the determination of the shear strength behaviour, the microstructure of the samples was analysed before and after alkaline fluid circulation. Samples of cubic shapes and of volume comprised between 1 and 2 cm 3 were taken in from the cells after the circulation period, then freeze-dried and kept inside desiccators. These were used to determine the microstructure of the different materials by using mercury intrusion porosimetry. The pore size distribution before and after 12 months of alkaline water circulation are compared in Figure 3. It can be seen that the impact of the alkaline water circulation is directly related to the nature of the additive employed to improve the behaviour of the argillite. When MX-80 is present, an important increase in the amount of pores between 1 and 30 µm can be evidenced after the fluid circulation. This has been attributed to the dissolution of the particles of MX-80 bentonite under high pH environment [START_REF] Cuisinier | Shear strength behaviour of compacted clayey soils submitted to an alkaline plume[END_REF] . Conversely, the stability of the microstructure of the mixture of argillite and sand is remarkable for the same period of circulation.

From Figure 3, it can be seen that the pore size distribution of each structure is made of two pore classes, a macropore class and a micropore class. In order to quantify the alteration of the microstructure, the evolution of the micropore and of the macroppore volume over the circulation period can be monitored. These data are plotted in Figure 4, the volume being expressed as a micropore void ratio and a macropore void ratio. This Figure highlights the importance of the mixture composition in the reactivity of the material during the circulation of the alkaline fluid, and the trend of modification of microstructure. As an example, after 12 months of alkaline water circulation, the macropore void ratio of the MX-80 -argillite increased from 0.44 to 0.69 while over the same circulation period, the macropore volume of the mixture of lime and argillite decreased from 0.25 down to 0.18.

The specific surface of the samples was also determined. It can be seen that the specific surface of the argillite is directly related to the nature of the additive (Table 3). The modification of the specific surface of the mixtures was also monitored. It can be seen that some additives induced dramatic modifications of the specific surface of the mixtures. Indeed, one effect of the alkaline fluid circulation was to induce the dissolution of clay particles (e.g. [START_REF] Huertas | Effects of cement on clay barrier performance -Ecoclay project[END_REF] ), that could result in a decrease in the specific surface of the soil. The modification of the specific surface as a function of the circulation time is directly related to the nature of the additive. While it remained stable in the case of the sandargillite mixture after 12 months of circulation of alkaline water, it dropped from about 51,4 m²/g down to 12.2 m²/g after 12 months of alkaline water circulations.

Discussion of the experimental resultscoupling parameter(s)

These results showed that alkaline fluid circulation could induce important modification of the geomechanical characteristics of a compacted soil. The extent of the chemo-mechanical couplings was highly dependent on the nature of the additive mixed with the argillite. Indeed, while the geomechanical behaviour of the mixture of argillite and sand remained unchanged after the alkaline fluid circulation, intense modification of that of the lime-treated argillite was observed. Moreover, similar observations can be drawn from the results of the investigations made at the microscale. As an example, the MX-80 based mixture was highly reactive with a strong increase of the macropore void ratio while the lime-based mixture was only slightly modified at the microscopic level with a slight increase of the micropore void ratio. Moreover, while the specific surface of the MX-80 mixture decreased by almost 10% after 12 months of water circulation, it decreased by 400 % in case of the lime-argillite mixture.

Then, it is not straightforward to link the intensity of the modifications at the microscale, the nature of the additive, and the alteration of the geomechanical behaviour after the alkaline fluid circulation. Therefore, if these results permit to assess how a given mixture is likely to behave when submitted to an alkaline fluid circulation, it is difficult to gather these observations together in a unique framework. Such a framework could be used to predict, at least qualitatively, how alkaline fluid circulation could alter the geomechanical properties as well as the microstructure of a mixture made of a different proportion of the constituents considered in this study. Despite the extensive experimental program already carried out, additional results obtained from other mixtures would be necessary to improve the representativeness of the results and predict the behaviour of a mixture not considered in the experimental program.

The next section is devoted to the determination of such relationship.

MODELLING OF THE SOIL BEHAVIOUR USING EPR

This section presents an overview of the EPR technique and its application to modelling the shear strength behaviour of pure argillite and argillite containing three different additives under circulation of alkaline water.

Evolutionary polynomial regression

EPR is a data-driven method based on evolutionary computing, aimed to search for polynomial structures representing a system. A physical system, having an output y, dependent on a set of inputs X and parameters θ, can be mathematically formulated as:

) , ( θ X F y  ( ( 1
)
where F is a function in an m-dimensional space and m is the number of inputs.

In EPR the evolutionary procedure is conducted in the way that it searches for the exponents of a polynomial function with a fixed maximum number of terms. During one execution it returns a number of expressions with increasing numbers of terms up to a limit set by the user, to allow the optimum number of terms to be selected. The general form of expression used in EPR can be presented as [START_REF] Giustolisi | A Symbolic Data-driven Technique Based on Evolutionary Polynomial Regression[END_REF] :

    m j j a a f F y 1 0 ) ), ( , ( X X ( ( 2
)
where y is the estimated vector of output of the process; a j is a constant; F is a function constructed by the process; X is the matrix of input variables; f is a function defined by the user; and m is the number of terms of the target expression. The first step in identification of the model structure is to transfer Eq. (2) into the following vector form:

    T d d N T m j m N N N θ Z a a a Z I Z θ Y 1 1 0 1 1 ... ) , (          ( (3)
where Y N×1 (θ,Z) is the least squares estimate vector of the N target values; θ 1×d is the vector of d=m+1 parameters a j and a 0 (θ T is the transposed vector); and Z N×d is a matrix formed by I (unitary vector) for bias a 0 , and m vectors of variables Z j . For a fixed j, the variables Z j are a product of the independent predictor vectors of inputs,

X = <X 1 X 2 … X k >.
In general, EPR is a two-stage technique for constructing symbolic models. Initially, using a standard genetic algorithm (GA), it searches for the best form of the function structure, i.e. a combination of vectors of independent inputs, X s , s=1:k, and secondly it performs a least squares regression to find the adjustable parameters, θ, for each combination of inputs. In this way a global search algorithm is implemented for both the best set of input combinations and related exponents simultaneously, according to the user-defined cost function [START_REF] Giustolisi | A Symbolic Data-driven Technique Based on Evolutionary Polynomial Regression[END_REF] . The adjustable parameters, a j , are evaluated by means of the linear least squares (LS) method based on minimization of the sum of squared errors (SSE) as the cost function.

The global search for the best form of the EPR equation is performed by means of a standard GA over the values in the user defined vector of exponents. The GA operates based on Darwinian evolution which begins with random creation of an initial population of solutions.

Each parameter set in the population represents the individual's chromosomes. Each individual is assigned a fitness based on how well it performs in its environment. Through crossover and mutation operations, with the probabilities P c and P m respectively, the next generation is created. Fit individuals are selected for mating, whereas weak individuals die off. The mated parents create a child (offspring) with a chromosome set which is a mix of parents' chromosomes. In EPR integer GA coding with single point crossover is used to determine the location of the candidate exponents.

The EPR process stops when the termination criterion, which can be either the maximum number of generations, the maximum number of terms in the target mathematical expression or a particular allowable error, is satisfied. A typical flow diagram for the EPR procedure is illustrated in Figure 5.

In the evolutionary process of building EPR models, a number of constraints can be implemented to control the output models in terms of the type of functions used, number of terms, range of exponents, number of generations etc. In this process there is a potential to achieve different models for a particular problem which enables the user to gain additional information for different scenarios [START_REF] Rezania | An Evolutionary-Based Data Mining Technique for Assessment of Civil Engineering Systems[END_REF] . Applying the EPR procedure, the evolutionary process starts from a constant mean of output values. By increasing the number of evolutions it gradually picks up different participating parameters in order to form equations describing the relationship between the parameters of the system. Each proposed model is trained using the training data and tested using the testing data provided. The level of accuracy at each stage is evaluated based on the coefficient of determination (COD) i.e., the fitness function as:

                 N 2 N a a N 2 p a Y N 1 Y Y Y 1 COD (4)
where Y a is the actual value; Y p is the EPR predicted value and N is the number of data on which COD is computed. If the model fitness is not acceptable or the other termination criteria (in terms of maximum number of generations and maximum number of terms) are not satisfied, the current model goes through another evolution in order to obtain a new model.

Database used in developing the EPR model

Results

    
). The following equation represents the EPR model developed through the above mentioned procedure: One of the major advantages of EPR is that it is capable of representing the relationship between the parameters in the form of a mathematical expression; therefore, parametric studies can be conducted to investigate the contribution of different parameters involved and to assess whether the developed model has been able to capture the physical relationships between various parameters of the system.

                                                    m M a d a M d d a m M M a m m m M m a M M

SENSITIVITY ANALYSIS

A parametric study was carried out to evaluate the predictive capabilities of the proposed model and the effects of varying different input parameters on the output. This was done through a basic approach to sensitivity analysis which is to fix all but one input variable to their mean values and vary the remaining one within the range of its maximum and minimum values. This procedure was repeated consecutively for input parameters and the results are shown in Figure 8. This sensitivity analysis aimed at determining a possible coupling parameter that could explain the changes in shear strength behaviour observed after the circulation of the alkaline fluid.

The sensitivity analysis was carried out considering four parameters (dry density, macropore void ratio, micropore void ratio and specific surface). It can be seen that the dry density did not permit to explain differences in maximum shear strength that have been observed in the experimental study. Even though there is a tendency for the maximum shear strength to increase with an increase in dry density, the calculated variation is much more limited (165-190kPa) compared to what has been observed experimentally for the tested mixtures (from hundred kPa up to several thousands of kPa). The same conclusion can be drawn from the sensitivity analysis for the macropore void ratio (e M ). However, the results of the sensitivity analysis showed that the variation of the maximum deviatoric stress for all the tested samples can be explained mainly by the value of the specific surface of the samples, especially when its value is below 100 m 2 .g -1 and to a lesser extent by the micropore void ratio. From a physical point of view, specific surface can be used as an indicator of the nature of the material as it reflects its clay content and therefore its mechanical behaviour before circulation. The addition of MX-80 bentonite to the Manois argillite results in a high value of the specific surface, which results in lowering of the mechanical characteristics compared to the initial material (Table 3). The addition of lime leads to the flocculation and aggregation of the clay particles that is responsible for a decrease in the specific surface of the Manois argillite, and an increase of the micropore void ratio. This was associated with a strong effect on the shear strength behaviour because addition of lime led to the formation of cementitious compounds responsible for the increase in shear strength (Table 3) as evidenced by several authors (e.g. [START_REF] Little | Stabilization of pavement subgrades and base courses with lime[END_REF] ). The effect of the alkaline water circulation may also alter the specific surface of the tested material. Indeed, the main effect of the alkaline fluid circulation was to induce the dissolution of clay particles (e.g. [START_REF] Huertas | Effects of cement on clay barrier performance -Ecoclay project[END_REF] ), that could result in a decrease in the specific surface of the soil. The dissolved compounds may react with the calcium to form cementitious compounds responsible for the increase in shear strength. The formation of new cementitious compounds can lead to a closure of the smallest pores of the soil as evidenced by some researchers who studied lime-treated soils at microstructural level ( [28] [29] ).

The combination of the two parameters permitted to explain the behaviour of the mixture of Manois argillite and of MX-80 bentonite. This sensitivity analysis shows that the model is also able to capture and represent the behaviour of the individual mixture. In fact, in the case of this mixture, the circulation led to a strong decrease of specific surface (-25 m 2 .g -1 ) and an increase of the macropore void ratio without significant modification of the micropore void ratio.

CONCLUSION

A comprehensive study of the impact of alkaline fluid circulation on the mechanical behaviour of several mixtures made with Manois Argillite mixed with different additives has been carried out. Although this study permitted to assess the behaviour of the individual mixture, there was a need to develop a relationship able to describe the behaviour of the different mixtures in a single framework. The main objective of the paper was to present the development of this relationship, based on evolutionary polynomial regression (EPR), to model the complex hydromechanical behaviour of the soil during alkaline fluid circulation.

Overall, the results show that the EPR model, trained from pure experimental data, was able to capture the main trends of the behaviour of the different mixtures after the fluid circulation with a reasonable accuracy for both training and testing sets of data.

This model was then used to perform a sensitivity analysis in order to identify the best possible coupling parameters between chemical's effect and the shear strength behaviour. It appeared that the most appropriate parameter is the specific surface of the mixture. The specific surface is able to describe the differences at the initial state (before circulation), reflecting the composition of the mixture, i.e. the nature of the additive, and the differences in their shear strength behaviour. Moreover, specific surface is sensitive to the alkaline fluid circulation duration, and its impact on the shear strength behaviour of each individual mixture. To a lesser extent, the micropore void ratio appeared to be a coupling parameter, even though the shear strength behaviour of the different mixtures is less sensitive to it. This study showed that the mechanical alteration of the backfill can be assessed from monitoring of the specific surface modification of the material that is to be used. Additional research is however required to better characterise the physico-chemical processes responsible for the modification of the specific surface and the micropore void ratio, and the corresponding increase in shear strength.

Results

Individuals from mating pool selected Exponent vector selected New generation of Exponent vector -EPR model captured many trends of the behaviour of the different mixtures considered -The most appropriate coupling parameter is the specific surface of the mixture.

G A

*Research Hightlights

  from 33 consolidated undrained triaxial tests conducted on samples of argillite (MA), lime-MA, sand-MA, bentonite-MA mixtures, subjected to different periods of exposure to alkaline water circulation, were used for development and validation of the EPR model. Of the total of 33 cases, 29 cases related to different circulation times (i.e. 0, 3, 6, 12 months) were used for training of the EPR model. Of these 29 cases, 11 cases were related to no circulation (0 month of circulation), 5 cases to 3 months, 2 cases to 6 months and 11 cases to 12 months of circulation of the alkaline water through the samples. The remaining 4 cases (each relating to a different soil and circulation time) were kept unseen during the model development process and used to evaluate the developed model. Eight parameters were used as input for the EPR model. These included the dry density ( d  ), alkaline water circulation time ( t ), axial strain ( a  ), pore pressure ( u ) and the effective confining pressure ( 3   ). It was also required to introduce parameters that could account for the physic-chemical processes induced by alkaline water circulation. It has been shown that a deep reorganisation of the structure occurred over the circulation period. This reorganisation can be accounted for by the balance between macro-pores ( M e ) and porosity of micro-pores ( m e ) that have been thus integrated in the analysis. Moreover, the alkaline water circulation can also alter the specific surface (SS) of the tested material to a different extent as a function of the nature of the additive. This parameter is related to dissolution of clay particles. The only output was considered to be the deviatoric stress ( 3 1

Figure 6

 6 Figure 6 shows typical results of the training of the EPR model for sand-argillite, lime-

FIGURES CAPTIONSFigure 1 .

 1 FIGURES CAPTIONS

Figure 2 .

 2 Figure 2. Mechanical behaviour of the mixture of lime and MA argillite before (a) and after

Figure 3 .

 3 Figure 3. Influence of 12 months of alkaline water circulation on the pore size distribution of

Figure 4 .

 4 Figure 4. Evolution of the porosity due to the circulation of the alkaline water.

Figure 5 .

 5 Figure 5. Typical flow diagram for EPR procedure.

Figure 6 . 3 

 63 Figure 6. Typical training results of the EPR model: (a) Sand-argillite, 3   =569kPa, (b) Lime-

Figure 7 .

 7 Figure 7. Typical testing results of the EPR model: (a) Pure argillite, 3   =295kPa, No

Figure 8 .

 8 Figure 8. Sensitivity analysis considering effect of (a) dry density, (b) specific surface, (c)

  Figure captionsClick here to view linked References

  Figure 6 Click here to download high resolution image

Table 1 . Properties of the tested materials.

 1

		Liquid limit (%)	Plasticity index (%)	Solid density (Mg/m 3 )	Cation exchange capacity (meq /100 g)
	MHM argillite¹	31.0	17.9	2.70	6.0 to 14
	MA	51.0	11.2	2.68	23
	Calcareous sand	non plastic	non plastic	2.71	-
	MX-80²	520	458	2.65	97
	1: data of MHM argillite from Deroo (2002)
	2: data from Marcial et al. (2002) and Neaman et al. (2003)

Table 2 . Characteristics of the different mixtures after compaction (MA : Manois Argillite).

 2

	Mixture	MA	MA + 50 % Sand	MA + 20 % MX-80	MA + 4 % lime
	Optimum water content (%)	23.0	15.5	25.5	25.5
	Dry density (Mg.m -3 )	1.61	1.83	1.54	1.50

Table

Table 3 . Characteristics of the samples as a function of the circulation duration.

 3

	Mixture	Circulation duration (months)	Water content (%)	Dry weight (Mg.m -3 )	Specific surface (m 2 .g -1 )	Effective friction (°) angle	Effective cohesion (kPa)
		0	23.0	1.61	156.6	18.6	25.6
	Pure argillite
		12	27.5	1.54	142.0	22.2	9.5
		0	15.5	1.83	67.5	28.9	0
	50 % Argillite +	3	17.1	1.86	71.7
	50 % sand	6	14.9	1.85	67.5	26.8*	0*
		12	16.2	1.88	74.6
	80 % Argillite +	0	25.5	1.54	205.6	18.1	0
	20 % MX-80	12	34.5	1.34	182.6	13.5	30.0
		0	25.5	1.55	51.4	39.5	50.3
	96 % Argillite +	3	29.0	1.49	24.5
	4 % lime	6	29.9	1.47	19.6	53.7*	282.5*
		12	29.7	1.48	12.2

* : no significant influence of circulation duration on shear strength parameter.


Abdelwadoud Mehenni
email: a.mehenni@bouygues-construction.com

Olivier Cuisinier
email: olivier.cuisinier@univ-lorraine.fr

Farimah Masrouri
email: farimah.masrouri@univ-lorraine.fr

Impact of lime, cement and clay treatments on the internal erosion of compacted soils

Keywords: Soil stabilization, internal erosion, lime, cement, clay treatment

Introduction

Soil treatment with different types of clays, lime or cement is widely used in geotechnical engineering to improve soil characteristics and thus achieve a higher design strength, elastic modulus, and lower hydraulic conductivity, among other features. The main effects of these treatments on the geotechnical characteristics of compacted soils are relatively well known.

Clayey soils are often used as treatment product specifically to reduce the hydraulic conductivity of soils in order to reach a target design. The reduction of the hydraulic conductivity depends on the percentage and the nature of the clayey soil employed (e.g., Sivapullaiah et al., 2000[START_REF] Chapuis | The 2000 R.M. Hardy Lecture: Full-scale hydraulic performance of soil-bentonite and compacted clay liners[END_REF]. Several authors (e.g., [START_REF] Eades | A quick test to determine lime requirements for lime stabilization[END_REF][START_REF] Brandl | Alteration of soil parameters by stabilization with lime[END_REF][START_REF] Bell | Lime stabilization of clay minerals and soils[END_REF] showed that lime treatment improves mainly the soil workability as well as the soil strength. Lime effects on the hydraulic conductivity of soils depend on the compaction conditions [START_REF] Mccallister | Physical property changes in a lime-treated expansive clay caused by leaching[END_REF][START_REF] Le Runigo | Impact of initial state on the fabric and permeability of a lime-treated silt under long-term leaching[END_REF][START_REF] Cuisinier | Microstructure and hydraulic conductivity of a compacted lime-treated soil[END_REF]. Several studies showed the enhancement of mechanical characteristics of soils after cement treatment (e.g., Al-Amoudi 2002, Sariosseiri and[START_REF] Sariosseiri | Effect of cement treatment on geotechnical properties of some Washington state soils[END_REF]. The addition of cement is often related to a decrease of the hydraulic conductivity. However some authors (e.g., Bellezza and Fratalocchi, 2006) showed that the evolution of the hydraulic conductivity is mainly related to the nature of soil, and may increase or decrease after treatment.

Nevertheless, using compacted treated soils to build hydraulic earth structures (e.g., earth dams, levees, water retaining structures) raises some specific issues. Several objectives must be met simultaneously; specifically, low permeability and the overall stability of the earth structure are paramount. Moreover, these structures can be affected by different internal or external erosion phenomena. Erosion may affect soils, reducing the life of the structure or even leading to its destruction. Soil treatment with clays, lime or cement could be of interest to increase the strength and reduce soil erodibility, extending the service life of an earth structure. Therefore, the impact of these common treatments on erosion characteristics must be determined. After the description of the erosion process and its experimental characterization, a literature review on the effects of treatments on the internal erosion characteristics is presented.

Background

Erosion phenomena occurring on hydraulic earth structures can be classified into two categories: external erosion and internal erosion. External erosion concerns the outer surfaces of structures, whereas internal erosion occurs within the body of the structure. This study is focused on piping, which is an internal erosion mechanism. According to [START_REF] Fell | Time for development of internal erosion and piping in embankment dams[END_REF], the process of internal erosion and piping can be divided into four phases: the initiation of erosion on a concentrated leak, the continuation of erosion, the progression of erosion by the enlargement of the concentrated leak to form a pipe, and the formation of a breach leading to the destruction of the structure.

To study internal erosion and piping, Wan andFell (2002, 2004) developed the hole erosion test (HET), which was derived from other experimental erosion devices, such as the pinhole test [START_REF] Sherard | Pinhole test for identifying dispersive soils[END_REF]ASTM D-4647), the drill hole test [START_REF] Lefebvre | Erosivity of natural intact structured clay: evaluation[END_REF][START_REF] Rohan | A new technique to evaluate erosivity of cohesive material[END_REF], and the flow pump test [START_REF] Reddi | Comparison of internal and surface erosion using flow pump tests on a sand-kaolinte mixture[END_REF]. The HET characterizes the behavior of soils only once a pipe is completely formed. The test does not allow for the study of the initial phase of the internal erosion process (i.e., the formation of the pipe in the soil). In fact, in the HET, a preformed hole in a soil sample is created to simulate a pipe; erosion of the pipe is then studied by subjecting the hole to a flow of water. The growth of the diameter of the hole is monitored by measuring hydraulic parameters (i.e., the pressure drop between the areas upstream and downstream of the sample and the flow through the hole). The results determine the empirical erosion law of the material, which can be expressed as follows:

𝜀𝜀̇= 𝑘𝑘 𝑒𝑒𝑒𝑒 (𝜏𝜏 -𝜏𝜏 𝑐𝑐 ) (1)
where:

𝜀𝜀̇ is the erosion rate per unit surface area of the hole at time 𝑡𝑡 (kg/s/m 2 ), 𝜏𝜏 is the hydraulic shear stress along the hole at time 𝑡𝑡 (Pa), 𝑘𝑘 𝑒𝑒𝑒𝑒 is the coefficient of soil erosion (s/m), 𝜏𝜏 𝑐𝑐 is the critical shear stress (Pa).

The HET yields two parameters that characterize the soil erosion behavior: i) the critical shear stress 𝜏𝜏 𝑐𝑐 , corresponding to the minimum shear stress necessary to initiate the departure of soil particles; and ii) the coefficient of soil erosion 𝑘𝑘 𝑒𝑒𝑒𝑒 , which expresses the erosion rate during the erosion process.

To compute 𝜀𝜀̇ and 𝜏𝜏, it is necessary to determine the variation of the diameter of the hole during the test; several assumptions are made following this measurement. It is assumed that erosion occurs on a uniform circular cross section along the length of the soil sample during the test. [START_REF] Wan | Investigation of rate of erosion of soils in embankment dams[END_REF] proposed a method to determine the growth of the hole diameter during erosion based on the friction factor. This method required the determination of the final hole diameter after erosion, which is typically difficult to evaluate because the final shape of the hole is often irregular. [START_REF] Wan | Investigation of rate of erosion of soils in embankment dams[END_REF] assumed that the friction factor in laminar or turbulent flow varies linearly with time. Two problems with this approach can be noted: 1) during the drilling step, the hole is disturbed, and thus, the initial friction factor used in the calculations is not representative of the real situation; and 2) the hypothesis of the linear variation of the friction factor on the hole during the test is not realistic [START_REF] Haghighi | Improvement of hole erosion test and results on reference soils[END_REF]. Another method uses the turbidity signal to compute the growth of the hole diameter during the test [START_REF] Pham | Erosion et dispersion des sols argileux par un fluide[END_REF][START_REF] Indraratna | Modelling the erosion rate of chemically stabilized soil incorporating tensile force-deformation charcacteristics[END_REF][START_REF] Benahmed | Investigation concentrated leak erosion behaviour of cohesive soils by performing hole erosion tests[END_REF][START_REF] Haghighi | Improvement of hole erosion test and results on reference soils[END_REF].

To classify soils by their resistance to internal erosion, authors often use the method proposed by [START_REF] Wan | Investigation of rate of erosion of soils in embankment dams[END_REF], which is based on 𝑘𝑘 𝑒𝑒𝑒𝑒 . The Erosion Rate Index is calculated as ERI = -log(𝑘𝑘 𝑒𝑒𝑒𝑒 ), and soils are classified by extremely rapid erosion (1) to extremely slow erosion (6), as shown in Table 1. However, this method does not consider 𝜏𝜏 𝑐𝑐 , which qualifies the shear stress needed to initiate soil erosion along the pipe; this parameter is not included because the HET does not yield good repeatability of 𝜏𝜏 𝑐𝑐 [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF]. Indeed, for specimens prepared under the same conditions, different studies indicated that both 𝑘𝑘 𝑒𝑒𝑒𝑒 and 𝜏𝜏 𝑐𝑐 might vary from one test to another for the same soil. 𝑘𝑘 𝑒𝑒𝑒𝑒 can vary by a factor of 2 (e.g., [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF][START_REF] Farrar | Bureau of reclamation erosion testing for evaluating piping and internal erosion of dams[END_REF], whereas 𝜏𝜏 𝑐𝑐 may vary by a factor of more than 10 (e.g., [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF]. The use of the ERI allows for the classification of different samples of the same soil in the same erosion group even if 𝑘𝑘 𝑒𝑒𝑒𝑒 varies from one test to another. Another method to overcome these repeatability issues is to use all results for 𝜀𝜀̇ and 𝜏𝜏 𝑐𝑐 obtained from several specimens, prepared under the same conditions, to determine a unique erosion law [START_REF] Pham | Erosion et dispersion des sols argileux par un fluide[END_REF][START_REF] Haghighi | Caractérisation des phénomènes d'érosion et de dispersion des sols : développement d'essais et applications pratiques[END_REF].

In the last decade, several optimizations were made on the HET to allow for more control of the hydraulic parameters, such as the hydraulic pressure drop across the sample or the water flow rate [START_REF] Pham | Erosion et dispersion des sols argileux par un fluide[END_REF][START_REF] Indraratna | Modelling the erosion rate of chemically stabilized soil incorporating tensile force-deformation charcacteristics[END_REF]Whal, 2010;[START_REF] Benahmed | Investigation concentrated leak erosion behaviour of cohesive soils by performing hole erosion tests[END_REF][START_REF] Luthi | A modified hole erosion test (HET-P) device[END_REF][START_REF] Haghighi | Improvement of hole erosion test and results on reference soils[END_REF].

The HET was used to assess the impact of the compaction parameters (i.e., the initial water content, initial dry density and compaction energy) on the erodibility of compacted soils. [START_REF] Attom | The effect of compaction condition and initial water content on soil erosion[END_REF] showed that at the same initial dry density, soils exhibit a higher ERI if they are compacted on the wet side of optimum. This observation has been confirmed by [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF] and [START_REF] Lim | Experimental investigation of erosion in variably saturated clay soils[END_REF]. The increase in the initial dry density of soil induces a higher ERI [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF][START_REF] Lim | Experimental investigation of erosion in variably saturated clay soils[END_REF] and a higher 𝜏𝜏 𝑐𝑐 [START_REF] Benahmed | Investigation concentrated leak erosion behaviour of cohesive soils by performing hole erosion tests[END_REF]. For compacted soils, the effects of the initial dry density, on increasing the ERI, are also related to the initial water content; the increase of the ERI is more pronounced on the dry side of optimum [START_REF] Attom | The effect of compaction condition and initial water content on soil erosion[END_REF]. [START_REF] Wahl | A comparison of the hole erosion test and jet erosion test[END_REF] and [START_REF] Attom | The effect of compaction condition and initial water content on soil erosion[END_REF] have also shown that along the compaction curve, the highest ERI values are obtained for coupled values of the optimum initial water content and maximum dry density.

The percentage of clay particles can modify the internal erosion resistance of soils. Several authors showed that increasing the clay percentage induces a higher 𝜏𝜏 𝑐𝑐 14T and lower 𝑘𝑘 𝑒𝑒𝑒𝑒 [START_REF] Bennabi | Some elements of comparison between two laboratory devices for soil erosion testing[END_REF][START_REF] Benahmed | Investigation concentrated leak erosion behaviour of cohesive soils by performing hole erosion tests[END_REF][START_REF] Haghighi | Improvement of hole erosion test and results on reference soils[END_REF].

A limited number of studies considered the improvement of soil internal erosion characteristics by lime or cement treatment. To study the internal erosion of a specific soil, it is necessary to use a device capable of applying the necessary inlet pressure and hence the necessary pressure drop across the soil sample required to initiate the erosion of the studied soil. However, the existing devices are not convenient to fully characterize the impact of these treatments on erodibility due to technical limitations. In fact, the study of erosion in treated soils requires a high level of applied hydraulic shear stress that few devices can provide. [START_REF] Chevalier | Resistance to erosion of lime treated soils: a complete parametric study in laboratory[END_REF] showed that a 2% lime-treatment of silt can increase 𝜏𝜏 𝑐𝑐 14 T by a factor of at least 4. [START_REF] Herrier | Lime treated soil as an erosion-resistant material for hydraulic earthen structures[END_REF] studied a 2% lime-treated soil with a curing time of 14 days and observed that 𝜏𝜏 𝑐𝑐 16 T increased by a factor of 20, whereas 𝑘𝑘 𝑒𝑒𝑒𝑒 16T was reduced by a factor of 10. [START_REF] Indraratna | Modelling the erosion rate of chemically stabilized soil incorporating tensile force-deformation charcacteristics[END_REF] worked on silty sand treated with cement with a 𝜏𝜏 𝑐𝑐 near zero before treatment; they showed that treatment with 3% cement increased 𝜏𝜏 𝑐𝑐 by up to 50 Pa and decreased the 𝑘𝑘 𝑒𝑒𝑒𝑒 by two orders of magnitude. Additionally, they found that the decrease in erosion by cement treatment is directly related to the percentage of cement added to the soil.

Thus, the literature has shown that the HET could be a convenient method to quantify soil erodibility. Furthermore, treatment with clays, lime or cement could reduce soil erodibility.

However, the limited data available did not provide a sufficient amount of information regarding the impact of clay treatment on silty soils or the impact of lime and cement treatments, the curing time and the percentage of admixture.

The main focus of this study is to investigate the impact of soil treatments on erosion characteristics. A new enhanced HET device was created to quantify the effects of different treatments. In the following sections, the testing procedures and theoretical models are presented, and the modifications produced by the treatments are discussed.

Materials and experimental setup

In this section, details regarding the tested materials and experimental setup are presented.

Materials

The studied soil was a silty soil from northern France; Table 2 summarizes the main physical and geotechnical properties of the soil. Four types of commonly used treatment products were considered: kaolinite (2 and 9%), bentonite (2%), quicklime (1 and 3%) and cement (3 and 6%). These percentages refer to the quantity of the product calculated on a dry soil weight basis. These choices were made based on usual field uses of these treatment products in

France [START_REF] Setra-Lcpc | Traitement des sols à la chaux et/ou aux liants hydrauliques (GTS) -Application à la réalisation des remblais et des couches de forme[END_REF]. The bentonite was sodium smectite clay. The lime used in this study contained 94% of quicklime (CaO). The category of the cement was CEM II (i.e., 65% of clinker, 35% of limestone and fly ash).

Specimen preparation

First, the compaction characteristics of the different mixtures were determined. For specimen preparation, the water content of the soil was adjusted to the desired water content for compaction. After a storage period of 24 h, during which the moisture content homogenized, the soil and treatment product were thoroughly mixed. When the lime treatment was added, the mixture was left for 1 h in an airtight container before compaction. When the cement treatment was used, compaction was carried out within a few minutes (maximum of 30 min) after treatment to account for the setting time of the cement. When clay treatments were used, no specific time was needed between the mixing and compaction; the mixture was statically compacted in the cell test to the target dry density. When a curing period prior to testing was required, the compacted samples were wrapped in plastic sheets and kept at 20.0 ± 1.5°C to prevent any water loss. The compaction characteristic curves were determined for each type and percentage of treatment according to the standard ASTM-D 698; the results of these experiments are summarized in Table 3.

Experimental strategy

Several authors (e.g., [START_REF] Mitchell | Permeability of compacted clays[END_REF][START_REF] Benson | Influence of clods on hydraulic conductivity of compacted clay[END_REF][START_REF] Watabe | Influence of compaction conditions on poresize distribution and saturated hydraulic conductivity of a glacial till[END_REF] showed that soil hydraulic conductivity reaches its minimum value on the wet side of optimum; thus, the investigations were focused on one specific compaction state for each treatment and each percentage. This state was defined by w = wOMC+3% and ρd = 0.96 ρdmax, depending on the nature and percentage of each treatment, where w is the compaction moisture content (%), wOMC is the optimum moisture content (%), ρd is the dry density (Mg/m 3 ) and ρdmax is the maximum dry density (Mg/m 3 ) (Table 3). The soil, untreated and treated, was statically compacted directly in the cell mold. After treatment and compaction, the soil specimens were sealed in airtight bags and cured at 20°C for different periods (e.g., 0, 7, and 30 days) before starting the erosion tests.

Experimental procedures

In this section, the enhanced HET device, test procedure and theoretical calculation methods are described.

HET device

An enhanced HET device was developed to measure the internal erosion of the compacted treated soil specimens. This device included several optimizations that allowed for the application of a high inlet pressure, producing a large pressure drop across the sample and hence a high hydraulic shear stress along the hole.

Major improvements were made to the pressure system: the water pressure at the inlet of sample was applied with a special air-water reservoir; an electromechanically operated valve controlled the air flow in the air-water reservoir during the experiment; the inlet pressure was controlled by a sensor at the entrance of the sample; and the water pressure was also measured at the outlet of the sample. The maximum inlet pressure exceeded 650 kPa, producing the necessary pressure drop of 650 kPa between the flows upstream and downstream of the soil samples.

Other optimizations were made to the configuration of the testing cell. A detailed view of the testing cell is provided in Figure 1. The soil samples were strongly held within the testing device from both sides. This prevented any loss of pressure and water leakage and also allowed for better holding of the soil specimen during the test under the large pressure drops.

The water flow was measured with an ultrasonic flowmeter and ranged from 0 to 1.4 L/s. A turbidimeter was connected to the outlet of the test cell to measure the effluent turbidity to determine the diameter of the hole during the test.

The initial diameter of the hole in the sample was between 3 and 6 mm. The tested samples were cylindrical, with a diameter of 70 mm and a length between 70 and 150 mm. When using a sample of 70 mm length and an initial hole of 6 mm, the device allowed for the application of a hydraulic shear stress of approximately 10,000 Pa (calculated with equation 6).

Test procedure

The initial hole was made by drilling into the specimen center; the specimen was then placed in the testing cell. Saturation of the whole system ensured that no air bubbles were trapped inside the cell or pipes; this saturation was established under a low flow while the vent valves were open. The erosion test was conducted by controlling and monitoring hydraulic parameters (i.e., the pressure applied to the sample, the pressure drop between the upstream and downstream of sample, and the flow through the hole).

In the literature, three different methods were used depending on which hydraulic parameter was applied and controlled during the test: i) applying a constant pressure at the upstream side of the sample (Wan and[START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF][START_REF] Wan | Investigation of rate of erosion of soils in embankment dams[END_REF][START_REF] Muttuvel | Erosion rate of chemically stabilized soils incorporating tensile stressdeformation behaviour[END_REF]; ii) applying a constant drop pressure across the sample [START_REF] Pham | Erosion et dispersion des sols argileux par un fluide[END_REF][START_REF] Haghighi | Caractérisation des phénomènes d'érosion et de dispersion des sols : développement d'essais et applications pratiques[END_REF]; and iii) maintaining a constant flow through the hole (Bonelli and [START_REF] Benahmed | Etude expérimentale de l'érosion interne d'une kaolinite. 25e Rencontres de l'AUGC[END_REF][START_REF] Benahmed | Investigation concentrated leak erosion behaviour of cohesive soils by performing hole erosion tests[END_REF]. The methods of the tests performed are related to technical constraints only and do not influence the erosion law of the soil, which is independent of the hydraulic conditions of the test.

In this study, tests were performed by applying a constant pressure at the inlet of the sample. Some preliminary tests were performed for each treatment to define the necessary range of applied pressure to initiate erosion by detaching soil particles. This pressure was related to the nature of the soil and the treatment product used. Table 4 provides some examples of the

Theoretical model

Integration of the final shape of the hole

For all the successful tests, the cross section of the eroded hole at the end of the test was in general circular or in some cases slightly ellipsoidal. These observations lead to suggest a general theoretical model that accounts for the real final shape of the hole and accurately compute the erosion rate and hydraulic shear stress. This model was based on the two radii of the hole (a and b) to better approximate the real shape of the hole. a and b are equal for a circular hole.

The erosion rate 𝜀𝜀̇ (kg/s/m 2 ) for a given hydraulic shear stress can be expressed by the eroded soil mass (kg) per unit surface area of the hole at time t (s):

𝜀𝜀̇= 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑜𝑜𝑒𝑒𝑒𝑒𝑒𝑒 𝑚𝑚𝑜𝑜𝑠𝑠𝑠𝑠 𝑚𝑚𝑠𝑠𝑒𝑒𝑜𝑜𝑚𝑚𝑐𝑐𝑒𝑒 𝑚𝑚𝑒𝑒𝑒𝑒𝑚𝑚 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 ℎ𝑜𝑜𝑠𝑠𝑒𝑒 * 𝑠𝑠𝑢𝑢𝑠𝑠𝑡𝑡 𝑜𝑜𝑜𝑜𝑡𝑡𝑠𝑠𝑚𝑚𝑒𝑒 (2)
Considering an ellipsoidal hole, the cross section is 𝑆𝑆 = 𝜋𝜋𝜋𝜋𝜋𝜋 and the perimeter is 𝑝𝑝 𝑒𝑒 = 𝜋𝜋�2(𝜋𝜋 2 + 𝜋𝜋 2 ), where 𝜋𝜋 and 𝜋𝜋 are the smaller and larger radii of the ellipsoid, respectively.

The erosion rate can then be expressed as follow:

𝜀𝜀̇= 𝜌𝜌 𝑑𝑑 2 � 2 𝜋𝜋(𝑡𝑡) 2 +𝜋𝜋(𝑡𝑡) 2 [𝜋𝜋(𝑡𝑡)𝑑𝑑𝜋𝜋 + 𝜋𝜋(𝑡𝑡)𝑑𝑑𝜋𝜋] 1 𝑑𝑑𝑡𝑡 (3)
where 𝜌𝜌 𝑒𝑒 is the dry density of the soil (Mg/m 3 ); 𝜋𝜋(𝑡𝑡) and 𝜋𝜋(𝑡𝑡) are the radii of the hole (m) at time 𝑡𝑡; and 𝑑𝑑𝜋𝜋 and 𝑑𝑑𝜋𝜋 are the variations of the radii at the elementary time 𝑑𝑑𝑡𝑡.

Using the principle of the equilibrium forces carried out on a volume of fluid between the inlet and outlet of the hole, the applied shear stress to the surface of the hole is related to the hydraulic gradient measured across the hole (Wan andFell, 2002, Muttuvel, 2008).

Considering the ellipsoidal shape, the hydraulic shear stress along the hole can be expressed as:

𝜏𝜏 = 1 � 2 . ∆𝑃𝑃(𝑡𝑡) 𝐿𝐿 . 𝜋𝜋(𝑡𝑡).𝜋𝜋(𝑡𝑡) � 𝜋𝜋(𝑡𝑡) 2 +𝜋𝜋(𝑡𝑡) 2 ( 4
)
where 𝐿𝐿 is the length of the sample (m) and ∆𝑃𝑃(𝑡𝑡) is the pressure drop between the inlet and outlet of the sample (Pa). To solve this equation, it is assumed that there is a unique relationship between the radius of the hole: 𝜋𝜋 = 𝛼𝛼𝜋𝜋. The factor 𝛼𝛼 is measured at the end of the test and assumed to remain constant during the test. The final radii a and b are taken as the mean values along the hole, and only specimens with a uniform shape along the hole were considered. Then, 𝜀𝜀 ̇ and 𝜏𝜏 are obtained as follows:

𝜀𝜀̇ = 𝜌𝜌 𝑒𝑒 . � 2𝛼𝛼 2 𝛼𝛼 2 +1 . 𝑑𝑑𝜋𝜋 𝑑𝑑𝑡𝑡 (5) 𝜏𝜏 = ∆𝑃𝑃(𝑡𝑡) 𝐿𝐿 . 𝛼𝛼 � 2(𝛼𝛼 2 +1)
. 𝜋𝜋(𝑡𝑡)

Equations 5 and 6 account for the general shape of the hole. For a circular hole, 𝜋𝜋 is equal to 𝜋𝜋; the coefficient 𝛼𝛼 is then equal to 1.

Determination of the evolution of the radius

To compute the erosion parameters (𝑘𝑘 𝑒𝑒𝑒𝑒 , 𝜏𝜏 𝑐𝑐 ), it is necessary to determine the evolution of the radius of the hole (db) during the test, which can be related to the soil eroded mass.

Two assumptions were made to determine the evolution of the cross-sectional shape of the hole: i) erosion is uniform along the hole and ii) the factor 𝛼𝛼 relating the radii of the hole is constant. The soil eroded mass can be expressed as follows:

𝑑𝑑𝑑𝑑(𝑡𝑡) = 2. 𝛼𝛼. 𝜋𝜋. 𝜌𝜌 𝑒𝑒 . 𝐿𝐿. 𝜋𝜋(𝑡𝑡 -1). 𝑑𝑑𝜋𝜋

where 𝑑𝑑𝑑𝑑(𝑡𝑡) is the soil eroded mass (kg).

The soil eroded mass during time step 𝑑𝑑𝑡𝑡 is also related to the measured concentration of soil particles as follows:

𝑑𝑑𝑑𝑑(𝑡𝑡) = 𝑐𝑐(𝑡𝑡). 𝑞𝑞(𝑡𝑡). 𝑑𝑑𝑡𝑡 (8)
where 𝑐𝑐(𝑡𝑡) is the concentration of soil particles (kg/m 3 ) and 𝑞𝑞(𝑡𝑡) is the water flow passing through the hole (m 3 /s).

From Equations 7 and 8, 𝑑𝑑𝜋𝜋 is given as follows:

𝑑𝑑𝜋𝜋 = 𝑐𝑐(𝑡𝑡).𝑞𝑞(𝑡𝑡) 2.𝛼𝛼.𝜋𝜋.𝜌𝜌 𝑑𝑑 .𝐿𝐿 . 𝑑𝑑𝑡𝑡 𝜋𝜋�𝑡𝑡-1� (9)
Given the initial radius of the hole 𝜋𝜋 0 , the radius at each time step can then be expressed as

[𝜋𝜋(𝑡𝑡) = 𝜋𝜋(𝑡𝑡 -1) + 𝑑𝑑𝜋𝜋].
The turbidity signal was used to determine the instantaneous concentration during the erosion process [START_REF] Reddi | Comparison of internal and surface erosion using flow pump tests on a sand-kaolinte mixture[END_REF][START_REF] Pham | Erosion et dispersion des sols argileux par un fluide[END_REF][START_REF] Muttuvel | Erosion rate of chemically stabilized soils incorporating tensile stressdeformation behaviour[END_REF]. The relationship between the concentration of soil and the turbidity of the effluent concentration was also calibrated. These calibrations were performed for each treatment and each percentage of treatment product.

Some examples of the relationship between the soil concentration and turbidity signal are

given in Figure 3. The results indicated a bilinear calibration curve for all samples. Hence, two relationships were considered for each treatment, one for each concentration range. The threshold concentration value was approximately 1 kg/m 3 and depended on the nature and percentage of treatment.

Validation of the calculation method

Figure 4 presents the final eroded diameter measured at the end of the test versus the final diameter calculated from the turbidity signal (Equation 9). The slight differences could be due to inaccuracy of initial and final drilled hole measurements and the turbidity sensor response.

The results showed the relevance of the technique used to determine the hole diameter during the test using the soil detachment particles.

Impact of treatment on soil erodibility

Untreated soil

The repeatability of the erosion parameters obtained with the new HET device was verified with the untreated soil by performing several tests. Two sets of these results are shown in Figure 5. Table 5 shows the erosion parameters determined from six tests. 𝑘𝑘 𝑒𝑒𝑒𝑒 varies between 1.13 x 10 -04 and 3.04 x 10 -04 s/m, and 𝜏𝜏 𝑐𝑐 varies between 291 and 615 Pa. This dispersion is of the same order of magnitude as has been found in previous studies. As an example, [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF] found that 𝜏𝜏 𝑐𝑐 for a clay-sandy soil varied between 85 and 327 Pa. [START_REF] Haghighi | Caractérisation des phénomènes d'érosion et de dispersion des sols : développement d'essais et applications pratiques[END_REF] also found a similar range of 𝜏𝜏 𝑐𝑐 (i.e., from 152 to 541 Pa) for a similar silty soil.

The mean values for the untreated silt are 𝑘𝑘 𝑒𝑒𝑒𝑒 = 2.07 x 10 -04 s/m and 𝜏𝜏 𝑐𝑐 = 429 Pa. The ERI of the tested materials is between 3.59 and 3.95; therefore, all untreated specimens are classified as having "moderately rapid" erosion according to the classification of [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF].

The studied silt has a high critical shear stress compared to the values typically observed in untreated soil [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF]. This can be related to the high percentage of the fine particle fraction (< 80 µm), 99.2%, and to the compaction conditions. In fact, compaction on the wet side of optimum allows for a better rearrangement of the soil particles, than the dry side of the optimum, and reduces porosity (e.g., [START_REF] Benson | Influence of clods on hydraulic conductivity of compacted clay[END_REF]. Compaction on the wet side (w = 17.5%) combined with the high dry density (ρd = 1.73 Mg/m 3 ) induces a dense arrangement of soil particles. This arrangement hinders the detachment of soil particles and requires a greater level of hydraulic shear stress to start the erosion. However, once this value of hydraulic shear stress is reached, the erosion rate progresses rapidly, as shown by the value of the coefficient of soil erosion (𝑘𝑘 𝑒𝑒𝑒𝑒 = 2.07 x 10 -04 s/m).

Clay treatment

Examples of the erosion law for the clay treatments are shown in Figure 6, and Table 6 summarizes the erosion characteristics for the clay-treated soil obtained from the other tests.

The use of 2% kaolinite did not significantly change 𝑘𝑘 𝑒𝑒𝑒𝑒 ; however, 𝜏𝜏 𝑐𝑐 tended to increase and had a mean value of 607 Pa. The addition of 9% kaolinite decreased 𝑘𝑘 𝑒𝑒𝑒𝑒 by one order of magnitude and induced a small increase in 𝜏𝜏 𝑐𝑐 with a mean value of 513 Pa. According to the classification of [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF], the silt treated with 2% kaolinite was in the "moderately rapid" group, whereas the silt treated with 9% kaolinite became "moderately slow". Therefore, 9% kaolinite was required to significantly modify soil erodibility.

For the treatment with 2% bentonite, 𝑘𝑘 𝑒𝑒𝑒𝑒 is of the same order of magnitude for all samples and is approximately 5.00 x 10 -05 s/m (Table 6) which is less than one order of magnitude less than that of the untreated soil. 𝜏𝜏 𝑐𝑐 varies between 123 and 483 Pa, which are included within the interval of variation of 𝜏𝜏 𝑐𝑐 for untreated silt. Thus, the silt treated with 2% bentonite is classified as "moderately slow".

The enhancement of the silt internal erosion characteristic induced by clay treatment is then characterized mainly by the reduction of the coefficient of soil erosion depending on the nature and percentage of the added product.

Lime treatment

To study the internal erosion of lime-treated silt, it was necessary to apply an inlet pressure of up to 250 kPa for 1% and 300 kPa for 3% lime to initiate soil erosion.

A typical set of results for the lime-treated silt is presented in Figure 7. Two phases can be distinguished during the test. The erosion process was initiated for a hydraulic shear stress of 926 Pa with a slope 𝑘𝑘 𝑒𝑒𝑒𝑒 of 3.76 x 10 -04 s/m until a maximum erosion rate of 0.30 kg/s/m² was reached. Then, the erosion rate decreased and stopped for a hydraulic shear stress of 1,500 Pa with a mean slope of approximately 2.65 x 10 -04 s/m. This behavior was observed with 1% and 3% lime after 7 days of curing time. 𝜏𝜏 𝑐𝑐 at the ending phase was higher than during the starting phase (Figure 7), indicating that the soil resisted erosion more at the end of the test than at the beginning. Such behavior could be explained by the type of material used. Indeed, lime-treated soil can be considered as brittle material. The drilling is a mechanical process which may damage the brittle material by inducing small cracks. A small area of the soil around the hole could have been damaged during the drilling of the hole after the curing period. This damaged part is then the first to be easily eroded during the test, leading to a higher erodibility than the undisturbed part of the sample.

To verify this hypothesis, some tests were conducted on 3% lime-treated silt with the hole being drilled just after sample compaction. The drilled sample was then left for curing until the erosion test was performed. The results in Figure 8 showed that the effects of the drilling process were less pronounced when the drilling was made before the curing time; this observation was made for both specimens after 7 days of curing (Figure 8.a and 8.b) and after 30 days of curing (Figure 8.c and 8.d). The results also indicated that the erosion law was identical at the starting and ending phases of the test. In fact, soil drilling occurred before the beginning of the formation of the cementitious compounds that bind the soil particles. The soil is then less disturbed by the drilling just after compaction than when drilling occurs after curing. The drilling step after curing significantly modified the soil and produced different behaviors between the starting and ending phases of erosion. To examine the impact of treatment on the erosion characteristics, only the characteristics stemming from the ending phase will be considered; these characteristics correspond to the erosion of the soil that is less disturbed by the drilling process.

Some results for lime-treated soil are shown in Table 7, and examples of the erosion law are shown in Figure 9. For treatment with 1% lime, 𝑘𝑘 𝑒𝑒𝑒𝑒 varied between 1.09 x 10 -04 and 2.87 x 10 - 04 s/m regardless of the curing time. For treatment with 3% lime, 𝑘𝑘 𝑒𝑒𝑒𝑒 varied between 1.54 x 10 -04 and 4.52 x 10 -04 s/m regardless of the curing time. Lime treatment increases the 𝜏𝜏 𝑐𝑐 of the untreated soil, yielding values of up to 1,770 Pa for 1% lime and 1,889 Pa for 3% lime.

Moreover, 𝜏𝜏 𝑐𝑐 tended to increase with curing time but not in any common progression.

The erosion rate during the test was higher for samples treated with 3% lime than for the untreated silt (Figure 9). This observation could be explained by the maximum coarse grain size in the treated soil, which is higher than the untreated soil; furthermore, during the erosion of the treated soil, the detachment of particles accompanied aggregates of larger size than for the untreated silt. It was also observed during the tests, that the erosion of lime treated soil particles often occurs by the detachment of aggregates of soil and not only individual soil particles. Hence, the erosion rate will be higher for the lime-treated silt.

The erosion of the lime-treated silt is classified as "moderately rapid," as is that of the untreated soil, regardless of the lime percentage or curing time. The impact of the lime treatment cannot be observed in these tests because the classification of [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF] does not consider the critical shear stress.

The improvement of the silt internal erosion characteristics induced by lime treatment is then characterized mainly by the increase in the critical shear stress regardless of the curing time and the percentage of the added product.

Cement treatment

Some results from the HET for the cement treatment are shown in Table 8. For 3% cement, it was necessary to apply an inlet pressure of up to 450 kPa to initiate the soil erosion; this is reflected by the high hydraulic shear stress. For most tested samples, erosion started at a hydraulic shear stress of 2,000 Pa, and the erosion was mainly independent from the curing time. For the lime-treated silt, the erosion law exhibited a different behavior between the beginning and end of the tests. Therefore, the same selection process was used to compute the erosion parameters for each step as described for the lime treatment cases. In case of treatment with 6% cement, the applied inlet pressure was more than 500 kPa, and the length of samples was reduced to 70 mm to apply a higher hydraulic shear stress. Detachment of soil particles started after 4,000 Pa for most specimens; however, the erosion law exhibited nonlinear characteristics for most specimens, especially after 7 days of curing. Thus, it was difficult to compute 𝑘𝑘 𝑒𝑒𝑒𝑒 for some samples, and in such cases, 𝜏𝜏 𝑐𝑐 was reported as the threshold of the hydraulic shear stress at the end of erosion.

Examples of the erosion law for cement-treated specimens are shown in Figure 10. For treatment with 3% cement, 𝑘𝑘 𝑒𝑒𝑒𝑒 varied between 1.13 x 10 -04 and 1.67 x 10 -04 s/m, regardless of the curing time. For treatment with 6% lime, 𝑘𝑘 𝑒𝑒𝑒𝑒 varied between 1.00 x 10 -05 and 8.00 x 10 - 05 s/m. Furthermore, for treatment with 3% cement, 𝜏𝜏 𝑐𝑐 varied between 2,850 and 3,619 Pa, and the curing time appeared to have no effect on the evolution of 𝜏𝜏 𝑐𝑐 . In the case of 6% cement, 𝜏𝜏 𝑐𝑐 varied between 4,400 and 6,800 Pa for a curing time of 1 day and reached 9,893 Pa after 7 days of curing. Thus, the erosion of cement-treated silt is classified as "moderately rapid" for 3% cement and "moderately slow" for 6% cement.

The main enhancement of the silt internal erosion characteristic induced by cement treatment is then characterized by the increase in the critical shear stress and may also result in a decrease in the coefficient of soil erosion depending on the percentage of the added product.

Discussion

Figure 11 shows the erosion characteristics (𝑘𝑘 𝑒𝑒𝑒𝑒 and 𝜏𝜏 𝑐𝑐 ) obtained for both treated and untreated silt.

The addition of kaolinite induced a relatively small increase in 𝜏𝜏 𝑐𝑐 , and the addition of bentonite did not significantly change the range of 𝜏𝜏 𝑐𝑐 . This observation can be explained by the fact that the initial compaction states (ρd, w) were similar for untreated and clay-treated silt.

The addition of 9% kaolinite and 2% bentonite decreased 𝑘𝑘 𝑒𝑒𝑒𝑒 by one order of magnitude. The nature of the clay particles appears to have an important effect on the erosion behavior of soil.

Indeed, the addition of only 2% bentonite has similar effects to the addition of 9% kaolinite.

A higher percentage of clay fraction increases the resistance of sandy soils to erosion (e.g., [START_REF] Bennabi | Some elements of comparison between two laboratory devices for soil erosion testing[END_REF][START_REF] Benahmed | Investigation concentrated leak erosion behaviour of cohesive soils by performing hole erosion tests[END_REF]. This study showed that this trend is also valid for silty soil.

The lime treatment resulted in an increasing 𝜏𝜏 𝑐𝑐 , even with a short curing time (1 day) (Figure 11), indicating that the increase in erosion resistance by lime treatment is a short-term effect of lime. The development of longer-term (pozzolanic) reactions for increasing the hydraulic shear stress appears to be less important. One of the short-term mechanisms responsible of these changes could be the flocculation-aggregation of soil particles caused by lime addition.

The coefficient of soil erosion 𝑘𝑘 𝑒𝑒𝑒𝑒 maintained the same level with or without lime treatment regardless of the lime percentage or curing time, indicating that once erosion has started, the rate of soil detachment will be the same as untreated silt. This result can be explained by the detachment process of soil particles. As it was observed during the tests, the erosion of soil particles often occurs by the detachment of aggregates of soil and not only individual soil particles; this indicates that erosive forces can break more easily the weaker bonds between aggregates.

The main impact of cement treatment was the increase in 𝜏𝜏 𝑐𝑐 in proportion to the percentage of cement (Figure 11). Furthermore, the increasing cement percentage was accompanied by a decrease in 𝑘𝑘 𝑒𝑒𝑒𝑒 . The enhancement of the erosion characteristics induced by cement treatment is related to the cement setting and the development of cementitious compounds, which induce strong bonds between soils particles. The detachment of soil particles becomes more difficult, making the soil more resistant to erosion. When treated with 3% cement, this improvement came in early ages of curing time, and no further significant modification was observed until 30 days. At higher percentages of cement (6%), an increase in the critical shear stress was obtained with increasing curing time. This behavior could be related to the soilcement setting time, which may be longer for a higher amount of cement treatment.

Conclusion

The main objective of this paper was to study the impact of soil treatment on the internal erosion resistance of treated compacted silt. A new enhanced HET device was developed to determine the erosion characteristics of treated materials. A modified interpretation method that considered the final shape of the eroded hole has been proposed.

The results showed that the studied silt had a higher critical shear stress compared to the values typically observed in the untreated soil. This higher critical shear stress was primarily caused by the compaction conditions on the wet side of the optimum water content. However, the mean coefficient of soil erosion was 2.07 x 10 -04 s/m, which may accelerate detachment due to erosion once it has begun. The solution to this issue could be the treatment of soil by adding clay, lime or cement. The following conclusions can be made based on the results obtained:

• The use of clay soil as a treatment product reduces the erosion rate of soil particles.

This effect is related to the nature of the clay, and the use of bentonite further reduces the coefficient of soil erosion than the use of kaolinite at the same percentage.

• Lime treatment increases the internal erosion resistance of soil by increasing the critical shear stress. This effect is similar to either 1 or 3% lime treatment and occurs rapidly, with no significant change with increased curing time up to 30 days.

• Cement treatment increases the critical shear stress, and the increase in erosion resistance is proportional to the percentage of cement used. A high percentage of cement will also provide a smaller coefficient of soil erosion. The enhancement occurs rapidly for 3% cement, whereas higher percentages require additional curing time to yield greater erosion resistance.

Additional works are required to relate the evolution of erosion characteristics to the microstructure of the treated soil which may give more information about the factors responsible of the changes of the erosion characteristics.

Although the short-term effects seemed to be positive, the use of these treatments for hydraulic earth structures should consider their long-term behavior. In fact, such products require specific studies to assess the evolution of these parameters over time when the structures are submitted to climatic conditions, which can affect the sustainability of treatments and decrease their initial performances. The impact of these different treatment products on other hydro-mechanical parameters (e.g., hydraulic conductivity, unconfined compression strength) of the tested soil should also be considered in the framework of the selection of the most cost effective treatment product.

Tables Table 1: Method of the classification of soils based on their erodibility [START_REF] Wan | Investigation of internal erosion and piping of soils in embankment dams by the slot erosion test and the hole erosion test[END_REF]. The lines represent the erosion law drawn from the best fit line for the selected data of each treatment. The lines represent the erosion law drawn from the best fit line for the selected data of each treatment. The lines represent the erosion law drawn from the best fit line for the selected data of each treatment. The vertical small bar refers to samples where the coefficient of soil erosion was calculated at the starting phase only.

  initial inlet pressure necessary to begin erosion for different treatment products and percentages. The erosion starting process was determined by the turbidity signal which increased when the erosion of soil particles increased. A typical set of the test parameters monitored during a test performed on untreated soil is shown in Figure 2. The system can maintain a constant inlet pressure (Figure 2.a). At the beginning of the test, and after the clear-out of the hole from the soil particles detached during the drilling process, soil erosion increased rapidly due to the large pressure drop between the inlet and outlet of the sample (Figures 2.b and 2.c). Detachment of soil particles produced an increase in the hole diameter, creating an increase in the flow through the sample (Figure 2.d) and a progressive decrease in the pressure drop between the inlet and outlet of the sample.The evolution of the erosion rate is shown in Figure2.e. After the increasing soil erosion phase, as also shown in the turbidity signal, the detachment of soil particles decreased progressively, likely because of the decrease in the hydraulic shear stress along the hole (Figure2.f). The changes of the hydraulic shear stress are related to both the increase of the hole diameter and the decrease of the pressure drop between the inlet and outlet of the sample (equation 6). The test ended when the turbidity reached a lower asymptote. At the end of each test, the cell was dismantled, and the hole in the specimen was filled with liquid wax to determine the final characteristics of the eroded hole.

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Example results for a HET performed on a sample of untreated silt. a) pressure applied at the inlet of the sample. b) pressure drop between the inlet and outlet of the sample. c) turbidity signal of the effluent. d) flow through the sample. e) erosion rate. f) hydraulic shear stress.

Figure

  Figure 3: Relationship between the soil concentration and turbidity signal. a) concentration interval [0 -1.5 kg/m 3 ]. b) concentration interval [0 -15 kg/m 3 ]. The best fit line is drawn for each treatment.

Figure 4 :

 4 Figure 4: Representation of the calculated and measured diameter at the end of the different tests.

Figure 5 :

 5 Figure 5: Two examples of erosion law for the untreated silt.

Figure 6 :

 6 Figure 6: Impact of clay treatment on the erosion characteristics of silt.

Figure 7 :

 7 Figure 7: Example of the erosion law for lime-treated silt (1%) with 7 days of curing.

Figure 8 :Figure 9 :

 89 Figure 8: Impact of the drilling on the erosion law of 3% lime-treated soil. a) drilling after 7 days of curing b) drilling before 7 days of curing c) drilling after 30 days of curing d) drilling before 30 days of curing

Figure 10 :

 10 Example of the erosion law for cement-treated silt.

Figure 11 :

 11 Effect of treatment on the erosion characteristics of the studied silt.

Table 2 : Physical and geotechnical properties of the studied soil.

 2

Table 3 : Optimum Proctor compaction characteristics for silt treated with different products and the compaction characteristics for the HET samples. Table 4: Example of the range of the initial applied inlet pressures for different treatments.Table 5 : Erosion characteristics for the untreated silt calculated from 6 tests. Table 6: Impact of clay treatments on the erosion characteristics of the studied silt.

 35

Table 7 : Erosion characteristics of lime-treated silt calculated from several tests after 1, 7 and 30 days.

 7

Table 8 : Erosion characteristics for cement-treated silt calculated from several tests after 1, 7 and 30 days.

 8

Table 1 : Method of the classification of soils based on their erodibility (Wan and Fell, 2002).

 1

	Group number	Erosion Rate Index (ERI)	Description
	1	<2	Extremely rapid
	2	2 -3	Very rapid
	3	3 -4	Moderately rapid
	4	4 -5	Moderately slow
	5	5 -6	Very slow
	6	>6	Extremely slow

Table 2 : Physical and geotechnical properties of the studied soil. 694

 2

	Soil properties	Value
	Liquid limit (%)	28.5
	Plastic limit (%)	20.5
	Plasticity index (%)	8.0
	Passing sieve 80 μm (%)	99.2
	Clay size content (<2 μm) (%)	6.0
	Specific gravity Gs (-)	2.64

Table 3 : Optimum Proctor compaction characteristics for silt treated with different products and the compaction characteristics for the HET samples.

 3

			Optimum Proctor	Compaction characteristics
	Types of		compaction characteristics	for the HET samples
		Notation
	treatment		ρdmax
				wOMC (%)	ρd (Mg/m 3 )	w (%)
			(Mg/m 3 )
	Silt	S	1.82	15.0	1.73	17.5
	+ 2% kaolinite	SK 2%	1.80	15.5	1.73	18.0
	+ 9% kaolinite	SK 9%	1.79	15.8	1.74	18.3
	+ 2% bentonite	SB 2%	1.78	15.3	1.74	17.8
	+ 1% lime	SL 1%	1.75	17.5	1.70	20.0
	+ 3% lime	SL 3%	1.73	17.5	1.68	20.0
	+ 3% cement	SC 3%	1.81	15.0	1.75	17.5
	+ 6% cement	SC 6%	1.82	15.0	1.75	17.5

Table 4 : Example of the range of the initial applied inlet pressures for different treatments. Type of treatments Range of initial applied pressure (kPa)

 4

	Silt

Table 5 : Erosion characteristics for the untreated silt calculated from 6 tests.

 5

			Mean values
	Test	𝒌𝒌 𝒆𝒆𝒆𝒆 (s/m)	𝝉𝝉 𝒄𝒄 (Pa)
			𝒌𝒌 𝒆𝒆𝒆𝒆 (s/m)	𝝉𝝉 𝒄𝒄 (Pa)
	1	2.58 x 10 -04	391
	2	2.16 x 10 -04	389
	3	2.22 x 10 -04	398
			2.07 x 10 -04	429
	4	1.13 x 10 -04	491
	5	3.04 x 10 -04	615
	6	1.30 x 10 -04	291

Table 6 : Impact of clay treatments on the erosion characteristics of the studied silt.

 6

	707

Table 7 : Erosion characteristics of lime-treated silt calculated from several tests after 1, 711 7 and 30 days. 712

 7

	0	500	1000	1500	2000	2500	3000

Acknowledgment

The authors would like to thank S. Bonne, E. Lavallée and T. Le Borgne from DTP -Bouygues Construction -France for their valuable support of this study.

.