Climate data for the last ten years were provided to us by the National Civil Aviation and Meteorology Agency (ANACIM, 2020). This data includes dominant wind direction, average wind speed (m/s), insolation in hours, cumulative rainfall in mm, average maximum and minimum temperature in degrees Celsius, maximum humidity in %, minimum humidity in %, and evaporation in mm. These values constitute input data in the VADOSE/W program at the level of the window representing the climatic parameters (see Table 1).

Table 1. The climatic parameters used in the modeling.

After modeling the soil-atmosphere interaction over time, several parameters can be collected, such as the degree of saturation, suction, water content, infiltration, etc. In this study, we will model soil movements for a period of fourteen months, i.e. from November 2019 to December 2020. Regarding the monthly precipitation measured in the study area during the period between November 2019 and December 2020, we see that the majority of rainfall occurred between June, July, August, September and October 2020 with a maximum of 206.3 mm/m² in September (see Figure 1). Regarding infiltration, we note that the negative values are between December 2019 and June 2020 with a maximum of −12.99 mm/m² in February 2020. In this interval, values are recorded very high suction. The evolution of the curve in Figure 2 shows that the month of June 2020 marks the start of the infiltration which reaches its maximum in the month of July (50.45 mm/m²). Figure 3 allows us to make the correlation between rainfall, infiltration and radiation over the fourteen months. In this curve, we see that precipitation and infiltration evolve in the same direction while radiation partly governs evaporation.

We also note that the months of February, March, April, May and June record the greatest drying times. During this period, the soils will be subjected to intense evaporation phenomena leading to desiccation cracks in the surface areas and significant settlements which can cause cracks in the supported structures.

Figure 1. Precipitation variation curve for the last fourteen months.

Figure 2. Infiltration curve.

Figure 3. Correlation between rainfall, infiltration and solar radiation.

The different stages of modeling soil volume variations using MEBM are grouped into three main stages:

Laboratory tests which consist of determining the geotechnical and physical characteristics of the soil, namely the natural water content, the elastic modulus in the saturated state E s a t from the oedometric test as well as the water content. water in the saturated state w s a t , the void index in the natural state e 0 and the porosity n. The permeability k is determined from the variable volume permeability test and finally for what concerns the thermal conductivity, we took the theoretical value of these soils.Modeling with the VADOSE/W software of soil-atmosphere interaction which makes it possible to determine the variation in suction from climatic data and soil characteristics already determined in laboratory tests. In fact, the suction data comes from the soil-water characterization curve after projection of the volumetric water content values.The determination of ground movements by MEBM is obtained using the general volume variation equation d ε v = Δ V v V 0 = m 1 s d ( σ − u a ) + m 2 s d ( u a − u w ) , but before that preliminary calculations will be carried out to determine the modulus of elasticity in the unsaturated state E u n s a t , the coefficient of total change with respect to changes in suction m 2 s , the modulus of elasticity with respect to change in suction H.

Figure 4 shows the different stages of the modeling.

For the soil parameters, we will need the saturated modulus of elasticity, dry density, porosity, solid grain density, permeability, thermal conductivity and degree of saturation. The values of these parameters are grouped in Table 2.

Figure 4. The different stages of modeling.

Table 2. Characteristics of marl.

Figure 5 shows the profile of the crumbly marl, the surface layer, and the climatic conditions.

The strain variable associated with volume change was rationally interpreted in terms of two independent variables of the stress state of unsaturated soils, namely, matric suction ( u a − u w ) and net normal stress ( σ − u a ) (Fredlund and Morgenstern, 1976).

with m 1 s coefficient of volume variation linked to the net stress, m 2 s coefficient of volume variation linked to the matric suction ( σ − u a ) , the net normal stress ( u a − u w ) is the matric suction and d the thickness of the layer.

Figure 5. Profile of the crumbly marl.

The constitutive volume change relationship (Equation (1)) can be presented graphically as a three-dimensional constitutive surface (Figure 6).

Figure 6. The constituent surface of the soil (modified after Fredlund et Rahardjo, 1993) [8].

However, for lightly loaded pavements and structures, the influence of mechanical stress (i.e. net normal stress) is not significant and can be neglected (Fredlund et al., 1980) [9]. In this case, only the change in matric suction will mainly contribute to the change in soil volume; Equation 1 reduces to Equation (2) as shown below.

In the first steps, we first determine the Volumetric Water Content which describes the volume of water a material can store based on the pore water pressure. These values are automatically generated by the software after entering climatic data and soil characteristics. These values vary with time and depth. The corresponding suctions are then determined from the soil-water characteristic curve (SWCC) in Figure 7.

Figure 7. Soil-water characterization curve.

In the general equation (Equation (2)), the coefficient of total volume change with respect to suction changes m 2 s is determined from the elastic modulus with respect to suction change H by the following formula.

The modulus of elasticity H is obtained using the modulus of elasticity in the unsaturated state according to Equation (4).

with E u n the modulus of elasticity in the unsaturated state taken from the modulus of elasticity in the saturated state according to Equation (5).

In modeling for each given water content, we will have to determine from the equations above the modulus of elasticity in the unsaturated state E u n s a t , the coefficient of change in total volume with respect to changes in suction m 2 s and finally the vertical movement of the ground d h i .

For the three meters thick of the crumb marl, we will arbitrarily take a few depths of which we will determine according to climatic variations over time their volumetric water contents, their suctions and their recorded vertical movements. Table 3 gives the different values of volumetric water content as a function of time and depth.

Table 3. Variation in the volume of water contained (m³/m³) over the last twelve months.

Depending on these variations in water volume, the corresponding suctions are determined in Table 4 from the soil-water characterization curve (see Figure 7).

Table 4. The variations in suction power as a function of depth and time.

From these data, the ground displacements can be calculated at each depth from Equation (2). The values of the two parameters of the model, namely the coefficient α, are taken equal to 1/14 and β to 2.

The value β = 2 is used for expansive soils regardless of the value of the plasticity index Ip. The parameter was assumed to be (1/14) in order to provide reasonable shrink-swell prediction values.

The results reveal that the suction variations are higher in the superficial part. In the lower parts, the values gradually decrease. Subsequently, the vertical movements of the crumb marl with a thickness of three meters are calculated and grouped in Table 5.

Table 5. The swelling and shrinking of crumbly marl depending on climate variations and depth.

Figure 8 shows the evolution of the shrinkage-swelling of crumb marl as a function of the variation in suction.

Figure 8. Shrinkage-swell curve of the crumbly marl.

The study of the movement curve shows that the crumb marl is in continuous movement over time. The shape of this curve shows the existence of a shrinkage phase and a swelling phase. The shrinkage phase begins in November and continues until March while swelling takes place during the other remaining months.

In this specific case, if we consider the month of November 2019 as the initial state, the curve is characterized by a sequence of shrinkage and a sequence of swelling and during this period, the vertical movement of the ground is marked by a total contraction of the order of 0.047 m and a swelling of 0.0233 m. The comparison we made shows that more than half of the ground movements are recorded at the superficial part. For example, if we take the example of the month of February, the total displacement of the marl is estimated at 5.32 × 10⁻² m and that of the surface is 3.58 × 10⁻² m or 67% of the total movement of the ground. This percentage is explained by the fact that the suction variations are greater at the surface of the ground and that the deeper we go, the more the suction decreases until it reaches more or less constant values.

Figure 9 illustrates the vertical movements of the ground according to climate variation and depth for December 2019, February 2020, November 2020, August 2020, and May 2020.

Figure 9. Variation of the volumetric changes of crumbly marl according to the evolution of suction with depth.

We note that ground movements are more accentuated at the upper parts of the layer as we wrote previously. The shrinkage-swellings are intense in the superficial parts then gradually decrease going deeper. For swelling phenomena, the contents are greater in the higher levels which justifies the higher swellings at this level. In the lower parts, under the effect of infiltration, the water can reach certain levels but with low contents. In fact, these water contents fluctuate slightly at depth, leading to more or less stable volume changes in the soil layer. In the same way, the contraction of the soil is synonymous with an increase in suction which is greater in the upper parts of the marl layer and weaker in the lower levels. In this specific case, the swelling is therefore a recovery of the thickness of the layer already lost during contraction, which is why in countries like Senegal where the dry season is longer around nine months out of twelve, the contraction phenomenon must be studied more rigorously before the implementation of construction projects. These two parameters, namely water content and suction, are controlled on the one hand by climatic data which are more reactive in the superficial parts and on the other hand by the characteristics of the soil in question without also forgetting the time parameter.

In addition to determining the amplitudes of vertical movements of the ground over a period of fourteen months, the model also gives us an idea of the depth of the active zone which extends up to 1.25 m from the surface ( see Figure 9). In the following lines, we will carry out this same study on foliated marl with attapulgite and sandy clay with calcareous concretion.

Like crumb marl, we will proceed in the same way to model a layer of foliated attapulgite marl with a thickness of three meters. The geotechnical characteristics of the laminated marl are shown in Table 6.

Table 6. Characteristics of the layered marl.

The vertical displacement values after calculation are grouped in Table 7:

Table 7. Vertical displacement as a function of depth.

The evolution of shrinkage-swelling phenomena during these fourteen months is shown in Figure 10.

Figure 10. Curve of vertical displacements.

This curve shows an intense contraction in the first three months followed by a more or less stable movement from February until May and once in June, the swelling begins to gain momentum before falling again in the month of December.

Figure 11 allows you to visualize the shrinkage-swelling phenomena as a function of the depth of a few months (December 2019, February 2020, July 2020, August 2020 and December 2020).

Figure 11. Variation of volumetric changes of the laminated marl according to the evolution of suction with depth.

Unlike the crumb marl, the active zone of foliated marl is located at a depth of 1.5 m and the movements are more accentuated in the upper parts.

As in the two previous cases, we will model the shrinkage-swelling of sandy clay with limestone concretion with a thickness of 3 m. The characteristics of the clay are shown in Table 8.

Table 8. Some characteristics of blackish sandy clay.

After carrying out the different calculation steps, the vertical movements found are grouped together in Table 9.

Table 9. Vertical displacements as a function of depth.

Figure 12 shows us the evolution of the vertical movements of the sandy clay over time.

Figure 12. Shrinkage-swelling curve of the blackish sandy clay.

The study of the movement of sandy clay as a function of depth shows a significant contraction in the superficial parts which gradually decreases going deeper before reaching a stable value. The values in Table 9 reveal that the active zone of the sandy clay is located at a depth of 1.5 m. Under the effect of solar radiation and evaporation, water contents gradually decrease at depth. This drop in water content implies an increase in suction depending on the depth and consequently, the movements will be more accentuated in the superficial parts. Subsequently, the month of April marks the end of the contraction and once in May the evolution of swelling becomes rapid then remains constant between May, June and July before beginning another sequence of swelling between August, September, October, November and December. During this period, under the effect of rainfall and infiltration, suction decreases considerably. Once the volume variations of these three soils have been studied, we will proceed in the following lines to make a comparison of these vertical movements over time.