Soil improvement techniques include a set of interventions to positively modify the properties of a soil, either through physical actions such as vibration [9] or by introducing more robust materials [10] in soil or their mixing with existing soil. As civil engineer John X. Wang pointed out, “Vertical drains are essential components in soil treatment for road projects, as they help to regulate groundwater levels and reduce the risk of soil saturation. This is essential to maintaining stable pavements and sustainable roads” [11] .

The specific choice and application of these techniques are closely linked to the composition and grain size of the soils to be treated.

Since its introduction in California around 1930, the use of vertical drains to accelerate the consolidation of compressible soil deposits has become a common practice in civil engineering. This method was initially based on an empirical design of sand drainage systems [12] by pushing the soil back. However, thanks to the pioneering work of Barron (1947) [13] , who developed abacus for the dimensioning of drainage systems, and the synthesis of Moran et al. (1958) [14] and Johnson (1970) [15] , which has been the result of decades of sand drain use, this technique has been stabilized and internationally recognized.

At the same time, new techniques for sand drainage have been developed such as open pipe drilling, full or hollow boring, and the use of prefabricated drains [16] [17] made from various materials such as cardboard, plastic, non-woven, sand or rope. Despite the diversity of drainage methods and devices available on the market, as well as the scale of work carried out each year in the world, uncertainties remain about the merits of different approaches and the prediction of soil consolidation rates.

Since the first work of Johnson (1970), problems of vertical drainage consolidation have been addressed in several specialized international conferences, such as the conference of the ASCE (American Society of Engineers) Purdue, the Bangkok International Soft Clay Symposium, and the Sixth Asian Regional Congress on Soil Mechanics and Foundation Work in Singapore. These conferences helped to develop a “body of doctrine” guiding drainage practices, highlighting the importance of accurate assessment of soil consolidation parameters, preference for prefabricated drains, and the need to verify on-site the actual efficiency of drains under specific conditions. RPD can effectively reduce excess pore pressure in loose soil and shorten the consolidation period. The pore pressure in the RDF-reinforced area is less than 20.0 kPa at the end of construction, compared to 52.0 kPa at the beginning of construction [18] .

The first type of drain used consists of sand drains that can contribute to reducing settlement and are a better option for depth by working properly during primary consolidation of the soil to be improved but limited because of their duration life weak favored by their obstruction [19] . Then there are the prefabricated vertical drains which are suitable in case of a depth limited by their equipment [20] .

A PVD is defined as any material (or product) consisting of a synthetic filter envelope surrounding a plastic core that has the following characteristics:

The water collected in the drain is carried along the drain, via the core, to a draining level.

Pre-fabricated Vertical Drains (PVD) are preferred for their quick execution, best drainage properties [21] .

There are a variety of improvement techniques and their use depends largely on the size of the soil in place. While dynamic consolidation causes compaction of granular soils, vibroflottation is mainly applied to non-coherent granular soils, resulting in a temporary phenomenon of liquefaction of the soil surrounding the vibrator.

Rigid inclusions are suitable for all types of work on compressible soils of any kind. The ballasted column technique is an extension of vibroflottation to soils with silt or clay layers whose elements cannot be rearranged by vibration.

Drains facilitate the evacuation of interstitial overpressures in compressible soils to reduce the consolidation time from several decades to a few months or weeks depending on the spacing of the drains and the thickness of the drained layer a new load on cohesive floors (clays, silts...) saturated generates first of all an interstitial overpressure. It is during the dissipation of this interstitial overpressure that the solid skeleton of the soil can in turn be charged. The natural process of dissipation of these interstitial overpressures in clay soils is very slow because of their weakly permeable character and over several years. Vertical drains are set up in order to accelerate this phenomenon by facilitating the flow of water under overpressure to allow the loading of the soil grains themselves [22] . The use of vertical drains that are set up in a regular mesh, with a spacing not often exceeding 3 m, causes horizontal drainage over a smaller distance, compared to the thickness of the compressible layer, but with a horizontal permeability greater than the vertical permeability ( Figure 1 ). This results in a high degree of acceleration of horizontal consolidation and, as a result, a considerable reduction in the time needed for consolidation [23] .

Figure 2 shows a cut of a precast vertical drain: geodrain.

Two theories have been developed.

Barron puts forward the hypothesis of horizontal drainage (unidimensional consolidation), an applied load assimilated to constant vertical stress and a uniform vertical deformation.

$\frac{\partial u}{\partial t}=C_h\left[\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}\right]$(1)

u: excess interstitial pressure.

C_h: Horizontal consolidation coefficient.

Equivalent diameter: D_e (drain regular mesh) ( Figure 3 ).

$C_h=\frac{k_h{E}_{oed}}{{\gamma }_w}\text{with}\{\begin{array}{l}1\le \frac{C_h}{C_v}\le 5\hfill \\ E_{oed}\text{oedometrique module of soil}\hfill \\ k_h\text{Phorizontal soil permeability of the disturbed area}\hfill \\ {\gamma }_w\text{the density weight of water}\hfill \end{array}$(2)

$T_h=\frac{C_{h}t}{D_e^2}\text{with}\{\begin{array}{l}t\text{the consolidation time in month}\\ D_e\text{is the equivalent diameter}\end{array}$ (3)

The coefficient n is determined from the Barron abacus ( Figure 4 ):

$U_h=f\left(T_h,n\right)\text{with}n=\frac{D_e}{d}$(4)

U_h is degree of horizontal consolidation.

d is drain diameter.

The Carillo theory concerns 3D consolidation (C_h and C_v). It consists of a set of the combined Terzaghi and Barron solutions: (3 abaques). Knowing the horizontal and vertical consolidation coefficients Ch and C_v from laboratory tests and knowing the consolidation time t, the coefficient T_v is determined, from which the value of the vertical degree of consolidation U_v is estimated using the Terzagui diagram shown in Figure 5 .

Knowing U_v and the degree of consolidation required U, U_h can be easily deduced from the Carillo formula:

$1-U=\left[1-U_h\right]\left[1-U_h\right]$(5)

For an efficient calculation, the spacing will be optimized:

$U_h=f\left(T_h,n\right)\text{avec}n=\frac{D_e}{d}$(6)

the procedure is as follows:

$T_h=\frac{C_{h}t}{D_e^2}\text{avec}\{\begin{array}{l}t\text{the consolidation time in months}\\ D_e\text{is the equivalent diameter}\end{array}$(7)

and $D_e={\left(\frac{C_{h}t}{T_h}\right)}^{1/2}$.

Our study is on part of the ROCADE project, the bypass of the city of Porto Novo, political capital of Benin. The Porto-Novo Bypass project is a major road infrastructure that bypasses the city of Porto-Novo to reduce rush hour traffic, located on the southern coast of Benin near the border with Nigeria. The planned development crosses a marshy area with a very compressible soil between PK 0 + 850 and PK 1 + 275 of the Bypass. It will be built next to an existing platform. The plan view of the study area is shown in Figure 6 .

The control of the hydrogeological context of the study environment allows to study the capacity of drains to lower the water table, reduce soil saturation and improve the stability of road infrastructure. The town of Porto-Novo is located near the coast, which means that it is influenced by coastal hydrology related phenomena such as tides, fluctuations in the groundwater level in response to tidal movements, and the possibility of saline intrusion into coastal aquifers. The city is crossed by several rivers and streams that can influence natural drainage and groundwater movements.

In addition, the climate of Porto-Novo is tropical with two dry seasons and two rainy seasons. During the rainy season, the region experiences heavy rainfall and unstable weather conditions, which affect soil stability and road infrastructure sustainability.

The region’s water table is medium to shallow in some areas, making soils prone to saturation and waterlogging, especially during the rainy season.

In the course of our study, the different layers will be composed of several materials selected with specific thicknesses. The physical and mechanical properties of these materials will be determined from geotechnical studies carried out in the project area. These geotechnical studies will provide a precise characterization of locally available materials, providing essential data such as compressive strength, permeability, plasticity, and other relevant parameters. This information will provide a solid basis for the design and construction of the different layers of the project, ensuring their suitability to site-specific conditions.

The pavement studied has a flexible structure. Among the different types of pavements, the flexible pavement is a solution widely adopted in many road projects. Designed to provide a flexible and resistant running surface, the soft pavement is composed of several layers of laminate, each layer having a specific role in the distribution of loads, fatigue resistance and long-term preservation of road quality.

In the particular case of the Bypass project, it consists of a BB bituminous concrete bearing layer, a base layer and a foundation layer made of untreated GNT gravel all resting on clay soil.

The environment is characterized by complex geological formations including alluvial sediment layers, volcanic rocks and limestone formations. These geological formations can have an impact on the permeability and drainage capacity of soils.

The types of soils encountered in the study area vary. Examples include clay, sandy and silty soils. The composition and geotechnical properties of soils can have a significant impact on the stability and performance of infrastructure.

Numerical modeling uses various techniques such as finite elements and finite differences to analyze the stability, deformation and impact of several parameters at all points in the model effectively. In this chapter, we will present the PLAXIS 2D finite element code [25] version 8.2 that will be used in our study, focusing on the description of some behavior models available in this code.

The PLAXIS user interface consists of four sub-programs:

PLAXIS allows different types of finite element calculations. The calculation program only deals with deformation analysis and allows to perform a plastic calculation (Plastic Calculation), consolidation analysis (Consolidation Analysis), calculation of safety coefficients (Phi/C Reduction) or dynamic calculation (Dynamic Calculation).

The main results of a finite element calculation are the shifts at nodes and constraints at stress points. In addition, when a finite element model includes structural elements, forces are calculated in these elements.

The curves program (Curves) can be used to draw load-or time-displacement curves, stress-strain curves, stress paths or deformation for selected points in geometry. These curves represent the evolution during the different phases of calculation, and this gives an overview of the global and local behaviour of the soil [26] .

The values of these parameters used in modelling the different layers are presented in Table 1 .

The applied load represents the traffic loads that will be borne by the road during its lifetime, including vehicle weight, traffic effects, environmental conditions and other external loads. This study identifies all traffic loads at a vertical load evenly distributed over a length of 0.25 m on either side of the track axis. The values of the loads are summarized in Table 4 .

In our case study we used the drain function defined in PLAXIS version 8.2.

Figure 7 shows the in-situ execution of PVD in real image.

In this initial phase, the objective is to evaluate the deformation of the soil before the installation of vertical drains. This involves modelling the soil, applying loads and conditions to the boundary. Adding the layer and interstitial pressures. This data provides a thorough understanding of the study model before treatment. Scenario 0 is explained in Table 5 .

After the drainless parameters are evaluated, vertical and horizontal drains are installed in specific areas identified as having drainage or soil stability problems. This phase of the study is designed to observe changes in geotechnical characteristics of soils after drains are installed. The same parameters as those initially assessed are measured again to assess the impact of drains on soil behaviour. Scenarios 1, 2 and 3 are explained in Table 6 .

Figures 8-10 illustrate the different geometries of the mesh for vertical drains, respectively 0.5 m 0.5 m; 1 m 1 m and 1.5 m 1.5 m. For different scenarios: scenario 0 (without drain), scenario 1 (with a drain of 0.5 m 0.5 m), and scenario 2 (with a 1 m × 1 m drain), including variations in the placement of drains.

This section is devoted to the exposure and analysis of results obtained during this research, providing a detailed and structured description of data and highlighting key findings and trends.

The results of phase 0 in terms of deformed mesh give a total displacement of 78.17 × 10⁻³ m.

The results of phase 1 in terms of vertical displacements, vertical effective stresses and vertical strain corresponding to modelling with vertical drains on a 0.5 m 0.5 m mesh are presented respectively in Figures 11-13 .

The results of phase 2 corresponding to the modelling with vertical drains on a mesh of 1 m 1 m and materializing the plastic areas are presented respectively in Figure 14 .

The results of phase 3 in terms of shear deformation, vertical effective stress and vertical strain corresponding to the modelling with vertical drains on a 1.5 m 1.5 m mesh are presented respectively in Figures 15-17 .

Table 7 summarizes the output data from the different modelling scenarios.

Results show a trend of reduction in extreme total displacement with increasing mesh size for drain scenarios. However, the extreme vertical effective stresses and extreme shear deformations appear to vary in a non-linear manner depending on the mesh size. Extreme vertical displacements are also affected by the presence and size of drains. These results suggest the importance of choosing the mesh size and location of vertical drains wisely to optimize the performance of the pavement in terms of stability and deformation ( Figures 18 ).

When analyzing the results, we see that the minimum of deformations is actually found in phase 1, where the pavement is modelled with a vertical drain mesh of 0.5 m * 0.5 m. This suggests that this particular configuration offers better resistance to deformation than other mesh configurations.

After a thorough analysis of the results obtained, it is clear that the mesh configuration with vertical drains of 0.5 m * 0.5 m in phase 1 is distinguished by its ability to minimize road deformations. This finding suggests that this specific configuration offers increased resistance to deformation compared to other tested configurations. Several factors can explain this promising result. First, a denser mesh allows for more uniform stress distribution, which helps to reduce strain. In addition, the use of vertical drains helps effectively drain water and reduce pore pressures, which plays a crucial role in limiting strain.

Thus, based on these conclusions, it is reasonable to recommend the configuration of phase 1, characterized by a mesh size of 0.5 m * 0.5 m for vertical drains, as well as the incorporation of horizontal drains. This configuration appears to be an optimal choice for ensuring maximum stability and minimum road deformations. These findings highlight the critical importance of proper vertical drain and mesh design in road infrastructure planning to ensure optimal performance in terms of stability and durability.

For validation study, we deal with numerical calculation using Carrillo theory.

The permeability coefficients of clay 1 allow to deduce the horizontal and vertical consolidation coefficients using Equation (2): C_h = 0.288 m²/month and C_v = 0.187 m²/month.

Using T_v = C_vt/H₂ and T_h = C_ht/H₂, we find respectively using the Terzaghi’s Abacus ( Figure 3 ), we find U_v = 0.72 and U_h = 0.62.

The formula of carillo (Equation (5)) allows to find U = 0.29.

This means that for 9 months, the consolidation will only be 69%.

The results of 9-month consolidation calculations for the different clay layers are shown in Table 8 .

The results in Table 8 show low consolidation rates over a period of 9 months. For this reason, it will be impossible to achieve final clay layer consolidation within an acceptable time [23] . Why vertical drainage improvements are needed.

$T_h=\frac{C_{h}t}{D_e^2}=\frac{0.288\times 1}{{1.13}^2}=0.255$ considering a time of 1 month with $D_e=1\times 1.13=1.13$, $n=\frac{D_e}{d}=\frac{1.13}{0.2}=5.65$.

The Projection of T_h on n curve (Terzaghi abacus) gives U_r = 82%.

$T_h=\frac{C_{h}t}{D_e^2}=\frac{0.288\times 1}{{0.565}^2}=0.9$ considering a time of 1 month with $D_e=0.5\times 1.13=0.565$, $n=\frac{D_e}{d}=\frac{0.565}{0.2}=2.825$.

The Projection of Th on n curve (Terzaghi abacus) gives U_r = 94%.