Salt production is an income-generating activity for salt growers . To produce salt, three (3) methods are traditionally used: (i) rock salt extraction , (ii) solar salt production or solar evaporation and (iii) salt production by cooking . At the Djègbadji site in Benin Republic, salt is produced by cooking the brine obtained through leaching the salty soil scraped from the salt marsh floor . The crystallization of salt is a phenomenon that occurs during the favorable and greatest salt-growing periods, from November to March . The soil of the Djègbadji salt marshes is a salty and silty soil considered to be hydromorphic . The phenomenon that leads to salting the Djègbadji salt marsh soil surface can be declined in two processes: (a) the capillary ascent of brackish water from a shallow underground water reached around 50 - 60 cm under the surface and (b) the crystallization of salt at the surface . In porous locations, underground movement of water can be governed by hydraulic gradients . But osmotic gradients , temperature gradients , electrical gradients , chemical gradients , gravity and matrix pressure gradients and even vapor pressure (evapotranspiration) , are all forces that can also induce groundwater flow in these environments . In the particular case of capillary ascent of water in the ground, fluid movement is governed by hydraulic head gradients . From the point of view of static and dynamic properties, the movement of a liquid in a capillary tube presents some interesting characteristics and, it is therefore an ideal setting for the application of surface tension effects (contact angle, notion of wettability and adhesion of liquids to solids) as well as some of the fundamental concepts of fluid mechanics . In the past decades, several authors have focused on the phenomenon of fluid ascent by capillary action and salt crystallization . In particular, the liquid column inside a tube was described by Menon and Agrawal as a mechanical system with variable mass. Thanks to Newton's equation, the phenomenon was understood under simplifying assumptions leading to theoretical solutions. Similarly, in 1996, Duarte et al described the motion of viscous fluid in a capillary tube, using the fundamental laws of hydrodynamics and the Navier-Stokes equations. In 2004, Lu and Likos developed a rigorous closed-form analytical solution for the analysis of capillary rise rates in soils. This new solution, compared with Terzaghi's classical one and with a series of experimental data from open-tube capillary rise tests, was found to provide more realistic and practical predictions of the capillary rise rate in unsaturated soils. As for the crystallization phenomenon, in 2022, the shape and size distribution of salt crystals induced by the crystallization mechanism were studied by Alireza et al, and, their results opened avenues for controlling the size and shape of crystals and patterns formed in a complex gel matrix. For Mancigotti and Hamilton , the constraints exerted by crystallization on the porous host, the growth rate of mirabilite crystals as well as the degree of supersaturation of the solution with respect to mirabilite at the initiation temperature were studied. They observed that mirabilite crystallization is exothermic and then, they related the heat produced to the quantity of crystals formed. They also discovered that mirabilite-ice eutectic point occurs at -3°C and results in complete solidification of the remaining liquid phase accompanied by damage. The problem of crystallization is addressed by Veran-Tissoires et al, , with a view to understand the choice of a location of crystallization on the surface of a porous site. For Shah et al, (2011) , soil salinization would be the result caused by upward capillary flow of poor-quality groundwater; for Norouzi Rad and Shokri (2012) , salt distribution and precipitation depend significantly on salinity as the concentration gradient generates flow in and around the column of highly saline water. According to Nachshon et al (2011) , salt dissolved in groundwater is transported by capillary action in arid zones, eventually precipitating and accumulating near the soil surface in the vadose zone. To our knowledge, no work has yet been carried out on the capillary rise of water in the soil and the crystallization of salt at the soil surface in Benin. In this work, the capillary rise of water in the soil of the Djègbadji salt marsh, the influence of different soil parameters (hydraulic conductivity at soil saturation (ks), soil porosity (n), liquid-solid contact angle (i) and rate of decrease in hydraulic conductivity (a) on this phenomenon and the crystallization of salt at the soil surface were analyzed using the fundamental laws of hydrodynamics and the Navier-Stokes equation. The study explains the physical processes taking place during the capillary rise at the Djègbadji salt marsh. The next sections of the paper present respectively, the material and the method, the obtained results and the concluding remarks

Djègbadji is a district within the municipality of Ouidah, located in the Southwest of Benin, a country in West Africa, at coordinates 6°17’54” to 6°20’48” North latitude and 1°58’36” to 2°7’39” East longitude This region has a climate characterized as subequatorial bimodal. Throughout the year, it experiences four distinct seasons: a major rainy season from April to mid-July, a minor rainy season covering the period from mid-August to mid-October, a major dry season from mid-October to March, and finally a minor dry season observed from mid-July to mid-August . According to research conducted by Boko in 1988 , precipitation during the major season varied between$\mathrm{ }40\mathrm{ }$and$\mathrm{ }65\%$, while that during the minor season ranged from$\mathrm{ }18\mathrm{ }$to$\mathrm{ }30\%$. Furthermore, in 2018, statistical data provided by ASECNA indicates that the annual cumulative precipitation ranges between 950 and 1300 mm . Regarding the winds, an average speed of about$\mathrm{ }4 {m.s}^{-1}\mathrm{ }$is recorded with a predominant direction oriented towards the Southwest . The average temperature is around$\mathrm{ }28.04 °C$; the average humidity reaches$\mathrm{ }81.04 \%$; the average radiation amounts to about$\mathrm{ }50.21{\mathit{ kWh}.m}^{-2}$; the average insolation is estimated at around$\mathrm{ }4.98 {\mathit{kWh}.m}^{-2}\mathrm{ }$ while the average net radiation is estimated at around$\mathrm{ }401.35 {W.m}^{-2}\mathrm{ }$401.35 W.m^-2. The local topography is marked by the presence of the coastal lagoon also known as Djessin lagoon, which houses salt marshes and swamps. Endowed with a soil classified as silty and hydromorphic , this region is home to various plant species such as mangroves and coconut trees.

The figure 1 below illustrates the capillary rise process in soil. It shows the soil profile delineated by three distinct zones. The First zone (zone 1) is below the water table. The pore water pressure is considered positive in that zone. The second zone is located under the air inlet head$\mathrm{ }\left(h_a\right)\mathrm{ }$but above the water table. The pore water pressure is negative to atmospheric pressure. Finally, the third zone is unsaturated and is above the air intake head, where the capillary water rises like a series of fingers connected or disconnected at a maximum height$\mathrm{ }\left(h_c\right)$. The capillary rise’s water phenomenon in unsaturated soil is therefore described as the movement of pore water from a lower to a higher level. This phenomenon of water circulation occurs due to the hydraulic head gradient acting across the interface between pore air and curved pore water .

To analyze capillary rise’s phenomenon, Terzaghi equation governing the one-dimensional motion of capillary rise of water in homogeneous, rigid porous media (soil) with varying degrees of saturation (equation 1) , is used:

In this equation, the maximum height of capillary rise$\mathrm{ }\left(h_c\right)$, is given by equation (2) :

where$\mathrm{ }{k}_s$, is the hydraulic conductivity at saturation of the soil;$\mathrm{ }n$, the soil porosity;$\mathrm{ }i$, the liquid-solid contact angle;$\mathrm{ }{\rho }_w$, the density of water;$\mathrm{ }g$, the earth's gravity;$\mathrm{ }\eta$, the kinematic viscosity of water;$\mathrm{ }\gamma$, the surface tension and$\mathrm{ }{h}_a$, the air entry height or height of the saturated part of the capillary fringe given by relation (3) :

Physical parameters used in this work come from the work of Duarte et al . These values correspond to those of water considered at 30°C. They include parameters such as: gravity intensity$\mathrm{ }g=980\mathit{ cm}.s^{-2}$; surface tension$\mathrm{ }\gamma =70\mathit{ dyn}.{\mathit{cm}}^{-1}$; water density$\mathrm{ }\rho =1g.{\mathit{cm}}^{-3}$and kinematic viscosity$\mathrm{ }\eta =8,31.{10}^{-3}\mathit{Pl}$. The constant$\mathrm{ }c\mathrm{ }$was set at$\mathrm{ }c=20$.

Data from soil parameters such as hydraulic conductivity at soil saturation$\mathrm{ }{(k}_s)$, rate of decrease of hydraulic conductivity$\mathrm{ }\left(a\right)\mathrm{ }$and soil porosity$\mathrm{ }\left(n\right)\mathrm{ }$were used .

To better understand how salt crystallizes on the soil surface of the Djègbadji salt marshes, we resorted to the Navier Stokes equation given in relation (4).

with$\mathrm{ }\rho$, density;$\mathrm{ }\mu$, dynamic viscosity;$\mathrm{ }\vec{p}$, pressure field and$\mathrm{ }\vec{g}\mathrm{ }$gravitational force.

By applying the law of conservation of mass and considering that water is incompressible everywhere and at all times, i.e. :

The Navier Stokes equation given by relation (2) reduces to :

Since water subject to evapotranspiration must evaporate, we assume that water on the ground surface is not under the influence of turbulent acceleration or external forces (pressure field and gravitational forces). Under these conditions, we have:$\mathrm{ }\left(u.\nabla \right)u=\vec{0}$,$\mathrm{ }\nabla \vec{p}=\vec{0}\mathrm{ }$and$\mathrm{ }\rho \vec{g}=\vec{0}$. Relation (6) above can thus be rewritten in the following form:

If$\mathrm{ }D=\frac{\mu }{\rho }$, the relation (7) becomes:

Equation 8 is the diffusion equation with D, the diffusion coefficient. Thus, for unidirectional flow, equation (8) can be written as:

Since we're addressing about evaporating issue, the water wetting the surface of the Djègbadji salt marshes, i.e. considerably reducing its water content$\mathrm{ }\left(W\right)$, we pose$\mathrm{ }u=W\mathrm{ }$and equation (9) becomes:

Equation (10) is well known as Fick’s second law equation, which solution is obtained by making the following assumptions :

(1) water flows by diffusion only;

(2) surface water content is equal to equilibrium;

(3) the diffusion coefficient and product temperature are assumed to be constant and;

(4) sample shrinkage is negligible.

Based on these assumptions, the analytical solution of Fick's second law is given by relation (11) :

The simplified form of this equation (11) is obtained by taking the first term of the series assuming$\mathrm{ }{W}_e = 0$. Then :

and when we pose$\mathrm{ }A=\frac{8}{{\pi }^2}\mathrm{ }$and$\mathrm{ k}=\frac{{\pi }^{2}D}{4L^2}$, we’ve obtained the equation (13) below:

In order to better understand capillary rise's phenomenon of salt water in the Djègbadji salt marshes and salt crystallization's mechanism at soil surface, the solution of equation (1) and the Navier Stokes equation were required. Equation (1) was solved in the MATLAB environment and the solution is described by relation (14):

where$\mathrm{ }{k}_s$, is the hydraulic conductivity at soil saturation;$\mathrm{ }n$, soil porosity;$\mathrm{ }a$, rate of decrease in hydraulic conductivity;$\mathrm{ }{h}_c$, maximum height of capillary rise and finally,$\mathrm{ }c$, a constant set at$\mathrm{ }c=20$.

The influence of soil parameters such as soil saturation hydraulic conductivity$\mathrm{ }{(k}_s)$, soil porosity$\mathrm{ }\left(n\right)$, hydraulic conductivity decrease rate$\mathrm{ }\left(a\right)\mathrm{ }$and liquid-solid contact angle$\mathrm{ }\left(i\right)\mathrm{ }$on water capillary rise in the soil was analyzed.

Figure 2 illustrates the temporal evolution (during a day) of capillary rise of water in the soil under the influence of hydraulic conductivity at saturation$\mathrm{ }\left(k_s\right)$. The analysis reveals an increase in capillary rise for$\mathrm{ }{k}_s<k$, with$\mathrm{ }k=2.{10}^{-4}{\mathit{cm}.s}^{-1}$, the hydraulic conductivity of unsaturated silty soil and a decrease in the level of rise for$\mathrm{ }{k}_s>k$. For$\mathrm{ }{k}_s\mathrm{ }$smaller than$\mathrm{ }k$, the capillary rise level is higher, indicating that the soil is close to saturation. On the other hand, for$\mathrm{ }{k}_s\mathrm{ }$greater than$\mathit{ k}$, the level of capillary rise is lower than for$\mathrm{ }k$, indicating that the soil moving further away from saturation. The saturation hydraulic conductivity$\mathrm{ }{k}_s\mathrm{ }$is therefore a soil parameter that strongly influences the capillary rise of water in soils and in the Djègbadji salt marshes particularly.

The parameter$\mathrm{ }{k}_s\mathrm{ }$is expressed in$\mathrm{ }\mathit{cm}.s^{-1}\mathrm{ }$and its value will be taken equal to$\mathrm{ }1.5\times {10}^{-3}\mathit{cm}.s^{-1}\mathrm{ }$for the next sections because the dry season is the favorable period for the activities of salt production in Djègbadji and, during this period, the soil is not saturated; at the angle of contact$\mathrm{ }i$, we assign the value$\mathrm{ }0°$, because it is under this angle that the liquid perfectly wets the wall of the pore .

Figure 3 indicates the influence of the rate of decrease in hydraulic conductivity$\mathrm{ }\left(a\right)\mathrm{ }$on capillary rise of water in Djègbadji salt marsh soil. During the analysis, the parameters of hydraulic conductivity at soil saturation$\mathrm{ }{(k}_s)$, soil porosity$\mathrm{ }\left(n\right)\mathrm{ }$and liquid-solid contact angle $\left(i\right)\mathrm{ }$have respectively been set to$\mathrm{ }{k}_s=1.5{\times 10}^{-3}\mathit{cm}.s^{-1}, i=0° \mathrm{and} n=0.03$. The rate of decrease in hydraulic conductivity$\mathrm{ }\left(a\right)\mathrm{ }$was then varied from$\mathrm{ }10\%\mathrm{ }$to$\mathrm{ }60\%$. The analysis shows that the capillary rise of water in the soil decreases as the rate of decrease in hydraulic conductivity increases. For a hydraulic conductivity reduction rate of$\mathrm{ }10\%$, we observe a minimum capillary rise of 30.07 cm and a maximum rise of 83.87 cm for an average of 73.24 cm, compared with minimum, maximum and average capillary rises of 19.19 cm, 28.16 cm and 26.39 cm respectively for a hydraulic conductivity reduction rate of 60% in one day. The analysis reveals that the hydraulic conductivity rate (a) is a parameter that also strongly influences the capillary rise in Djègbadji salt marsh soils.

Figure 4 illustrates the evolution of capillary rise of water in Djègbadji salt marsh soil under the influence of porosity$\mathrm{ }\left(n\right)$. Analysis of this figure shows that capillary rise of water in Djègbadji salt marsh soil increases as porosity increases. Thus, the minimum value of capillary rise is$\mathrm{ }21.31 \mathit{cm}\mathrm{ }$at$\mathrm{ }n=1\%\mathrm{ }$and its maximum value is$\mathrm{ }104.6 \mathit{cm}\mathrm{ }$at$\mathrm{ }n=10.6\%\mathrm{ }$for an average of$\mathrm{ }80.19 \mathit{cm}\mathrm{ }$at$\mathrm{ }n=5.8\%$. The analysis thus indicates that soil porosity is also a major parameter influencing capillary rise of water in Djègbadji salt marsh soil.

Figure 5 shows the evolution of capillary rise of water in Djègbadji salt marsh soil under the influence of the liquid-solid contact angle$\mathrm{ }\left(i\right)$. Analysis of this figure shows an oscillatory (sawtooth) upward trend in capillary rise, with a minimum value of$\mathrm{ }27.86 \mathit{cm}\mathrm{ }$and a maximum value of$\mathrm{ }81.53 \mathit{cm}$, for an average of$\mathrm{ }60.33 \mathit{cm}$. This parameter therefore also influences the phenomenon of capillary rise in the soil.

Equation (13) is the Henderson and Pabis drying model obtained by modification of the Page model in 1962 . From the analysis of this equation (13) involving the function$\mathrm{ }{(\mathrm{e}}^{-x})$, it emerges that the water content of the soil surface in the Djègbadji salt marshes decreases over time. This implies a probable progressive oversaturation of the soil with salt, culminating in evapotranspiration-induced salt precipitation followed by accumulation on and near the soil surface of these salt marshes.

Djègbadji is a wetland in the municipality of Ouidah , which, like any other area of this type, undergoes cycles of moistening and drying resulting in transient interactions between groundwater and surface water . As Shah et al, (2011) and van der Zee et al, (2014) have already pointed out, groundwater can be a real source of water and salt for these wetlands. In fact, dissolved salts in the shallow groundwater at Djègbadji rises to the surface by capillary action, creating an induced flow towards the root zone where the salt accumulates. As shown in the analysis of Figures 2, 3, 4 and 5, the upwelling of a fluid (water) by capillarity in the soil is a phenomenon that is influenced by several soil parameters such as the saturated hydraulic conductivity of the soil$\mathrm{ }{(k}_s)$, the porosity of the soil$\mathrm{ }\left(n\right)$, the rate of decrease of the hydraulic conductivity$\mathrm{ }\left(a\right)\mathrm{ }$and the liquid-solid contact angle$\mathrm{ }\left(i\right)$, i.e. it varies according to the nature of the soil characterized by the values of these parameters. In addition, the profiles in Figures 2, 3 and 4 are identical to those obtained by Duarte et al, (1996) in their study of capillary rise in open tubes. Moreover, the results from the analysis of Figure 2 differed from those found in 2015 by Coulibaly regarding the infiltration of water in the soil, which is a phenomenon opposed to water capillary rise in the soil. In Djègbadji salt marshes, water is able to rise by capillary action from the shallow$\mathrm{ }50-60 \mathit{cm}\mathrm{ }$brackish water table to the soil surface , carrying with it the salt dissolved in it. During drought (a favorable period for salt production in Djègbadji), the demand for water from the atmosphere met by evapotranspiration is generally higher than the flow of liquid water . The transport of salt from groundwater to the surface follows an increase in liquid water saturation at the surface. This saturation is followed by salt crystallization as saline efflorescence and salt accumulation on and around the soil surface is governed by its water content values .

In addition, it should be noted that the climatic conditions on the site of Djègbadji (mean temperature:$\mathrm{ }28.04°C$; mean humidity:$\mathrm{ }81.04\%$; mean radiation:$\mathrm{ }50.21 {\mathit{kW}h.m}^{-2}$; mean insolation:$\mathrm{ }4.98{ \mathit{kW}h.m}^{-2}$; mean wind speed:$\mathrm{ }4.24 {m.s}^{-1}\mathrm{ }$; mean net radiation:$\mathrm{ }{401.35 W.m}^{-2}$), are favourable for soil surface drying. Indeed, the climatic conditions listed above contribute to the gradual loss of soil moisture over the time. This loss of humidity leads to the oversaturation of the soil, to a certain precipitation of the salt followed by its accumulation on the soil surface. This could justify obtaining brine at Djègbadji by leaching sand from soil scraping.

Equation 13 provides a better understanding of salt crystallization at the Djègbadji site in Benin. It highlights a progressive decrease in water content at the soil surface over time. This reduction leads to saturation, or even supersaturation, of the upper soil layers with salt, which, under the effect of evapotranspiration, results in salt precipitation followed by its accumulation. This phenomenon likely explains the extraction of brine through the leaching of saline soils in this region.

Similar results were obtained in 2008 by Guglielmini et al. and in 2014 by Zhang et al. . In their work, Guglielmini et al. (2008) considered the evaporation rate as constant and the liquid saturation in the porous medium as uniform, varying only with time. Under these conditions, they found that salt tends to accumulate in a narrow region adjacent to the surface where evaporation occurs, while in the rest of the site, salt remains evenly distributed and decreases over time due to a purely convective effect. Similarly, Zhang et al. observed through their simulations that the presence of a hydraulic connection between the soil’s surface layer and the underground water source promotes vaporization near the surface, with salt exclusively depositing as efflorescence. In contrast, in the absence of such a hydraulic connection, the vaporization front moves downward, and salt settles as subflorescence. Efflorescence, being more abundant, has a significantly greater impact on evaporation than subflorescence during the soil drying process. The characterization of the evaporation process in bare saline soils, whether or not they have a fixed water table, can be achieved by identifying the different evaporation stages based on the location of the vaporization front and salt accumulation.

During salt crystallization, the phenomenon depends on heat transfer , whose coefficient is itself influenced by free convection in a liquid water film and a saline vapor gas . The phase transition rate is closely linked to the thickness of the liquid layer available in the salt solution Furthermore, the hydrophilicity of the surface significantly affects the evaporation rate , as does the location of hydrated salt crystal formation. When significant heat fluxes are present within the droplet and the thin layer, natural convection occurs, leading to enhanced evaporation . Initially, an increase in supersaturation is observed (the actual concentration increases over time). Crystallization behavior is influenced by the location of the first crystal nucleus, the rate of evaporation and crystallization, and the temperature distribution at the free surface of the layer . The phenomena of capillary rise and crystallization of salt on the surface of the salt marshes of Djègbadji site have ensured the continuation of food salt production in this district for centuries.

Iodized salt helps address iodine deficiency and associated disorders; plays an important role in physical and mental activity for adults, in the normal growth and development of the fetus, infant and child . Moreover, salt production in Djègbadji is a revenue generating activity for the salt workers in this locality . It thus contributes to the achievement of the Sustainable Development Goals, in particular SDG 8, which stipulates “to promote sustained, shared and sustainable economic growth, full productive employment and decent work for all” and SDG 9, which stipulates that “build resilient infrastructure, promote sustainable industrialization that benefits everyone, and encourage innovation.”

A solution of the one-dimensional Terzaghi equation governing the capillary rise of water in soil was obtained and the capillary rise of brackish water in the Djègbadji salt marshes was analyzed. The appearance of fine salt crystals on and near the soil surface maybe due to the evapotranspiration on soils soaked in brackish water. In short, the phenomena of capillary rise and crystallization in the process of salt precipitation and accumulation on the soil surface are important for understanding the process of salinization rate at the Djègbadji salt marsh.

Armel Vitouley: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Validation, Visualization, Writing – original draft

Guy Hervé Hounguè: Conceptualization, Data curation, Formal Analysis, Methodology, Software, Supervision, Validation, Visualization, Writing – review & editing

Gabin Koto N’gobi: Conceptualization, Data curation, Formal Analysis, Methodology, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing

Julien Adounkpè: Formal Analysis, Funding acquisition, Investigation, Resources, Supervision, Validation, Writing – review & editing

Waliou Amoussa Hounkpatin: Validation, Visualization, Writing – review & editing

Clément Ahouannou: Investigation, Supervision, Validation, Visualization

Basile Kounouhéwa: Conceptualization, Data curation, Formal Analysis, Funding acquisition, Methodology, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing

The authors gratefully acknowledge funding from the Federal German Government, particularly its Ministry of Higher Education and Scientific Research BMBF, through the FOCUSED-Benin project, the work package V of the YES PV NIGBEN project, Grant TTB/B1806.05.18.

The authors declare no conflicts of interest.