Estimation of the Hydraulic Parameters of a Stratified Alluvial Soil in the Region of El Haouareb—Central Tunisia. Experiments, Empirical, Analytical and Inverse Models

Abstract

Modeling water flow and contaminant transport in the unsaturated zone is a difficult task that relies heavily on good hydrodynamic soil characterization. This article presents a complementarity between experimentation, direct modeling and inverse modeling in order to provide a better estimate of the hydrodynamic parameters of stratified alluvial soil in the El Haouareb region of the Kairouane plain in Tunisia. A field sampling campaign was carried out. The samples collected underwent particle size analysis, bulk density measurements and infiltration tests using a mini-Muntz. In parallel, simple evaporation tests were applied to separate strata. In addition, a 2 m soil column was reconstituted and fitted with sensors to monitor water content, tension, temperature and electrical conductivity. An internal drainage test was performed on this monolith. Three methods were applied using experimental data to estimate soil hydrodynamic parameters. In the first method, pedotransfer functions were used (Rosetta platform) based on granulometric results and bulk density. In the second, water tension and water content monitored during the simple evaporation test were used to plot the soil–water retention curve (SWRC) using SWRC-Fit. In the third method, inverse modeling was applied to the internal drainage test. A comparison of the results showed that the inverse method had the lowest RMSE. Uncertainty analysis has been implemented for both the experimental and numerical set up.

1. Introduction

Water scarcity is the most serious threat worldwide due to climate change and anthropogenic actions, particularly in arid and semi-arid regions [1]. According to recent reports on climate projections [2], climate change manifests primarily in the disruption of rainfall patterns, reduced frequency of precipitation and runoff, and increased rates of evapotranspiration [3]. This situation will be further worsened by anthropogenic factors, notably population growth [4]. These changes influence the hidden parts of the hydrological cycle, such as the soil–water content, groundwater levels and quality [5].

The direct consequence of these global changes is the intensification of irrigation, leading to the misuse of available freshwater and non-conventional resources (salt water, treated wastewater). As a result, agricultural activities can affect soil and groundwater quality by causing salinization [6].

In this context, it is important to explore the path of water in the unsaturated zone, as this is an intermediate zone between the atmosphere and the aquifer where key processes that control both subsurface water fate and geochemical transfers take place.

This study focuses on the Kairouane plain in central Tunisia, which has been the subject of several studies [7,8,9,10]. The area suffers from intensive irrigation [11] and is known for its alluvial deposits, which generally have very different grain sizes and vary from one place to another. Irrigation in these soils is often difficult, as the stratification of the materials affects the dynamics of water and salts [12].

It is therefore crucial to explore the buried phenomena that occur in these types of soils in order to make appropriate decisions regarding future agricultural practices under the impact of climate change on natural resources, mainly soil and water.

Our work began in the field with soil sampling in the irrigated perimeter of El Haouareb. These samples were characterized by various analyses, such as textural fractions and bulk density. In addition, simple evaporation tests and infiltration tests were carried out on the soils of each stratum.

Meanwhile, a 2 m soil column (monolith) was reconstituted at the National Institute of Rural Engineering Water and Forests at Tunis (INRGREF), equipped with sensors to monitor water content, water tension, temperature and electrical conductivity. The monolith went through various phases, including an internal drainage test.

In order to model the flow and transport of salts through the unsaturated zone, the results of the tests carried out were used to determine the hydrodynamic parameters of the soil, specifically the retention curves θ (h) and the hydraulic permeabilities K (h) of the soils. As these curves θ (h) and K (h) are the most common representation of water behavior in soil, they were formulated and modeled using various equations and methods.

In this work, three parameter estimation methods were applied to identify the hydrodynamic parameters of the sampled soils.

In the first, we used a user friendly platform of pedotransfer functions, namely the Rosetta model [13], using textural fractions and soil bulk density. In the second, we used the SWRC-Fit platform [14,15], fitting the soil retention curve to water content and pressure measurements during simple evaporation tests. In the third method, an inverse approach was adopted based on data from the internal drainage test on the monolith. Infiltration tests were used to determine saturation hydraulic conductivities.

The first two direct estimation methods are widely used. Rosetta, among plenty of PTFs [16,17,18,19] which are based on the artificial neural network approach, is implemented in the Hydrus code [20] and has been enhanced to determine the soil characteristics [21]. Seki’s method using the SWRC-Fit platform was useful for many study cases [22,23]. The third method has consisted of applying inverse modeling, which is a useful numerical tool for estimating parameters from experimental measurements. It has been widely used and applied over the last few decades in hydrology to determine hydraulic transmissivities in groundwater and unsaturated soil parameters as well as boundary conditions of the studied domain [24,25,26,27,28,29,30,31,32].

The final part of the paper presents a critical analysis and comparison of the obtained results. A Monte Carlo simulation has been implemented in order to assess the accuracy of the estimated parameters. This hydrodynamic characterization is a first step towards forecasting water dynamics and salt transport during simulations of climate change scenarios.

2. Materials and Method

2.1. Study Area

Sampling was carried out in the Haouareb perimeter, located 7 km downstream of the Haouareb Dam (Figure 1). This perimeter is part of the alluvial plain of Kairouane, which is a vast and deep collapse basin filled with more than 700 m of plio-quaternary detrital sediments brought from the Tunisian ridge by numerous wadis, the most important of which are the Zeroud and Merguellil wadis [33].

The study focused on the northern part of the plain: the Wadi Merguellil sub-basin. The altitude of the plot is around 170 m above mean sea level in the western part, falling gently towards the east by less than 20 m. The semi-arid climate is characterized by highly variable rainfall in space and time. Average annual precipitation is less than 300 mm, with the wettest months being October to April. Average annual evaporation is around 1700 mm. Surface runoff is highly intermittent. Before the construction of large dams in the 1980s, the wadis of Merguellil and Zeroud ran across the plain.

2.2. Field Sampling and Soil Characterization

Based on previous work carried out on a representative plot belonging to a farmer in the Haouareb irrigated zone, a profile was selected. It corresponds to an auger hole called STE 1.2 in previous studies that has been renamed PFS.

This profile, 2.2 m deep and 2.0 × 1.0 m wide, was dug in layers of 10 cm of thickness. The work was undertaken during a dozen field trips between January and March 2020. At the base of the profile, an auger hole was made that was up to 4.0 m deep, with samples being taken every 20 cm, allowing for a characterization of the soil between 0 and 6.0 m. Thus, 40 soil samples were collected, 20 between 0 and 2 m and 20 between 2 and 6 m. The soil description was carried out in the profile and completed between 2 and 4 m on the samples taken. Samples with similar characteristics were grouped into distinct diagnostic layers based on color, texture, structure, biological activity and soil.

Standard methods were used for the determination of soluble salts, pH measurements, electrical conductivity of saturated paste extract and organic matter and granulometric analysis. The granulometric fractions of the soils have been determined in a laboratory in triplicate.

Soil samples were first stripped of organic matter by oxidation with hydrogen peroxide (H₂O₂). Flocculent ions were then removed by washing with Potassium Chloride (KCl). The clay fraction was dispersed with sodium hexametaphosphate. The different particle size fractions were then determined by pipetting, drying and sieving.

Bulk density (BD) was determined in situ for undisturbed soil every 10 cm to 2 m depth. It was completed in the laboratory with the reconstituted soil. The cylinders used have a volume of 100 cm³. Three repetitions per layer of 10 cm were performed. For the reconstituted soil in the laboratory, several wetting–drying cycles were performed to allow for partial restructuring of the soils. This was carried out for all layers of the monolith.

2.3. Infiltration Test

The fixed-load water infiltration test was carried out using a Mariotte vase apparatus (Mini-Müntz) (Figure 2). It consists of two transparent Plexiglas columns 10 cm high and 5.5 cm in radius, fitted with a geotextile filter. In case of disturbed soil (in laboratory), a sample is placed in the first column, which contains a drainage hole. The second column, containing two tubes (3 cm and 1.5 cm long) for supplying water to the soil, is glued in place. Water is poured into the second column up to a height of 3 cm. The soil in the column is slowly saturated with water from the top. Each trial (3 replicates) was monitored for three days.

Meanwhile, in case of undisturbed soil (in field), the upper column is placed above the sample where infiltration is taking place.

2.4. Simple Evaporation Test

The evaporation test involves monitoring, over a 21-day period, the variation in water tension and water content for each layer separately, with 3 repetitions being undertaken to achieve satisfying outcomes. Each pot is filled with about 1500 g of undisturbed soil and equipped with a WaterMark probe (Campbell Scientific, Inc., Logan, UT, USA), which provides measurements on a regular (daily) time step (Figure 3).

2.5. Internal Drainage Test

A 2 m monolith of reconstituted soil profile brought from the Haouareb site was fitted with a battery of sensors for water tension, water content and apparent soil salinity, some of which were connected to continuous recording stations (DATA Logger) (Figure 4). Eight distinct horizons were instrumented with the following sensors:

-

WaterMark tensiometer: Monitoring of water tension. Only six sensors are connected to a data logger with a 4 h time step while the two other sensors are read manually.

-

EM50 sensors: These allow for the measurement of three parameters: volumetric water content (θ_v), according to the TDR electromagnetic method, apparent electrical conductivity of the soil (ECa) and temperature. The sensors are connected to recording stations with a 4-h time step.

Measurement uncertainty can be linked to sensor calibration, assuming factory calibration. Sensor calibration was tested under steady-state conditions, with sensors installed in jars filled with water only. In addition, on-line calibration was applied through numerical modeling to compare measured and simulated data [34].

The monolith was gradually saturated with tap water from July to the end of August 2020, then with distilled water until the end of September 2020. To check water flow and the saturation status of the monolith, methylene blue was mixed with the water as a tracer at a low concentration of 0.01 g/L. After checking the various soil layers, we carried out the internal drainage test on 6 October 2020, closing the monolith at the top with a Plexiglas cover. This test lasted 68 days until reopening on 13 December 2020. From this date, the monolith was exposed to atmospheric conditions until 25 June 2021, when the addition of distilled water was restarted at a volume of 20 L/day. The monolith then underwent a period of irrigation, during which borehole water from the study area and saline water were added.

3. Results

3.1. Pedotransfer Functions-Rosetta PTF

Pedotransfer functions (PTFs) are empirical equations describing the relationships between soil parameters. Introduced in the literature by [35,36,37], the general goal of PTF development is to set up predictive models or algorithms using soil property datasets that contain adequate predictors (core soil properties) and desirable estimands (inferred soil properties) [38]. It is mainly for this purpose that the Rosetta PTF, implemented in Hydrus 1D, was used in this study [20]. These PTFs use the percentage of clay, silt, sand and bulk density of the soils to calculate the parameters of the retention models of Books and Corey [39] or Van Genuchten [40].

The results of the soil textural analysis study enabled us to distinguish four layers (

Table 1. Soil properties:’ bulk density and granulometric fractions.

Layer	Thickness (cm)	BD (Kg/m3)	STD (Kg/m3)	Sand (%)	Silt (%)	Clay (%)	STD (%)
1	0–60	1530	36	36	16.5	47.5	5
2	60–120	1450		29	15	56
3	120–170	1500		19.5	45.2	35.3
4	170–200	1490		82	3.8	14.2

). The top layer, 60 cm deep, is dominated by clay and has the highest bulk density, 1530 kg/m³. The second layer, from 60 to 120 cm deep, has the highest percentage of clay (56%) and less sand than the top layer. The third layer, 50 cm deep, has the highest percentage of silt (45.2%) and a significant amount of clay (35.3%). On the other hand, it contains the lowest percentage of sand, 19.5%. The bottom layer, 30 cm thick, is predominantly sand, 82%.

The accuracy of the measurements was assessed by calculating the standard deviation of each variable. The apparent density has a standard deviation of 36 Kg/m³, taking into account the uncertainties associated with the calibration of the balance (supplied by the factory).

Table 2. Estimated hydrodynamic parameters—Rosetta Empirical Model.

Layer ID	Parameters	Mean	STD	θr (cm3/cm3)	θs (cm3/cm3)	α (1/cm)	n (-)	Ks (cm/d)
1	θr (cm3/cm3)	0.088	0.002	1
	θs (cm3/cm3)	0.416	0.011	0.912	1
	α (1/cm)	0.021	0.001	0.617	0.835	1
	n (-)	1.226	0.018	0.646	0.882	0.973	1
	Ks (cm/d)	7.573	1.221	0.885	0.993	0.824	0.884	1
2	θr (cm3/cm3)	0.096	0.002	1
	θs (cm3/cm3)	0.411	0.129	0.059	1
	α (1/cm)	0.02	0.001	0.603	0.001	1
	n (-)	1.242	0.016	0.084	0.011	0.006	1
	Ks (cm/d)	10.609	2.116	0.003	0.023	0.1	0.761	1
3	θr (cm3/cm3)	0.086	0.002	1
	θs (cm3/cm3)	0.425	0.007	0.549	1
	α (1/cm)	0.011	0.001	0.344	0.003	1
	n (-)	1.413	0.022	0.211	0.06	0.93	1
	Ks (cm/d)	5.04	0.635	0.11	0.751	0.178	0.391	1
4	θr (cm3/cm3)	0.062	0.002	1
	θs (cm3/cm3)	0.41	0.013	0.331	1
	α (1/cm)	0.027	0	0.362	0.084	1
	n (-)	1.903	0.233	0.882	0.072	0.671	1
	Ks (cm/d)	149.855	57.645	0.665	0	0.882	0.925	1

shows the soil hydrodynamic parameters estimated by the Rosetta PTF model [13]. Residual water content (θ_r) varies between 0.0632 and 0.0910 cm³/cm³. Water content at saturation (θ_s) varies between 0.4158 and 0.4375 cm³/cm³. Saturated hydraulic conductivity (K_s) is lowest in the third layer and highest in the fourth, composed mainly of sand. α and n are the Van Genuchten model parameters.

The uncertainties of the model were evaluated (see Table 2), and the results are also presented in Table 2. The neural network method adopted by [13] has low standard deviations where 75% of parameters have a coefficient of variation under less than 5%, while the R² correlation coefficient is high or even very high, with a few exceptions [41].

The saturated hydraulic conductivity of each sample, as determined by Rosetta on the one hand and by the infiltration test (Mini-Muntz experiment) on the other, were compared. Figure 5 illustrates the relationships between Ks-In Situ, Ks-Lab and Ks-Rosetta.

The highest coefficient of determination R² is that of the relationship between Ks-Lab and Ks-In Situ, which is equal to 0.94. This can be explained by the use of the same test. The correlation between Ks-Rosetta and Ks-Lab seems to be significant (R² = 0.87). However, a moderate correlation is seen in the relationship between Ks-Rosetta and Ks-In Situ.

But the order of magnitude of the two types of determination is very different. Indeed, Ks-Lab and Ks-In Situ vary between 39 and 1666 cm/day, whereas the values given by Rosetta vary between 5 and 60 cm/day. Compared to Ks measured in the field, Ks measured in the laboratory seem to be slightly overestimated. This could be due to the disturbed state of soil, which is slightly compacted and therefore lax. On the other hand, the moderate correlation between Ks-Rosetta and Ks-In Situ could be explained by the fact that the program (Rosetta) does not take account the soil structure, which influences in situ infiltration.

3.2. Soil–Water Retention Curve SWRC Fit

The retention curve describes the variation in water content as a function of tension (pressure head, h). Virtually all studies of unsaturated soils require the soil–water retention curve to be continuous so that it can be easily incorporated into numerical models simulating water flow and solute transfer. Experimental results are often represented by a cloud of points scattered over the suction range.

Several mathematical models are proposed for correlating experimental data. In this case, only a few experimental points are needed to obtain the complete curve. In this study, the retention curve was plotted online using the web-based interface SWRCfit provided by [14,15]. In this program, ten smoothing models are proposed to fit the experimental points of the soil–water retention curve along the drainage paths. Among them, four models are considered widely used and known, namely Brooks and Corey [39], Van Genuchten [40], Kosugi [42,43] and Fredlund and Xing [44]. Their expressions are given below while their parameters are described in details in

Table 3. Equations of some retention models utilized by SWRC Fit.

Model	Equation	Parameters
Brooks and Corey 1	(1)	θs 2
		θr 3
		hb 4
		λ 5
Van Genuchten 6	(2)	θs 2
		θr 3
		α 7
		n 8
Kosugi 9	(3) and (4)	θs 2
		θr 3
		hm 10
		σ 11
Fredlund and Xing 12	(5)	θs
		θr
		a 13
		m 14
		n

Note: ¹ Ref. [39], ² θ_s is the volumetric saturated water content (L3/L3), ³ θ_r is the volumetric residual water content (L3/L3), ⁴ h_b is the absolute value of the air-entry potential (kPa), ⁵ λ is the Brooks–Corey pore-size distribution index; ⁶ Ref. [40], ⁷ α is the air entry parameter (1/L), ⁸ n is the pore size distribution parameter (-); ⁹ Refs. [42,43], ¹⁰ h_m is a parameter of the Kosugi model related to the median pore radius (L), ¹¹ σ is a parameter of the Kosugi model related to the width of the pore radius distribution (-); ¹² Ref. [44], ¹³ a is mainly related to the air-entry value (-), ¹⁴ m is mainly related to the residual degree of saturation on the soil–water characteristic curve (S_r).

.

$$S_e=\left\{\begin{array}{c}{\left(h/h_b\right)}^{-\lambda } (h>h_b)\\ 1 ( h\le h_b)\end{array}\right.$$

(1)

$$S_e={\left[{\displaystyle \frac{1}{1+{(\alpha h)}^n}}\right]}^m with (m=1-1/n)$$

(2)

$$S_e=\left[{\displaystyle \frac{\mathit{ln}\left(h/h_m\right)}{\sigma }}\right]$$

(3)

$$Q\left(x\right)=erfc\left(x/\sqrt{2}\right)/2$$

(4)

$$S_e={\left[{\displaystyle \frac{1}{\mathit{ln}\left[e+{\left(h/a\right)}^n\right]}}\right] }^m$$

(5)

To apply this parameter estimation process, we used data (water content and tensions) from the simple evaporation test (Section 2.3).

In order to correctly estimate the hydrodynamic parameters of the four soil layers, 10 analytical models were used to test the quality of the experiments/observations and to obtain the best SWRC result. This is important in regard to obtaining realistic scenarios of the impacts of climate change on natural resources (water and soil).

Table 4. Analytical models performances.

Layer ID	Model	Brooks and Corey	Van Genuchten	Kosugi	Fredlund and Xing	dual-BC-CH
1	R2	0.975	0.986	0.987	0.989	0.975
	AIC	−165.280	−177.460	−179.190	−179.590	−163.270
2	R2	0.979	0.979	0.980	0.980	0.979
	AIC	−167.040	−167.040	−167.550	−165.530	−165.040
3	R2	0.803	0.827	0.830	0.834	0.803
	AIC	−122.590	−125.160	−125.470	−123.960	−120.590
4	R2	0.948	0.948	0.947	0.950	0.948
	AIC	−160.210	−160.290	−160.190	−159.130	−158.220
Layer ID	Model	VG1BC2-CH	dual-VG-CH	dual-BC	dual-VG	dual-KO
1	R2	0.986	0.986	0.983	0.988	0.961
	AIC	−175.460	−175.460	−168.680	−177.020	−152.920
2	R2	0.981	0.987	0.988	0.991	0.991
	AIC	−166.570	−174.110	−173.610	−178.860	−179.020
3	R2	0.827	0.827	0.870	0.889	0.837
	AIC	−123.160	−123.160	−126.830	−130.000	−122.310
4	R2	0.950	0.950	0.948	0.950	0.950
	AIC	−159.160	−159.080	−156.320	−157.090	−157.210

illustrates the performance of the 10 analytical models in estimating the hydrodynamic parameters of soil layers. The coefficient of determination (R²) [45] and the Akaike information criterion (AIC) [46] were taken into account when selecting the most suitable model. Both of them evaluate how well the model fits the data. R² defines the correlation between observation and model simulation and AIC uses a number of independent parameters to build the model and the maximum likelihood of the model [47].

The results show that, for each layer, a different analytical model is appropriate. For the upper layer, composed of clayey sand, the retention curve was well represented by the Fredlund and Xing model (Figure 6a,b).

The hydrodynamic parameters are better estimated by the Van Genuchten dual permeability model (Dual-VG) for both layer 2 and 3. This can be explained by the high degree of macroporosity characteristic of swelling clay (Figure 6 and Figure 7). Retention of the fourth layer is best represented by the Van Genuchten model, as it is a purely sandy layer (Figure 7a,b).

Most of the models proved effective in approximating experimental measurements, with high correlation coefficients. Since the aim of this study is to compare the results obtained by different methods, we have adopted the Van Genuchten model (

Table 5. Estimated hydrodynamic parameters -VG model.

Layer ID	Parameter ID	Mean	STD	θr (cm3/cm3)	θs (cm3/cm3)	α (1/cm)	n (-)
1	θr (cm3/cm3)	0.0265	0.0020	1.0000
	θs (cm3/cm3)	0.5676	0.0061	0.1918	1.0000
	α (1/cm)	0.0081	0.0008	0.1340	0.5640	1.0000
	n (-)	2.0278	0.0541	0.3352	0.8226	0.4502	1.0000
2	θr (cm3/cm3)	0.0400	0.0016	1.0000
	θs (cm3/cm3)	0.4996	0.0450	0.7957	1.0000
	α (1/cm)	0.0110	0.0919	0.8501	0.9940	1.0000	0.9158
	n (-)	1.2626	0.2170	0.8686	0.9880	0.9158	1.0000
3	θr (cm3/cm3)	0.0152	0.0083	1.0000
	θs (cm3/cm3)	0.4595	0.0181	0.1340	1.0000
	α (1/cm)	0.0034	0.0023	0.2959	0.0697	1.0000
	n (-)	1.4355	0.4860	0.1875	0.9643	0.1482	1.0000
4	θr (cm3/cm3)	0.0148	0.0135	1.0000
	θs (cm3/cm3)	0.3437	0.5990	0.5256	1.0000
	α (1/cm)	0.0077	0.0166	0.5640	0.9940	1.0000
	n (-)	1.3637	0.1430	0.8836	0.7379	0.9235	1.0000

), which will be used in the inverse approach. For the same reason, and for the sake of harmony, the standard deviation and correlation coefficient have been determined for the parameters of the Van Genuchten model only.

3.3. Inverse Method

Inverse modeling is a method increasingly used in hydrology and soil parameter estimation. It involves determining the input parameters of a numerical flow model (known as the direct problem) by exploiting experimental measurements of model outputs such as pressures or water content [26].

In this work, we used an inverse model developed by [31] to estimate the parameters of the alluvial soil of Haouareb soil using measurements from the internal drainage test carried out on the soil column (Section 2.5).

3.3.1. Direct Problem Formulation

The water flow in unsaturated soil column is described by the following equation, known as the Richard equation [48]:

$${\displaystyle \frac{\partial \theta \left(h\right)}{\partial t}}+div\left[-K\left(h\right) \overrightarrow{\nabla }\left(h-z\right)\right]=0$$

(6)

where

h is the soil pressure head (L);

θ is the volumetric water content (L³/L³);

K is the soil hydraulic conductivity (L/T);

t is the time (T);

z is the vertical coordinate (positive downward).

The resolution of Equation (6) requires the determination of the initial and the boundary conditions as well as the soil hydraulic properties.

The following initial condition was set:

$$h\left(z,0\right)=h_0\left(z\right)$$

(7)

where h₀ is the measured initial soil pressure head [L].

During the internal drainage test, the upper boundary of the monolith was wrapped with a Plexiglas cover. Thus, a nil flux was assumed.

$$q_{top}=-K\left(h\right){\displaystyle \frac{d\left(h-z\right)}{dz}}=0$$

(8)

The free drainage at the bottom boundary was represented by a unit-gradient [49] as follows:

$$q_{bottom}=-K\left(h\right){\displaystyle \frac{d\left(h-z\right)}{dz}}=-K(h)$$

(9)

since

$${\displaystyle \frac{d\left(h-z\right)}{dz}}=1$$

(10)

The soil–water retention curve θ(h) and the soil hydraulic conductivity are described using the following Mualem–Van Genuchten functional equations:

$$S_e={\displaystyle \frac{\theta \left(h\right)-{\theta }_r}{{\theta }_s-{\theta }_r}}=\left\{\begin{array}{c}{[1+{\left|\alpha \left.h\right|\right.}^n]}^{-m} \\ 1 \end{array}\begin{array}{c}if h<0\\ if h\ge 0\end{array}\right.$$

(11)

$$K\left(h\right)=K_s{S}_e^l{\left[1-{\left(1-S_e^{{\displaystyle \frac{1}{m}}}\right)}^m\right]}^2$$

(12)

where:

θ_r is the volumetric residual water content (L³/L³);

θ_s is the volumetric saturated water content (L³/L³);

K_s is the saturated hydraulic conductivity (L/T);

α is the air entry parameter (1/L);

n is the pore size distribution parameter (-);

l is the pore connectivity parameter (-) which is always taken as 0.5;

m is a soil parameter given by: m = 1 − 1/n [50].

A cell-centered finite difference based model was developed by [31] using the Matlab tool to solve the Richard equation (Equation (6)) in a vertical soil profile with the cited above initial conditions (Equation (7)) and boundary conditions (Equation (9)).

3.3.2. Inverse Problem Formulation

The inverse method is used to estimate hydraulic properties θ_r, θ_s, K_s, α and n.

This method compares the measured volumetric water contents to those calculated by the direct problem model.

The objective function to be minimized, J, is the root least square error, calculated by

$$J\left(A,X\right)=W_{\theta } {‖A_{\theta }^c\left(X\right)-A_{\theta }^{\ast }‖}_2^2$$

(13)

$A_{\theta }^c\left(X\right)$ are the computed values of water content;

$A_{\theta }^{\ast }$ are the measured values of water content;

X = (θ_r, θ_s, K_s, n, α) is the vector of the optimized parameters;

$W_{\theta }={\displaystyle \frac{1}{\max \left(A_{\theta }^{\ast }\right)-min\left(A_{\theta }^{\ast }\right)}}$ is the weight associated with the measurements.

The optimization problem seeks to find the vector X(θ_r, θ_s, K_s, n, α) that minimizes the objective function J(A,X). The model can also use pressure or flow measurements if they are available.

With regard to the ill-posed property of the water flow problem in partially saturated soils [51], soil hydraulic property values are not unique. A powerful optimization algorithm is adopted in the Bound Optimization BY Quadratic Approximation (BOBYQA), which is an iterative algorithm that aims to find the smallest value of an objective function whose parameters are subject to simple bounds. This method does not require explicit calculation of the first derivative of the objective function [52].

3.3.3. Application to Soil Column

In order to avoid interference, the number of parameters has been reduced. The following assumptions were adopted [53,54]:

-

Firstly, and since the irreducible water content θ_r is considered the least sensitive parameter to model calibrations [55], it was set a priori at θ_r = 0.07.

-

Secondly, we noticed from the Rosetta results that the parameters of the Van Genuchten relationship, α and n, have minor variations from layer to layer. For this reason, we chose to identify a single value for each of the parameters for the entire column.

The parameters to be determined are K_s and θ_s for each of the four layers and α and n. Altogether, we have 10 parameters to identify.

Analysis of the measured pressure head data shows a lack of precision and inconsistencies that lead to unrealistic behavior. These are probably due to a malfunction of the measuring instruments. Therefore, we have only considered water content measurements for inverse identification.

The EM50 sensor did not respect the planned time step. In addition, as the hourly variation rate is insignificant, we only considered the measurements recorded every 10 days, thus reducing the mass of data supplied to the model.

We had to discard some measurements that we deemed to be inconsistent with the test, keeping only the water content measurements at depths of 5 cm, 55 cm, 90 cm, 140 cm, 160 cm and 190 cm. Therefore, we have at least one measurement point in each layer. Figure 8 shows the water content measured with a time step of 10 days. These data were used in the inverse identification. The variation in water content strengthened the subdivision of the soil column into layers, as the illustration clearly shows.

The 2 m column was subdivided into 20 cells. For the simulations, the time step varied from 60 to 7200 s. Rosetta results were used to set the upper and lower limits of the parameters to be provided to the optimizer (

Table 6. Inverse method parameters.

Parameter	Limit-Inf	Limit-Sup	Computed Value	STD
n (-)	1	2	1.5044	0.0178
α (1/cm)	0.01	0.04	0.0251	0.0007
poro1 (cm3/cm3)	0.4	0.5	0.4501	0.0114
poro2 (cm3/cm3)	0.4	0.5	0.4498	0.0129
poro3 (cm3/cm3)	0.4	0.5	0.4496	0.0066
poro4 (cm3/cm3)	0.25	0.35	0.2995	0.0134
k1 (m/s)	1.00 × 10−6	1.00 × 10−4	7.35 × 10−5	2.00 × 10−6
k2 (m/s)	1.00 × 10−6	1.00 × 10−4	6.97 × 10−5	2.00 × 10−6
k3 (m/s)	1.00 × 10−6	1.00 × 10−5	7.82 × 10−6	4.00 × 10−7
k4 (m/s)	1.00 × 10−6	1.00 × 10−3	8.04 × 10−4	3.00 × 10−5

). The optimizer converged after 36 iterations, with an objective function value of J_θ = 0.76. The pre-set parameter intervals were respected. None of the calculated values reached the thresholds. The final trust region radius RHOEND was 10⁻².

The identified parameters are shown in the fourth column of Table 6 (poro1, 2, 3, 4 and k1, 2, 3, 4 are, respectively, θ_S and K_s for layers 1, 2, 3 and 4.

3.3.4. Simulation Under Matlab

The inverse method provided the findings shown in Figure 9. It illustrates the calculated rate of soil saturation as a function of time and depth. In the initial state, the upper layer is saturated, then dries out. Layers 2 and 3, which are initially unsaturated, increase in saturation on the first day of internal drainage, then gradually desiccate. The bottom layer, made up of sandy soil, delaminates as water percolates through the overlying layers. Overall, the saturation profiles are consistent with an internal drainage test between day 1 and day 60.

In Figure 10, the pressure head are plotted as a function of time and depth. The initial profile is the data used in the simulations. It was determined from measurements on the column. The evolution of the profiles over time confirms the conclusions drawn from the saturation results plotted in Figure 9.

In Figure 11, we have chosen to represent the calculated water content profiles and experimental measurements for days 1, 20, 40 and 60. A comparison of these profiles shows that the values of the parameters identified by the inverse procedure correctly approximate the experimental measurements.

The calculations made it possible to reproduce variations in water content from one layer to the next. In addition, soil layers differentiated on the basis of textural data were also distinguished by soil moisture profiles.

4. Discussion

4.1. Comparison Using COMSOL Simulation

Modeling the water flow using the previously estimated parameters could not only provide information on the most efficient method but also allow for comparison of the impact of the model’s input data on its simulation.

The model used for this comparison is developed under COMSOL and adopts the finite element method for discretization of the partial differential equation here, the Richards equation (Equation (6)). It also takes into account the Van Genuchten retention model in order to simulate internal drainage.

Similarly, it considers the same initial and boundary conditions as those used previously in Matlab with a few minor variations. Indeed, COMSOL does not accept the zero value of the pressure head h as the saturation state of the soil, so we have considered an initial condition of −0.001 m at the top layer. On the other hand, no flow boundary condition was applied to the upper boundary of the monolith, assuming a closed surface. At the lower boundary, on the other hand, an outflow boundary condition was imposed in relation to the free drainage (Figure 12).

The estimated parameters were tested with this model for the same monolith. Three simulations were carried out in which the model inputs were, successively, the hydrodynamic parameters estimated by each method, namely Rosetta (Section 3.1), SWRC Fit (Section 3.2), and inverse problem (Section 3.3).The curves in Figure 13 illustrate the observed and simulated water content profiles as a function of depth. The curve shapes are suitable for the performed numerical test.

The discrepancies between the results can be explained by the state of the soil used for identification, as it was either undisturbed or reconstituted. In the case of the inverse problem, the behavior of the entire column was taken into account, including the influence of the assembly of the four layers, i.e., the effect of the flow from one layer to the next.

However, a particular problem is observed in the middle of the monolith, precisely at a depth of 100 cm. At this level, the observations do not match the simulations. This could be linked to the anomaly that occurred when the column was reconstituted. Indeed, at the start of COVID-19 confinement in March 2020, only 100 cm of soil were put in place; the soil remained dry for 1 month, and then the rest of the column was reconstituted. At this point, a dry clay crust formed. Also, this could be linked to the settling problem due to the silt component, which may create discontinuities in the middle of the soil column.

Consequently, this incident had an influence on the lower layers, layers 3 and 4, which could explain the differences between the calculated and measured values.

4.2. RMSE Calculation

To assess the performance of the models used in this study, the root mean square error (RMSE) is calculated for each method in

Table 7. RMSE for the three utilized methods.

Time (d)	Inverse Problem	Rosetta	VG-SWRC-Fit
1	0.049	0.070	0.069
10	0.054	0.064	0.063
20	0.060	0.050	0.066
30	0.058	0.048	0.068
40	0.059	0.074	0.099
50	0.055	0.078	0.116
60	0.050	0.082	0.129
Mean	0.055	0.067	0.087

. The RMSE ranges from 0.048 to 0.13 cm³/cm³. Overall, the lowest values were obtained using the inverse method, while the highest were generated by the VG-SWRC-Fit method. The lowest mean value is given by the inverse method.

4.3. Monte Carlo Simulation

The Monte Carlo simulation was carried out using typical Excel tools. Using the analysis tool, random variables were generated, 100 samples in the case of this study. The normal distribution was selected to provide a variable distribution based on the mean and standard deviation of the variable.

The uncertainty was assessed for all the estimated parameters. However, only the probability distribution of the parameters estimated by the inverse method is presented in this paper in order to reduce the number of figures.

Figure 14 shows that the rate of the estimated parameters is equal to the mean. The illustrations provide a better understanding of the accuracy of the estimated parameters from a statistical point of view.

5. Conclusions

This study aimed to characterize the alluvial soils of Haouareb and estimate their hydrodynamic behavior. We began with field sampling, extracting several soil samples from a 6 m deep auger hole.

Different analyses were applied to this soil. First, a granulomeric analysis was performed. Next, the bulk density was determined. The results of these tests were used to determine the SWRC parameters using Rosetta PRFs. Secondly, infiltration tests were carried out using Mini-Muntz to measure saturated hydraulic conductivity. Thirdly, simple evaporation tests were performed on undisturbed soil samples. During these tests, water content and water tension were monitored, enabling the SWRC to be determined using Seki’s SWRC-Fit platform [14]. Finally, a 2 m stratified column of reconstituted soil was constructed and fitted with sensors to monitor water content, EC and water tension across the soil layers. This monolith underwent an internal drainage test for two months. The measured data were used to solve an inverse unsaturated water flow problem to estimate the monolith’s hydrodynamic parameters.

An important task carried out in this work was to sort out the experimental values and critically examine reliability defects and measurement errors.

All these methods produced different results, partly due to the soil disturbance condition used in each method. To compare them, the internal drainage test was simulated using the estimated parameters of each method.

Overall, the simulated profiles show good agreement with the measured water content values. The results of the inverse problem are more relevant, as they deliver the lowest mean RMSE value. This is explained by the fact that this method originally used internal drainage data to estimate the parameters, and the influence of the four-layer assembly is therefore taken into account. In contrast, for the other methods, the soil layers were tested and characterized separately. The Monte Carlo simulation therefore proved to be an accurate estimation of the parameters.

Once the soil hydrodynamics have been identified, the salt transport parameters will be identified in order to simulate climate change impact scenarios on natural resources, namely water and soil in central Tunisia.