**4. Results and discussion**

#### **4.1. The granulometric curve**

*C x EC* = 640 (23)

} with the restriction 0<*sm*=1−2*s* / *n* <1; where *Θ* is the

where *C* is the concentration given in *mg* / *l* and *EC* the electrical conductivity given in *dS* / *m*

To solve the Boussinesq equation, the van Genuchten model [25] for the water retention curve was used, along with a model of hydraulic conductivity of Fuentes [27] namely geometric

The m and n form parameters from the water retention curve are obtained from the granulo‐ metric curve [35] adjusted with the equation *F* (*D*)= 1 + (*Dg* / *D*)*<sup>N</sup>* <sup>−</sup>*<sup>M</sup>* , where *F* (*D*) is the cumulative frequency, based on the weight of the particles whose diameters are less than or equal to *D*; *Dg* is a characteristic parameter of particle size, *M* and *N* are two form empirical

To evaluate the capacity of the numerical solution of the Advection-Dispersion Equation, the experimental information presented by [36] is used. The characteristics of the drainage module and the soil parameters used in the simulation are: *hs* =120 *cm*, *D*<sup>0</sup> =25 *cm*, *L* =100 *cm*, *ϕ* =0.5695

and Fuentes [25,27]. The scale parameters (*ψ<sup>d</sup>* , *Ks*) are obtained from the inverse problem, using the experimental drained depth and the drained depth calculated with the numerical solution of the Boussinesq equation [10], given an error criterion between the previous and the new estimator (1*x*10−<sup>12</sup> *cm*), using a constant head test and fractal radiation condition with variable storage capacity and a nonlinear optimization algorithm [37]. The calculations were performed on a dual-core AMD Opteron machine with 2.6 GHz CPU and 8 GB RAM. The computational

In order to model the salt concentration in the soil profile, with the numerical solution of the solute transport, the hydraulic parameters obtained from the previous analysis were used. In the numerical solution, the unknown parameter is the dispersivity coefficient (*λ*), which is estimated by minimizing the sum of squares errors between the salt concentration measured and the salt concentration calculated with the numerical solution over time, using a Levenberg-Marquardt [37], given an error criterion between the previous and the new estimator (1*x*10−<sup>9</sup> *g* / *l*). The initial condition is the sample initial, taken as a constant in all the system and

, and *s* =0.7026. The hydrodynamic characteristics used are those of van Genuchten

parameters. These parameters are rewritten as follows: *M* =*m* and *N* = 1 / 2(1−*s*) *n*.

or *mmhos* / *cm*.

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**3.3. The hydrodynamic characteristic**

mean model {*K*(*Θ*)= *Ks* <sup>1</sup>−(1−*<sup>Θ</sup>* 1/*m*)*sm* <sup>2</sup>

**3.4. The granulometric curve**

**3.5. Inverse problem**

*cm*<sup>3</sup> / *cm*<sup>3</sup>

effective saturation defined by *Θ* =(*θ* −*θr*) /(*θ<sup>s</sup>* −*θr*).

time required to solve the inverse problem was 5 h.

radiation as the boundary condition applied in the drains.

The adjusted parameters are shown in Table 1. Figure 3 shows the experimental granulometric curve and best fit is obtained with *Dg* =36.2993 *μm* and *m*=0.3410 with a root mean square error *RMSE* =0.1477.


**Figure 3.** The experimental granulometric curve and adjusted with the model

#### **4.2. The hydrodynamic characteristic**

In order to obtain the values of *ψd* and *Ks*, the spatial and temporal increments used in all the simulation are *Δz* =0.0010 *cm* and *Δt* =5*x*10−<sup>5</sup> *h* . Figure 4 shows the experimental drained depth and the drained depth calculated with the finite difference solution [10], using a storage capacity variable, fractal radiation condition in the drains and the geometric mean model. To linearize the boundary condition, one generalization of the conductance coefficient is opti‐ mized (*κ*) [8,10]. The residual volumetric water content is considered to be zero (*θ<sup>r</sup>* =0.0 *cm*<sup>3</sup> / *cm*<sup>3</sup> ) [38].

**Figure 4.** Comparison between the experimental drained depth and the calculated drained depth.

#### **4.3. Analysis of the salt content**

The EC data are shown in Figure 5 using a 2.5% like correction factor. Applying equation (23) to the data shown in Figure 5, we obtain the concentration in grams per liter (see Figure 6). The initial condition using in the numerical solution is the sample initial (*Cini* =2.4*g* / *l*), taken as a constant in all the system and radiation as the boundary condition applied in the drains. The dispersivity value obtained is *λ* =91.80 *cm*, with RMSE = 0.1063 *g* / *l* between the experi‐ mental values and the values obtained from the numerical solution. The computational time required to solve the advection-dispersion model was 2.7 h. The dispersivity value found is only for this soil, because this value change with depth [39], increase with the flow rate and is a soil type function. This increase was explained by the activation of large pores at higher flow rates [40]. Figure 6 shows the experimental salt concentration evolution and the concentration obtained with the numerical solution.

Comparison shows that the salt concentration obtained with the numerical solution, according to RMSE, reproduce the experimental salt concentration. Figure 7 shows that in the short time, when the water flow increased, the salt concentration increases sharply, and in the long time tends to an asymptote, indicating that the system could not continue removing salts from the system. However, the value of the dispersivity obtained (*λ* =91.80 *cm*) overestimates the measured data in the long time. Second simulation was performed with the accumulated mass. To obtain the accumulated mass, it is necessary to obtain the solute flow, which is estimated by multiplying the water flow by the measured salt concentration in the time interval (see Figure 7). The cumulative solute mass is obtained by multiplying the solute flow by the time interval (see Figure 8).

**Figure 5.** Evolution of the electrical conductivity of drainage water

**4.3. Analysis of the salt content**

obtained with the numerical solution.

interval (see Figure 8).

**Cummulative drained depth (cm)**

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0 20 40 60 80 100 120 140 160 **Time (h)**

The EC data are shown in Figure 5 using a 2.5% like correction factor. Applying equation (23) to the data shown in Figure 5, we obtain the concentration in grams per liter (see Figure 6). The initial condition using in the numerical solution is the sample initial (*Cini* =2.4*g* / *l*), taken as a constant in all the system and radiation as the boundary condition applied in the drains. The dispersivity value obtained is *λ* =91.80 *cm*, with RMSE = 0.1063 *g* / *l* between the experi‐ mental values and the values obtained from the numerical solution. The computational time required to solve the advection-dispersion model was 2.7 h. The dispersivity value found is only for this soil, because this value change with depth [39], increase with the flow rate and is a soil type function. This increase was explained by the activation of large pores at higher flow rates [40]. Figure 6 shows the experimental salt concentration evolution and the concentration

Comparison shows that the salt concentration obtained with the numerical solution, according to RMSE, reproduce the experimental salt concentration. Figure 7 shows that in the short time, when the water flow increased, the salt concentration increases sharply, and in the long time tends to an asymptote, indicating that the system could not continue removing salts from the system. However, the value of the dispersivity obtained (*λ* =91.80 *cm*) overestimates the measured data in the long time. Second simulation was performed with the accumulated mass. To obtain the accumulated mass, it is necessary to obtain the solute flow, which is estimated by multiplying the water flow by the measured salt concentration in the time interval (see Figure 7). The cumulative solute mass is obtained by multiplying the solute flow by the time

**Figure 4.** Comparison between the experimental drained depth and the calculated drained depth.

Experimental Calculated

**Figure 6.** Comparison between the experimental and the calculated drainage water salt concentration with numerical solution

The results obtained with the numerical solution, the solute flow and cumulative mass evolution are shown in Figures 7 and 8, respectively, which demonstrate that the reproductions of the data were acceptable. The solute flow decreases rapidly, as seen in Figure 7 the concen‐ tration decreased 3.5 *g* / *l* after 20 hours. In the long time, the theoretical water flow and experimental water flow tends to be constant. Comparison showed that the solute flow and the cumulative mass evolution obtained with the numerical solution, according to RMSE, reproduce the experimental salt concentration. The RMSE values for estimating the solute flow and cumulative solute mass were 0.1842 *g* / *l* and 0.1104 *g*, respectively. The dispersivity value obtained is *λ* =98.03 *cm*, with *RMSE* =0.1010 *g* between the experimental values and the values obtained from the numerical solution. The dispersivity value for this new optimization (cumulative mass evolution) compared to the previous (salt concentration evolution) increases 6.2 cm.

**Figure 7.** Solute flow (g/h) in the drainage system.

**Figure 8.** Cumulative mass evolution: experimental and obtained with the numerical solution.

### **4.4. Using the solution to simulate the leaching of saline soils**

reproduce the experimental salt concentration. The RMSE values for estimating the solute flow and cumulative solute mass were 0.1842 *g* / *l* and 0.1104 *g*, respectively. The dispersivity value obtained is *λ* =98.03 *cm*, with *RMSE* =0.1010 *g* between the experimental values and the values obtained from the numerical solution. The dispersivity value for this new optimization (cumulative mass evolution) compared to the previous (salt concentration evolution) increases

> 0 20 40 60 80 100 120 140 160 **Time (h)**

> 0 20 40 60 80 100 120 140 160 **Time (h)**

**Figure 8.** Cumulative mass evolution: experimental and obtained with the numerical solution.

Experimental Calculated

Experimental Calculated

6.2 cm.

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0.0

0

10

20

30

**Cumulative mass evolution (g)**

40

50

60

**Figure 7.** Solute flow (g/h) in the drainage system.

1.0

2.0

3.0

**Solute flow (g/l)**

4.0

5.0

6.0

To recover saline soils it is necessary to apply irrigation so that the salts are transported to deeper horizons without harming the roots and are evacuated to other areas through the drainage channel. For purposes of illustrating the leaching of salts in the soil by applying the finite difference solution, we assumed a soil with hydraulic and hydrodynamic characteristics previously found. The initial soil concentration was 10 *dS* / *m* and the problem was reduced to finding the number of irrigation that must be applied to carry a given concentration.

The final average concentration obtained in the profile at the end of the first simulation was the initial concentration in the system for the next simulation, and so on. Figure 9 shows the reduced concentration of salts in the soil profile based on an initial concentration. The values shown are an average concentration in the soil profile at 1-m depth. Depth of drains was assumed to be 2.0 m.

**Figure 9.** Evolution of the salt concentration in the soil by applying the leaching.

The simulations were performed with 5, 10, 15, 20 and 25 m of drains distances. It can be seen that the decrease in the concentration of salts in the soil profile is similar in all the separations between drains after applying 6 leachings. However, the time of drainage in each system was different. For example, with 5 days and 5 m of separation a decrease of the water table profile was more than one meter, while in the system with separation of 25 m decreased gives only a few centimeters (see Figure 10), other simulations was realized with 5, 10, 15, 20 and 25 m of drains distances, but the depth of drains was assumed to be 1.5 m (see Figure 11), therefore the time of drainage of the soil was a function of the distance between drains

**Figure 10.** Decrease of the midpoint water table at different separations between drains under a drain depth of 2.00 m.

**Figure 11.** Decrease of the midpoint water table at different separations between drains under a drain depth of 1.50 m.
