**3. Application**

#### **3.1. Laboratory experiment**

To evaluate the descriptive capacity of the numerical solution, a drainage experiment was conducted in a laboratory. The drainage module (see Figure 1) is the one used by [8] and [10]. The module dimensions are: *L* =100 cm, *Hs* =120 cm and *Do* =25 cm. The drain diameter is *d* =5 cm and the drain length is ℓ=30 cm.

**Figure 1.** Drainage module

The module was filled with altered sample of salty soil of Celaya, Guanajuato, México (see Figure 2). Soil passed through a 2 mm sieve and was disposed on 5 cm thick layers, in order to maintain the bulk density at a constant value. The soil was saturated by applying a constant water head (no salt) on its surface until the entrapped air was virtually removed. Once the drains were closed, the water head was removed from the soil surface; the surface of the module was then covered with a plastic in order to avoid evaporation. Finally, the drains were opened to measure the drained water volume; the initial condition was equivalent to *h* (*x*, 0)=*hs* and the recharge was null *Rw* =0 during the drainage phase. Soil porosity (*ϕ*) was calculated with the formula *ϕ* =1−*ρ<sup>t</sup>* / *ρ<sup>o</sup>* (the bulk density was determined by the weight and volume of the soil of drainage module *ρ<sup>t</sup>* =1.14 *<sup>g</sup>* / *cm*<sup>3</sup> and the particles density *ρ<sup>s</sup>* =2.65 *<sup>g</sup>* / *cm*<sup>3</sup> , *ϕ* =0.5695 *cm*<sup>3</sup> / *cm*<sup>3</sup> was obtained). The soil fractal dimension obtained was equal to 0.7026.

**Figure 2.** Study site

observe the one-dimensional equation of solute transport, the dimensionless conductance coefficient (*κs*) must be zero by the advective component, however, the solution is allowed only

In reference [10] the authors discuss the selection of spatial and temporal increments pointing out a comparison of the depletion of the free surface for all time between the results obtained with the finite difference solution of the Boussinesq equation and the results obtained with an analytical solution reported in the literature. The same authors [10] concluded that the optimal interpolation that minimizes the sum of the squares errors are *γ* =0.5*Δx* (cm) and *ω* =0.98*Δt* (h),

To evaluate the descriptive capacity of the numerical solution, a drainage experiment was conducted in a laboratory. The drainage module (see Figure 1) is the one used by [8] and [10]. The module dimensions are: *L* =100 cm, *Hs* =120 cm and *Do* =25 cm. The drain diameter is *d* =5

for purposes of illustration to derive the boundary conditions.

**2.7. Selection of the space (***Δx***) and time (***Δt***) increments**

for space and time respectively.

**3.1. Laboratory experiment**

cm and the drain length is ℓ=30 cm.

**3. Application**

108 Agroecology

**Figure 1.** Drainage module

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

During the module drainage process (154 h), measurements of pH, temperature and electrical conductivity of water samples were made at defined time intervals (each hour the first 20 hours and subsequently increased to the range 2, 4, 6 and 8 h). The sensor used for measurement is a CONDUCTRONIC PC 18 sensor. The electrical conductivity at room temperature was recorded with it. However, in order to accurately quantify conductivity, it is important to consider a standard value of 25° C, which can be used to correct the values obtained. The correction factor used in accordance with [34] is 2-3 % for every Celsius degree that is measured under standard temperature. According with [34] (1964), the relationship between electrical conductivity and concentration is:

$$C = 640 \ge EC \tag{23}$$

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

#### **3.3. The hydrodynamic characteristic**

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 mean model {*K*(*Θ*)= *Ks* <sup>1</sup>−(1−*<sup>Θ</sup>* 1/*m*)*sm* <sup>2</sup> } with the restriction 0<*sm*=1−2*s* / *n* <1; where *Θ* is the effective saturation defined by *Θ* =(*θ* −*θr*) /(*θ<sup>s</sup>* −*θr*).

#### **3.4. The granulometric curve**

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

#### **3.5. Inverse problem**

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 *cm*<sup>3</sup> / *cm*<sup>3</sup> , and *s* =0.7026. The hydrodynamic characteristics used are those of van Genuchten 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 time required to solve the inverse problem was 5 h.

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 radiation as the boundary condition applied in the drains.
