## 1. Introduction

Soil salinity is a worldwide problem as well as in Central and Northern Mexico. Nearly 8.4 million ha worldwide are affected by soil salinity and alkalinity, of which about 5.5 million ha are waterlogged [1]. The problem worsens in arid and semiarid areas, in soils with insufficient drainage [2] and high evaporation [3]. In Mexico, there are 6.46 million ha irrigated mainly in the center and north territory areas [4]; 10-30 % of irrigated land is affected by salinity and nearly two thirds of this area is located in the North [5].

The salinization of these irrigated areas is an increasing problem and the lands are abandoned; therefore, a technical and economic alternative to recover this land is needed. Agricultural subsurface drainage is a solution which takes into account the technology by environment interaction, as well as lowering the water table levels along with the salt concentration in the soil profile [1].

The dynamics of water drainage systems has been studied by applying the Boussinesq equation [6] for unconfined aquifers using the finite element technique [7,8] and the finite difference [9,10], and the solutes dynamics has been studied by applying the Fick’s law [11,12,13]. These results in the advection-dispersion equation, namely by gravity and Fick’s law.

The solutes are also found in the gas phase and adsorbed by soil in the solid phase, the first phase is disregarded for purposes of transport modeling in water, but it is really important in terms of the amount of fertilizer transferred into the atmosphere at a given time [14,15,16], and incorporating the adsorbed substance in the solid phase. The relationship between the substance which transported by the water flow and the substance which adsorbed and exchanges in the soil solid structure is known as the adsorption isotherm [11,12,13].

A large number of models for simulating solute transport in the unsaturated zone are now increasingly being used for a wide range of applications in both research and management [17], some of the more popular models include SWAP [18], HYDRUS-1D [19], STANMOD (STudio of ANalytical MODels) [20], UNSATH [21] and COUP [22] but the majority of applications for water flow in the vadose zone requires a numerical solution of the Richards equation [23], also requires more calculation time in order to find the equation solution.

This study aims to solve the one-dimensional advection-dispersion equation using the technique of finite differences, coupled with the Boussinesq equation in order to model the transport of solutes in subsurface drainage systems, assuming that the solute is concentrated in the liquid phase.

## 2. Theory

### 2.1. The Boussinesq equation

In the study of the water dynamics in agricultural subsurface drainage systems using the Boussinesq equation, the variations in hydraulic head along the drain pipes (direction

where

The storage capacity, see [24], is:

### 2.2. The drainable porosity

To calculate the storage capacity and the drainable porosity it is necessary to provide the soil water retention curve. The model of van Genuchten [25] was accepted in field and laboratory studies:

The saturated volumetric water content can be assimilated to the soil porosity

### 2.3. Initial and boundary conditions

To study the agricultural drainage with equation (1), the initial and boundary conditions should be defined at the domain. The initial condition is established from the water table position at the initial time. Dirichlet and Neumann boundary type conditions can be used on drains to solve equation (1), the pressure head on the drains is required in the first condition whereas the drainage flux is required in the second one [8]. A third type of boundary condition is a linear combination of the precedent conditions; this condition includes a resistance parameter to the flow at the soil–drain interface. Null resistance corresponds to the Dirichlet condition and infinite resistance corresponds to Neumann condition. The third condition is a radiation type condition [26]. In the case of drainage, the radiation condition establishes that drainage flux is directly proportional to the pressure head on the drain and inversely proportional to the resistance in the interface between soil and the drainpipe wall in concordance to the Ohm law.

The hydraulic head measured above the impermeable barrier

where the positive sign corresponds to *L* is the distance between drains;

### 2.4. Solute transport equation

The advection-dispersion equation used to study the solute transport [28,29,30], in a one-dimensional form, is a result of the continuity equation,

where

The water soluble compounds which have a negligible vapor pressure can exist in three phases of soil: 1) dissolved in water, 2) as vapor in the soil atmosphere and 3) as stationary phase adsorbed to soil organic matter or in the clay mineral surfaces [11,12,13]. The total concentration of the compound (

### 2.5. Numerical scheme

The numerical scheme used is based on the assumption that the solute is concentrated mainly in the liquid phase. Thus, the advection-dispersion equation in one-dimensional is given by equation (3). To solve this equation, we use the same discretization scheme to transfer water in the Boussinesq equation [10], for which two interpolation parameters are introduced:

The dependent variable

while the intermediate time

The discretization of the temporal derivative in the equation (3) is:

(6) |

where:

The spatial derivative discretization in the continuity equation is:

According with the dynamic law:

According with the equation (4), the spatial interpolation is:

and according with the equation (5) the temporal interpolation is

The dependent variables involved in the advective term of the equations (9) and (10) are defined by:

while the dependent variables involved in the dispersive term of the same equations are defined by:

Equation (8) considering equations (9) and (10) can be written as:

where:

(16) |

Substituting equations (12)-(14) in equation (15) and associating similar terms allows obtaining:

(17) |

Substituting equations (6) and (17) in the continuity equation, the following algebraic equations system is obtained:

where

The water flow and the head are obtained from the Boussinesq equation solution, so that they should be included in the system (18). To find the solution of the water transfer equation, it is necessary to specify the initial and boundary conditions, equation (18) can be solved with the Thomas Algorithm, see [31,10].

The Thomas algorithm, also known as the tridiagonal matrix algorithm (TDMA), is a simplified form of Gaussian elimination that can be used to solve *tridiagonal* matrix systems (equation 18) [32]. It is based on

### 2.6. Linear radiation condition

The radiation boundary condition, or mixed condition, is used to accept a linear variation between the dispersive flux and concentration difference with the external medium (

### 2.7. Selection of the space ( Δ x ) and time ( Δ t ) increments

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

## 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:

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

### 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:

where

### 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

### 3.4. The granulometric curve

The m and n form parameters from the water retention curve are obtained from the granulometric curve [35] adjusted with the equation

### 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:

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

## 4. Results and discussion

### 4.1. The granulometric curve

The adjusted parameters are shown in Table 1. Figure 3 shows the experimental granulometric curve and best fit is obtained with

### 4.2. The hydrodynamic characteristic

In order to obtain the values of

### 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

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 (

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 concentration decreased 3.5

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

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

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.

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

## 5. Conclusions

The irrigation in the arid and semi-arid regions to sustain agricultural production against the unpredictable of the rainfall have resulted in the double problem of salinity in many hectares of good agricultural land. Subsurface drainage systems are used to control the depth of the water table and to reduce or prevent soil salinity.

The advection-dispersion equation was solved in order to model the temporal evolution of the concentration of salts removed through an agricultural drainage system with the method of finite differences. The solution requires the values of the flow of water previously obtained from the solution of the Boussinesq equation. The hydrodynamic characteristic were obtained by the inverse problem from the depth drained.

The optimization of the accumulated mass which gave better results in terms of mean square error criterion between the theoretical and experimental values, since it is a property integrated in the time and concentration observed at specific levels. The solution presented coupled to the Boussinesq equation, satisfactorily reproduced the measured data, both in the short time where the change in concentration was high, as in long times where the concentration values tended to an asymptote. This asymptotic value of the concentration depended on the distance between drains of the drainage system.

Finally, the solution of differential equations of transfer processes of water and solute transport, and hydrodynamic characterization of the soil in an agricultural drainage system, will be a useful tool for designing new systems for the optimal development of crops according to their water needs and the degree of tolerance to salinity. In addition, this study can be help us for quantify crop yield reductions due to salinity on irrigation areas, in order to prevent future problems such as food shortages.