Values of the adjusted parameters from the granulometric curve and the drained depth.
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 . The problem worsens in arid and semiarid areas, in soils with insufficient drainage  and high evaporation . In Mexico, there are 6.46 million ha irrigated mainly in the center and north territory areas ; 10-30 % of irrigated land is affected by salinity and nearly two thirds of this area is located in the North .
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 .
The dynamics of water drainage systems has been studied by applying the Boussinesq equation  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 , some of the more popular models include SWAP , HYDRUS-1D , STANMOD (STudio of ANalytical MODels) , UNSATH  and COUP  but the majority of applications for water flow in the vadose zone requires a numerical solution of the Richards equation , 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.
2.1. 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 ) are negligible with respect to head variations in the cross section (direction ). It is the one-dimensional Boussinesq equation which is a result of the continuity equation, , and the Darcy’s law, , namely:
where is the storage capacity, is the elevations of the free surface or hydraulic head above the impervious layer , and is a function the horizontal coordinate () and the time (), is the transmissivity given by , is the volume of recharge in the unit of time per unit of the aquifer , is the drainable porosity as a head function, and is the saturated hydraulic conductivity .
The storage capacity, see , is: , where is the saturated volumetric water content , and represents the water content evolution in the position , while the free surface decreases, and is the elevation of ground surface .
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  was accepted in field and laboratory studies: , where is the soil water potential defined by , is the pressure scale parameter , is the saturated volumetric water content , is the residual volumetric content , and are parameters (dimensionless) that determine the shape of the soil water retention curve. The introduction of this equation in the storage capacity results in the following expression for storage capacity: .
The saturated volumetric water content can be assimilated to the soil porosity , dimensionless, this is calculated with the formula , where is the bulk density and is the particles density ; the residual volumetric water content is considered to be zero.
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 . 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 . 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 is associated with the head measured from above the drains using: , where is the distance from the impermeable barrier to the drains . Transversal variation of h at the beginning is considered as the initial condition , where is the head on the drain in the initial time . The fractal radiation condition for the Boussinesq equations is given by :
where the positive sign corresponds to and the negative sign to .
2.4. Solute transport equation
where is the diffusion coefficient in the water ; is the total solute concentration in soil ; is the solute concentration in water ; and is the term which includes gains or losses of the solute due to chemical reactions and the extraction plant . Note that and are obtained from the water flow model. The diffusion coefficient in the water is calculated by , where is the dispersivity and the interstitial velocity of water calculated by .
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 (), expressed in units of mass per volume of soil can be written as: , where is the concentration of the adsorbed compound and is a function of the concentration of the solute in the mobile phase () and the adsorption constant of the solute to the stationary phase surface , , namely linear isotherm. Thus, the concentration of the substance compared to the volume of the porous medium () will be the result of a part that is in the water, air and the dynamic equilibrium with the phase that generates it. Generally, in studies in small time scales, such as irrigation and drainage in a porous medium, the gas phase is not considered . Thus, in this work, the concentration in the adsorbed phase, the concentration in the gas phase and the term are ignored.
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 , for which two interpolation parameters are introduced: and , where and ; and are the space and time indices, respectively.
The dependent variable in an intermediate node for all is estimated as:
while the intermediate time for all is estimated as:
The discretization of the temporal derivative in the equation (3) is:
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:
Substituting equations (6) and (17) in the continuity equation, the following algebraic equations system is obtained:
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
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 () and the border, for all time. The linear radiation condition is due originally to Newton, who postulated that the heat flow at the border of a body is proportional to the temperature difference between the body and the medium that surrounds it; the result is equivalent to a Ohm law into electricity. To linearize these conditions, we introduce a generalization of the dimensionless conductance coefficient (), as follows: . If we observe the one-dimensional equation of solute transport, the dimensionless conductance coefficient must be zero by the advective component, however, the solution is allowed only for purposes of illustration to derive the boundary conditions.
2.7. Selection of the space and time increments
In reference  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  concluded that the optimal interpolation that minimizes the sum of the squares errors are (cm) and (h), for space and time respectively.
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  and . The module dimensions are: cm, cm and cm. The drain diameter is cm and the drain length is cm.
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 and the recharge was null during the drainage phase. Soil porosity was calculated with the formula (the bulk density was determined by the weight and volume of the soil of drainage module and the particles density , was obtained). The soil fractal dimension obtained was equal to 0.7026.
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  is 2-3 % for every Celsius degree that is measured under standard temperature. According with  (1964), the relationship between electrical conductivity and concentration is:
where is the concentration given in and the electrical conductivity given in or .
3.3. The hydrodynamic characteristic
To solve the Boussinesq equation, the van Genuchten model  for the water retention curve was used, along with a model of hydraulic conductivity of Fuentes  namely geometric mean model with the restriction ; where is the effective saturation defined by .
3.4. The granulometric curve
The m and n form parameters from the water retention curve are obtained from the granulometric curve  adjusted with the equation , where is the cumulative frequency, based on the weight of the particles whose diameters are less than or equal to ; is a characteristic parameter of particle size, and are two form empirical parameters. These parameters are rewritten as follows: and .
3.5. Inverse problem
To evaluate the capacity of the numerical solution of the Advection-Dispersion Equation, the experimental information presented by  is used. The characteristics of the drainage module and the soil parameters used in the simulation are: , , , , and . The hydrodynamic characteristics used are those of van Genuchten and Fuentes [25,27]. The scale parameters are obtained from the inverse problem, using the experimental drained depth and the drained depth calculated with the numerical solution of the Boussinesq equation , given an error criterion between the previous and the new estimator , using a constant head test and fractal radiation condition with variable storage capacity and a nonlinear optimization algorithm . 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 , 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 , given an error criterion between the previous and the new estimator . 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.
4. Results and discussion
4.1. The granulometric curve
|Geometric mean model||1.5458||143.87||0.0616||0.2195|
4.2. The hydrodynamic characteristic
In order to obtain the values of and , the spatial and temporal increments used in all the simulation are and . Figure 4 shows the experimental drained depth and the drained depth calculated with the finite difference solution , 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 optimized [8,10]. The residual volumetric water content is considered to be zero ( ) .
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 , taken as a constant in all the system and radiation as the boundary condition applied in the drains. The dispersivity value obtained is , with RMSE = between the experimental 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 , 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 . 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 ( ) 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).
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 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 and , respectively. The dispersivity value obtained is , with 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.
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 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.
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
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.
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