Values of the adjusted parameters from the granulometric curve and the drained depth.
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]; 1030 % 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 advectiondispersion 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], HYDRUS1D [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 onedimensional advectiondispersion 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
2.4. Solute transport equation
The advectiondispersion equation used to study the solute transport [28,29,30], in a onedimensional 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 advectiondispersion equation in onedimensional 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:
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:
Substituting equations (12)(14) in equation (15) and associating similar terms allows obtaining:
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
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 23 % 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 AdvectionDispersion 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








(cm/h)  (cm)  (nondimensional)  (cm)  
Geometric mean model  1.5458  143.87  0.0616  0.2195 
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 1m 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 semiarid 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 advectiondispersion 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.
References
 1.
Ritzema HP., Satyanarayama TV., Raman S., Boonstra J. Subsurface Drainage to Combat Waterlogging and Salinity in Irrigated Lands in India: Lessons Learned in Farmers’ Fields. Agricultural Water Management 2008; 95 (3) 179189.  2.
Mousavi1 SF., MostafazadehFard B., Farkhondeh A., Feizi M. Effects of Deficit Irrigation with Saline Water on Yield, Fruit Quality and Water Use Efficiency of Cantaloupe in an Arid Region. Journal of Agricultural Science Technology 2009; 11 469479.  3.
RuizCerda E., Aldaco NR., Montemayor TJ., Fortis HM., Olague RJ., Villagómez GJ. 2007. Aprovechamiento y mejoramiento de un suelo salino mediante vermicomposta. Tecnologías Pecuarias en Mexico 2007; 45 (1) 1924.  4.
CONAGUA. Statistics on Water in Mexico (in Spanish). México. D.F; 2010.  5.
IMTA. Manual de Diseño e Instalación de Drenaje Parcelario en Zonas áridas y Semiáridas bajo Riego. México; 1998.  6.
Boussinesq J. 1904. Recherches The´oriques sur L’e´coulement des Nappes d’eau Infiltre´es Dans le Sol et Sur le De´bit des Sources. J. Math. Pure. Appl., 1904; 10 578.  7.
Verhoest N., Pauwels V., Troch P., de Troch F. Analytical Solution for Transient Water Table Heights and Outflows from Inclined DitchDrained Terrains. J. Irrig. Drain Eng., 2002; 128(6) 358–364.  8.
Zavala M., Fuentes C., Saucedo H. Nonlinear Radiation in the Boussinesq Equation of Agricultural Drainage. J. Hydrol. 2007; 332(3) 374380.  9.
Singh S., Ghosh NC., Pandey RP., GalkateRV., Thomas T., Jaiswal RK. Numerical Solution of 1D Boussinesq Equation for Water Table Fluctuation between Drains in Response to Recharge and ET in A Sloping Aquifer. Int. J. Eco. Econ. Stat. 2009; 14(9) 4554.  10.
Chávez C., Fuentes C., Zataráin F., Zavala M. Finite Difference Solution of the Boussinesq Equation with Variable Drainable Porosity and Fractal Radiation Boundary Condition. Agrociencia 2011; 45(8) 911927.  11.
Taylor GI. The Disperson of Matter in Turbulent Flow Through a Pipe. Proc. R. Soc. London, Ser. A. 1954; 223 446–48.  12.
Elder JW. The Dispersion of Marked Fluid in Turbulent Shear Flow. J. Fluid Mech., 1959; 5 544560.  13.
Fischer HB. The Mechanics of Dispersion in Natural Streams. J. Hydraul. Div., Am. Soc. Civ. Eng., 1967; 93(6) 187216.  14.
Holly FM. Two Dimensional Mass Dispersion in Rivers. Hydrologic papers, Colorado State University Press, Fort Collins, Colorado, 1975; pp. 78.  15.
Holly FM. Dispersion in Rivers and Coastal Waters, 1, Physical Principles and Dispersion Equations. In: Novak, P. (ed). Developments in Hydraulic Engineering, 3. Elsevier, New York. 1985; Chap. 1: 138.  16.
Rutherford JC. River mixing, Wiley, New York; 1994.  17.
Mirabzadeh M., Mohammadi K. A Dynamic Programming Solution to Solute Transport and Dispersion Equations in Groundwater. J. Agric. Sci. Technol. 2006; 8 233241.  18.
van Dam JC., Huygen J., Wesseling JG., Feddes RA., Kabat P., van Valsum PEV., Groenendijk P., van Diepen CA. Theory of SWAP, Version 2.0. Simulation of Water Flow, Solute Transport and Plant Growth in the Soil WaterAtmosphere Plant Environment, Department Water Resources, WAU, Report 71, DLO Winand Staring Centre, Technical Document 45, Wageningen; 1997.  19.
Simunek J., Sejna M., van Genuchten MTh. The HYDRUS1D software package for simulating the movement of water, heat, and multiple solutes in variably saturated media, version 2.0, United States Salinity Laboratory, USDAARS, Riverside, Calif; 1998.  20.
Simunek J., van Genuchten MTh., Sejna M., Toride N., Leij FJ. The STANMOD Computer Software for Evaluating Solute Transport in Porous Media Using Analytical Solutions of ConvectionDispersion Equation, Versions 1.0 and 2.0, IGWMC – TPS71, International Ground Water Modeling Center, Colorado School of Mines: Golden; 1999.  21.
Fayer MJ. UNSATH Version 3.0: Unsaturated Soil Water and Heat Flow Model. Theory, User Manual, and Examples. Pacific Northwest National Laboratory 13249; 2000, USA, 184 pp.  22.
Jansson PE., and Karlberg L. Coupled Heat and Mass Transfer Model for SoilPlantAtmosphere Systems, Royal Institute of Technology, Department of Civil and Environmental Engineering: Stockholm; 2001.  23.
Richards LA. Capillary conduction of liquids through porous mediums. Physics 1931; 1 318333.  24.
Fuentes C., Zavala M., Saucedo H. Relationship between the Storage Coefficient and the SoilWater Retention Curve in Subsurface Agricultural Drainage Systems: Water Table Drawdown. J. Irrig. Drain. Eng., 2009; 135(3) 279285.  25.
van Genuchten MTh. A ClosedForm Equation for Predicting the Hydraulic Conductivity of the Unsaturated Soils. Soil Sci. Soc. Amer. J. 1980; 44 892898.  26.
Carslaw HS., Jaeger JC. Conduction of Heat in Solids. Oxford University Press, Oxford; 1959.  27.
Fuentes C., Brambila F., Vauclin M., Parlange JY., Haverkamp R. Fractal modeling of Hydraulic Conductivity in NonSaturated soils. Hydraul. Eng. México 2001; 16(2) 119137.  28.
Abassi F., Simunek J., van Genuchten MTh., Feyen J., Adamsen FJ., Hunsaker DJ., Strelkoff TS., Shouse P. Overland Flow and Solute Transport: Model Development and FieldData Analysis. J. Irrig. Drain. Eng. 2003; 129(2) 71–81.  29.
Zerihun D., Furman A., Warrick AW,. Sánchez CA. Coupled SurfaceSubsurface Solute Transport Model for Irrigation Borders and Basin. I. Model Development. J. Irri. Drain. Eng. 2005; 131(3) 396406.  30.
Simunek J. Models of Water Flow and Solute Transport in the Unsaturated Zone. Encyclopedia of Hydrological Sciences. Edited by M G Anderson; John Wiley & Sons, Ltd., Chichester, England, 2005; 11711180  31.
Zataráin F., Fuentes C., Palacios VOL., Mercado E., Brambila F., Villanueva N. Modelación del Transporte de Agua y Solutos en el Suelo (In Spanish). Agrociencia 1998; 32(4) 373383.  32.
Freund RW., Hoppe RW. Stoer/Bulirsch: Numerische Mathematik 1. SpringerLehrbuch, Germany; 2007.  33.
Conte SD., De Boor C. Elementary Numerical Analysis: An Algorithmic Approach. McGrawHill, New York; 1980.  34.
Villareal E., Bello S. The concentration and electrical conductivity in aqueous solutions of electrolytes. Rev. Mex. Fis. 1964; 13(2) 5574.  35.
Fuentes C. Approche Fractale des Transferts Hydriques Dans les Sols Nosaturés. Tesis de Doctorado, Universidad Joseph Fourier de Grenoble, Francia; 1992.  36.
Chávez C. Solución numérica de las ecuaciones de transferencia de agua y solutos en riego y drenaje. Dr. in Eng. Thesis, Universidad Autónoma de Querétaro (In spanish). México; 2010.  37.
Marquardt DW. An Algorithm for LeastSquares Estimation of Nonlinear Parameters. SIAM J. Appl. Math. 1963; 11 431441.  38.
Haverkamp R., Leij FJ., Fuentes C., Sciortino A., Ross PJ. Soil Water Retention: I. Introduction of a Shape Index. Soil Sci. Soc. Am. J. 2005; 69 18811890.  39.
Simunek J., van Genuchten MTh. Using the HYDRUS1D and HYDRUS2D Codes for Estimating Unsaturated Soil Hydraulic and Solute Transport Parameters. pp. 1523–1536. In M.Th. van Genuchten. Simunek, J. and "Sejna, M. (ed.) Characterization and measurement of the hydraulic properties of unsaturated porous media. University of California, Riverside, CA; 1999.  40.
Feyen J., Jacques D., Timmerman A., Vanderborght J. Modelling Water Flow and Solute Transport in Heterogeneous Soils: A Review of Recent Approaches. J. Agric. Eng. Res. 1998; 70 231256.