APPLICATION OF MATHEMATICAL MODELS TO OBTAIN THE SOIL MOISTURE RETENTION CURVE
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Introduction
Soil and water are two fundamental resources of the agricultural environment. With advances in new agricultural technologies, large amounts of water and chemicals are applied to the soil each year. This excessive use has endangered our resources, in addition there is an increase in incidents of contamination of aquifers and surface waters, erosion, etc. from point and nonpoint sources.
For this, it is necessary to know the hydraulic properties of the soil, the soil moisture retention curve (CTH) and the hydraulic conductivity function. Generally, the number of points measured for the curve results in a reduced amount due to field and laboratory work, as well as the costs required, so if these data are adjusted to an analytical function, it is possible to estimate intermediate points within the range. experimental in which they have not been measured.
The empirical models of soil moisture retention can be divided into four categories according to their functional relationship: exponential, potential, hyperbolic cosine and error function. Many authors use a potential function to characterize soil moisture retention (Brooks and Corey, 1964; Van Genuchten, 1980). The Brooks and Corey model was the most popular model among researchers because of its simplicity since the curve can be represented as a line on a loglog graph. However, Van Genuchten and Nielsen (1985) point out that this model produces acceptable results only for sieved soils, of coarse texture, with a relatively narrow pore distribution, while Hillel (1998) states that it is valid for tensions that do not correspond to the wet range.
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Case in Cuba
In Cuba, different simulation models have been used in research and agricultural management, one of the fundamental parameters being the knowledge of HTC. However, field or laboratory measurement of this property is time consuming, costly, and often impractical due to the high degree of spatial and temporal variation.
Material y Methods
Soil moisture retention curve 570 curves of different types of Cuban agricultural soils obtained in the period 1990 – 2003 were collected and validated, which were determined in the laboratory from altered and undisturbed samples, with tension ranges between 0 and 150 m, using Richard’s sandbox, kaolin box and press (EIJKELKAMP, 2010).
Table 1 shows the types of soils in which the determinations were made and which appear grouped according to the predominant texture, the II Genetic Classification of Cuban soils and the FAOUNESCO classification (Instituto de Suelos.ACC, 1980), finding a great diversity of these.
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Analytical models
For the evaluation of the soil moisture retention models, the RETC software made by Van Genuchten et al. (2000), from which the parameters of saturation and residual humidity were determined, as well as α and n, which serve as adjustment to the analytical models used and which are presented below:

Van Genuchten’s model with constraints
m = 11/n (vG1) y m = 12/n (vG2), has the following equation:
Se = [1+(αh)n]m 
The Brooks and Corey (BC) model is based on the following equation:
Se = (αh)λ 
The Lognormal Distribution (DLN) (Kosugi, 1996) is represented by the model:
Se = 1/2erfc(ln h/ho)/21/2σ  Dual Porosity (PD) (Durner, 1994) is determined as:
Se = w1[1+( α 1h)n 1]m 1+ w2[1+( α 2h)n 2]m 2
Where: Se = (θθr)/( θsθr), effective saturation; θr and θs correspond to residual and saturation humidity (g g1); θ moisture content (g g1); α, α 1, α2 is an empirical parameter; n, m, n1, m1, n2, and m2 are curve fitting parameters; h tension (cm); erfc complementary error function; λ characteristic parameter of the soil and σ standard deviation.
Determination of the model parameters
For the estimation of the unknown parameters of the evaluated models, a nonlinear least square optimization process was used. It was based on the partition of the total sum of squares of the observed values in a part described by the adjusted model and a residual part that involves the observed values and those estimated by the model. Therefore, the curve fitting process consisted of finding an equation that maximizes the sum of squares associated with the model and on the other hand minimizes the sum of residual squares. This statistical parameter reflects the degree of fit and the contribution of random errors and will be referred to as the objective function O (b) in which b represents the vector of the unknown parameter and is solved by applying the Marquardt maximum neighbor method (Marquardt, 1963). The objective function for adjusting the soil moisture retention curve is the following:
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Where: θi and θˆi correspond to the observed and adjusted soil moisture contents (g / g), N is the number of retention data, wi is the weight factor for each data on the retention curve. Another adjustment parameter used is the coefficient of determination (r2), which is represented by the following equation:
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Where: yi and yˆ i are the fitted and observed values respectively. In addition, the standard error of the adjusted parameters was determined, taking into account the Objective function, the number of observations, the number of unknown parameters to be adjusted and the inverse matrix following the procedure of Daniel and Wood (1971) and they were used as a standard. of comparison the validity ranges established by Shaap and Leij (1998) for the parameters θr, θs, α and n and the average physical and hydraulic properties for each type of soil.
Results and Discussion
The performance of the 5 analytical models was evaluated from the sum of residual squares, showing in table 2 the values obtained for each type of soil. An analysis of these shows that the Van Genuchten model accurately describes the HTC, that is, there is a better correspondence between the retention curves obtained experimentally and those estimated, with the vG1 variant with an average value of 0.89×103 g g 1 the one that best describes the data obtained in the laboratory. Similar results have been obtained by different authors, including Ruiz et al. (2006) that obtained SCRs ranging between 0.31×103 and 4.09×103 g g1 when adjusting the moisture retention curves for different Cuban soils.
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The values of the coefficient of determination that appear in table 2 corroborate the previous results with respect to the adjustment that exists between the humidity values obtained in the laboratory for each stress point of the curve with those estimated by the analytical models evaluated. Being the Van Genuchten model for the conditions m = 11 / n and m = 12 / n, those with the highest average r2, with a value of 0.97, which together with the results of the SCR makes the vG1 model the most appropriate to describe the process of water retention in the soil.
Although the dual porosity model takes into account a heterogeneous pore structure, which is applicable to the evaluated soils, it showed the highest SCR values with an average of 74.61×103 g g1 and the lowest average r2 of 0.95 together with the lognormal distribution model, which is due to the fact that in a soil sample of 100 cm3 there is only an approximation of the macroporosity present in the soil.
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Table 3 shows the values of the Van Genuchten parameters estimated with the condition m = 11 / n for the soil groupings studied. An analysis of these shows that the residual humidity varies in each grouping, the minimum being 0.1021 g g1 for sandy soils; This value is very accurate due to the amount of silicon elements in the form of sand or quartz gravel present in these soils and the very low clay contents, which do not exceed 15% in the first 100 cm of depth. While the maximum was 0.2851 g g1 and was obtained for the Dark Plastic Gleysoso soil, due to the amount and type of predominant clay in it; which reaches values above 50% and belongs to the smectite group, considered a mineral that retains high moisture content.
In the case of Ferritic and Ferralitic soils the residual humidity is relatively high, which could be given by the clay content (between 50 and 80%), which means that these soils have a large amount of surface area available to retain water. at high voltage values. The saturation humidity is within the validity ranges established for Cuban soils, while the rest of the parameters were adjusted to the criteria defined by Shaap and Leij (1998).
Table 4 shows the standard error for each parameter of the Van Genuchten model, with the average values in residual humidity, saturation humidity, α and n of 0.0513 g g1, 0.0267 g g1, 0.0417 and 0.1273 respectively, This indicates the range of variation that these parameters may have and, in turn, they are considered acceptable if the variability of the same in the soil is taken into account, indicated by Warrick (2003).
The upper and lower limits of each parameter for 95% confidence and their correlation matrix were also determined, finding that θr is highly correlated with n, reaching an average value of 0.9 and to a lesser extent with α (0.7 ); In the case of θs, a moderate correlation was obtained with α and a low correlation with n, these being 0.7 and 0.4, respectively. While the curve fitting variables (α and n) had a correlation value of 0.8, which indicates that there is a strong dependence between the parameters that form the Van Genuchten model.
Model sensitivity
Figure 1 shows that during the estimation of the CTH with the Van Genuchten model, no appreciable differences were observed between the estimated saturation moisture contents and those determined in the laboratory, with r2 equal to 0.99 and the sum of squares residuals of 6.44×103 g g1. This shows that it is not essential to define the value of θs in the model, and it can be estimated like the other parameters from iterations in which the objective function is minimized.
On the other hand, if the moisture content at 0 tension is unknown in the experimental curve, it can be used to determine the parameters of the Van Genuchten model θr, θs, α and n since, although it is an extreme value, the results obtained are within of the permissible validity ranges. Similar results were obtained by Van Genuchten and Nielsen (1985) when stating that it is possible to estimate the θs with the model, since it has a high variability due to the humidity changes that occur in the soil as a result of agricultural management, the size of the sample and the moment it is taken. Warrick (2003) also refers to the spatial variability of the extreme values of the soil moisture retention curve, pointing out that they have a coefficient of variation that ranges between 10 and 50%, respectively. In the case that there are variations in the moisture content (> 0.1 cm3 cm3) in the range from 0 to 100 cm, the parameters θs and α which represent the pore size distribution index are mainly affected; reaching values above those recommended for soils.
In the case of θr, it can be determined from the extrapolation of existing moisture retention data or through the model, obtaining results within the ranges defined for the studied soils, being necessary to have tensions close to the lower moisture limit (above 1000 cm).
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Conclusions
The Van Genuchten model with the condition m = 11 / n is the one that best describes the hydraulic behavior of the studied soils, being the sum of average residual squares of 0.89×103 g / g.
The values of θr, θs, α and n obtained can be used to characterize the movement of water in the unsaturated zone, when there are no moisture retention curves in the soil. The upper and lower limits of moisture in the soil can be estimated by the PRTR model, for irrigation and drainage purposes, reaching precise and reliable values.