A Simple Method for Estimating the Convection- Dispersion Equation Parameters of Solute Transport in Agricultural Ecosystem

The convection-dispersion equation (CDE) is the classical approach for modeling solute transport in porous media. So, estimating parameters became a key problem in CDE. For statistical method, some problems such as parameter uniqueness are still unsolved because of more factors. Due to the advantage of clear physical concept and unique parameter values, the simple deterministic method became very useful alternatives. In this paper, a simple method was proposed to estimate both D and R, and the validity was verified by experiment, which can be applied in agriculture and environmental fields for predicting soil quality property.


INTRODUCTION
Transport of agri-chemicals through soils affects both crop production and groundwater quality. The classical approach of modeling solute transport in porous media uses the deterministic convection-dispersion equation (CDE). For this reason, estimating parameters is a key problem for applying CDE (Buchter et al., 1995;Kool et al., 1987).
Methods of estimating CDE transport parameters can be divided into statistical and deterministic approaches. Statistical parameter estimation techniques consist of the least sum of square, maximum likelihood and moment analysis methods (Bresler and Naor, 1987;Jury and Sposito, 1985;Parker and van Genuchten, 1984;Toride et. al., 1995).These methods are complicated and require a special program. Meanwhile, for different initial values, *Corresponding author: E-mail:shaohongbochu@126.com.
there are different fitting parameters. Some problems such as parameter uniqueness are still unsolved because of more factors. Controvertibly, the deterministic method has the advantage of clear physical base and unique parameter values and simple deterministic method are very useful alternatives. Rifai et al. (1956) presented a method to estimate the dispersion coefficient (D) from the slope of the breakthrough curve (BTC) at a time corresponding to the displacement of one pore volume. The disadvantage is that only one data point of BTC is used. Elprince et al. (1977) modified this method by using two parameters, dispersion coefficient and retardation factor (D, R), but there is still a shortage of few data used. When the Pacelet number is larger than 5, the BTC can be approximated by a normal distribution (Levenspiell and Smith, 1957;Fried and Combarnous, 1971) or a lognormal distribution (Rose and Passioura, 1971), and the parameters can be estimated by the method of probability property. Yamaguchi (1989) had modified Rifai's method (1956) using four points different to the estimated parameters. But the change of slope of BTC is more intensive; the estimation parameters are more different with other method (Li, 1999). BTC is a time-consuming method. Shao et al. (1998) presented an approximate method to estimate parameters by boundary-layer theory not needed to make BTC, but it is difficult to detect the boundary layer in practice. The objective of this study was to propose simple method to estimate both D and R of CDE simultaneously under the condition which the BTC were made or not. The results could be useful for applying CDE to predict the agrichemicals transport in soils.

THEORY AND ERROR ANALYSIS
One-dimensional transient solute transport through a homogeneous medium during steady-state water flow is traditionally described by CDE: Where, Cr(x,t) is the volume-averaged resident concentration of the solution in the liquid phase, u 0 is the mean velocity and assumed constant, D 0 is the dispertion coefficient and supposed constant, R is retardation factor, t is time and x is the distance.
The boundary and initial conditions for a semi-infinite column experiment are: Where, the initial solute concentration in the soil column is 0, and the concentration of solute supplied at the soil column inlet is c 0 . The solution of Equation (1) under the condition of Equation (2) Where, arcerf is the inverse operation of the error function, the values can be obtained from tables (Tables of the Error Function and Its Derivative, 1954) through inverse checking. If x equal fixed L, settingξ equal to: can change the relationship between BTC and time t: Similarly, there is similar solution for effluent curve: Where, the C f is the effluent concentration. Using same method, the Equation (10) (3), the reduced time is t (log scale) and Brenner number is B (log-log Scale)]. The total percentage was 100, the percentage of second term was gradually decreased with Brenner number increasing. When the Brenner number was more than 100, the percentage of the second term was less than 2. This showed that the second term of the analytical solution can be neglected under Brenner number more than 100 (Yamaguchi, 1989). parameters, and this was named intercept method. Because BTC is time-consuming, the parmeters were hoped to be estimated not through BTC. Equation (5) can be transferred to be: The setting of Y and X equal to: Equation (10) (Yamaguchi, 1989). the percentage contribution of the second term is less than 2% (Figure 1). This showed there was reason to neglect the second term of analytical solution under Brenner number more than 100. The relationship between relative error in estimating u and D using complete and negligibility of the second term analytical solution and Brenner number is shown in Figure 2. When Brenner number is more than 100, there are no difference between the parameters estimation without the second term and real parameter values.
Because in a wide range Peclet number 1< ud/D m <10 6 (where d is mean diameter of the soil particles, D m is the molecular diffusion coefficient), ud D ≅ ( Bear, 1979); the Brenner number was changed: So, the condition of the negligibility of the second term of the analytical solution can be satisfied through adjusting the length of soil column L and soil particle diameters. The relative error in estimating u and D is < 1% and < 0.5%, respectively, when the Brenner number is Figure 2). In experience, the soil was sieved with 2 mm mesh and if the length of soil column is more than 20 cm, the condition of B > 100 can be satisfied; the above method can be used to estimate the parameters of CDE.

MATERIALS AND METHODS
The experimental soil samples were loam soil. The pH and organic matter of soil were 8.12 and 1.27% respectively. The contents of different particles of > 0.25, 0.25 to 0.05, 0.05 to 0.01, 0.01 to 0.005, 0.0005 to 0.001 mm, < 0.001 were 1. 1, 5.6, 47.8, 15.0, 17.9 and 12.8% respectively. The experimental setup is schematically shown in Figures 1 and 2. The soil column was a plexiglass cylinder, 10 cm in diameter and 40 cm long, and perforated at 20 and 30 cm from the top. The air-dried soils sieved with 2 mm mesh were uniformly packed in layers and made the average dry density in the column (1.3 g/cm 3 ). The holes were used to insert the selective Cliron electrode and to get the Clconcentration in different times. A 0.16 N solution of CaCl2 was applied with a positive displacement after the soil column had been saturated with Mariotte bottle under 3 cm high water head. The mean water velocity was h cm u / 03 . 1 0 = , and the porosity was 0.51. Under the basis, the leaching solution was collected and the Clconcentration of the effluent solution was measured by titration with AgNO3 solution.

Parameters estimated by intercept method
Based on Equations 8 and 9, ξ of the three different Relative error (%) B Figure 3. The schematic diagram about Cltransport in a soil column experiment. positions was significantly linearly related with t ( Figure  4). The parameters D and R can be calculated by the intercept and slope of ξ and t. The parameters R estimated by intercept method were similar with that by CXTFIT and parameters D was more than 17% by CXTFIT (Table 1). After the second term was neglected, all contribution to the analytical solution was formed by first term; the parameters were amplified to meet these requirements, this maybe a reason for the parameters by the intercept method than the CXTFIT. No matter which method, the parameters D increased along with the length of soil column; this reason maybe that the soil porosity was more complex with the stretching; hydrodispersion is more intensive. In order to compare the precision of the estimated parameters, the simulated breakthrough curve for three positions by three methods were compared; the simulated results were no obvious different with the measured one although the simulated one was higher than others (Figures 5 and 6). Especially for the first term approximate method (neglect of the second term of the analytical solution) and intercept method, the simulated results were as same as the results by the measured CXTFIT. This showed that intercept method can be used to estimate parameters of CDE equation and had high precision to predict the solute transport in soils.

Parameters estimated by position-time method
Compared with the intercept method, the position-time method needs to measure the BTC. Figure 6 is the linear relationship between Y and X for a series C/C0 at different time at 20 and 30 cm positions. The parameters D and R calculated were 0.645 and 0.97 according to Equation 12, respectively; these approximated the mean values by the intercept method at these 2 points. This showed the parameters by position-time method were an average condition. Figure 7 Figure A, B, C were the measured points at 20cm, 30cm, and effluent, respectively. Because the intercept method and position-time method are derived from the basic solution of Con-vection-Dispersion Equation, they have clear physical concept, and is fast and easy to calculate. The simulated result by the calculated and measured parameters coin-cided. These can provide a simple method to approxi-mate parameters in CDE.

ACKNOWLEDEGEMENTS
This research was supported by the National Natural