Investigation the Dispersivity Coefficient of NaCl in Laboratory Columns under the Effect of Different Textures and lengths

Introduction
The movement and fate of solutes in the subsurface in affected by a large number of physical, chemical processes requiring a broad array of mathematical and physical sciences to study and describe solute transport. The dispersion is an important parameter in the advection – dispersion equation, which is used in assessing, and solving the problems related to the contamination and the groundwater resources protecting. The study of solute transport in porous media in the small scale (laboratory columns) and the large scale (farm) started since decades ago. In recent years, relevant studies on the physical and chemical phenomena in porous media, such as dispersion, diffusion, anion desorption, adsorption or exchange processes has significantly increased.
The characteristics of processes such as transport, advection and diffusion in soil and groundwater flow is essential for predicting and modeling of solute movement in the soil.
Researchers in their experiment showed that the small and large scale affected on dispersion and this difference related to length of the scale.  Based on their results, the average dispersion (α) was determined 0.8 and 0.87 cm for tritium and chloride tracers in small columns soils respectively, while it was 5 cm in large columns.
Evaluation of dispersion coefficient can be used in the different texture and scales, in order to protect and manage of pollutant in groundwater resources and the aquifers. Thus, the main objective of this study is to evaluate the effect of soil texture and sample scale on the dispersion coefficients and the compliance model for predicting solute transport in different laboratory columns. For this purpose, the amount of sodium chloride was measured in the output drainage water in 12 laboratory columns with 4 textures of sandy soil and 50, 80 and 110 cm in length and then the breakthrough curve was drawn. Finally, transport parameters and thrust curve were estimated using the analytical Brigham model and the CXTFIT program through inverse method.
 
Materials & Methods
Advection–dispersion equation:
The advection–dispersion equation (CDE) is one of the models governing solute transport in soil, which is used for non-reactive ions and solutes. In this model, the transition process was governed with the mass flow and diffusion and tow phenomena, including flow velocity in the pores and dispersion coefficient (1) are represented the transmission characteristics. For one-dimensional steady flow equations, it can be written as follows:



 

(1)




Where C: the concentration of salt in the liquid phase (M/L³), x: the distance along the flow direction (L), V: the average water pore velocity (M/L) and D: dispersion coefficient (L²/T).
In order to analytical solutions of CDE various methods have been proposed that involving partial differential equations based on different boundary conditions and initial values.  In this case, there is a column of sand saturated with water and a steady flow of water into the soil column. The tracer with known concentration enters into the soil column and at time t value of the output of the column (C) is measured.
Inverse method:
In equation (2) the C value is measured for different values of U, the initial values are assigned to the parameters V, D, and R, theoretical and measured breakthrough curves are drawn, and the error value based on statistical Criterion such as root mean square error is calculated. Then the difference between the measured and simulated sodium chloride breakthrough curves become minimal while the optimal values ​​of the parameters V, D and R are found using an inverse optimization method based on the Levenberg-Marquardt algorithm.
 
Brigham model:
This model developed a graphical method to calculate dispersion coefficient using data from miscible displacements in short laboratory cores. In this model we must Measure C for different various values of U (U is the pore volumes injected). The breakthrough relative concentration, C/Co, was plotted versus [(U-1)/U^1/2] on linear probability paper. If the data fit a straight line, then the use of the diffusion equation was validated, and the dispersion coefficient could be calculated from the slope of the line. This model Show that by breaking the flow system into segments in which width was a linear function of distance.
 
Test Method:
The 12 samples, including four sandy soil textures (100, 90, 80 and 70 percent blown sand and remain percentage natural sand) and three lengths (50, 80 and 110 cm) were selected and NaCl was used as injection with EC 5.5 mmohs/cm. The salinity of the output drainage water was measured at different intervals. This process was continued until reaching constant concentration in the drained water.
 
Results
Inverse method:
According to results, obtained breakthrough curves are coincided and the delay factor is close to 1 with the increasing percentage of natural sand at different lengths soil columns. The treatment for 110 cm length and 100% blown sandy soil has delay factor 1.805 that is the lower from other ones with different lengths. Delay factor reaches about one in 90% blown sandy soil and 110 cm long and it was not seen in other ones with different lengths. 
According to results, delay coefficient decreases of 100% blown sandy soil treatment for increasing length. This reduction was seen slight in other treatments.
Diffusion coefficient decreases in increasing amounts of natural sand.  Due to the length of the samples, many changes over the length can be seen only in 100% blown sandy soil treatment. The average values of the diffusion coefficient are 0.387, 0.276 and 0.496 cm²/min in soil columns with 50, 80 and 110 cm long, respectively.
In 100% natural’s sandy soil treatment, the velocity is bigger than the other treatments. The amount of natural sand in soil treatments is more; the difference between velocities in all of the lengths is low. The average velocity is 0.402, 0.397 and 0.344 cm/min in various tissues in lengths of 50, 80 and 110 cm respectively that it changes little.
 
Brigham model:
 
The results show that dispersion coefficient values ​​increase when the length of the sample increased by 80% blown sandy soil, which indicates that dispersion coefficient is dependent the sample length. But this process of texture 100, 90 and 70% sand and lengths of 80 and 110 are correct (table 1), but the dispersion coefficient of the sample greater than 50 cm in length somewhat greater. For samples of the same length and different textures in this study, there is a clear trend.
Table 1. The amount of dispersion coefficient, diffusion coefficient and speed of solutes in treatments using Brigham





 Texture
 
 Length(cm)   


100 % blown sand


90 % blown sand
10 % natural sand


80 % blown sand
natural sand 20 %


70% blown sand
30 % natural sand




α


D


v


α


D


v


α


D


v


α


D


v




50


0.945


0.292


0.309


0.519


0.128


0.247


0.473


0.130


0.276


0.551


0.215


0.390




80


0.279


0.075


0.268


0.338


0.139


0.409


0.548


0.140


0.254


0.400


0.113


0.283




110


2.868


0.669


0.233


0.465


0.192


0.392


0.740


0.339


0.459


0.496


0.165


0.332





 
Discussion & Conclusions
Comparing the values ​​obtained for the coefficient of α with the results of other researchers show that the ​​ range varies less than 1 cm, which corresponded to the reported results, except of 100% blown sand texture sample 110 cm in length that 2.868 cm.
The results showed that the estimated dispersion coefficient of CXTIFT has changed between 0.524 to 3.282 cm in treatments that was more than the corresponding estimated values ​​with Brigham.
Increasing the percentage of sand in the soil and sample length, delay coefficient decreased. In addition, the dispersion coefficient with increasing percentage of natural sand decreases. However, the average value of the dispersion coefficient has not many changes in soil columns with lengths of 50, 80 and 110 cm.
The minerals velocity of 100% blown sandy soil was higher than other treatments. The Flow rate of the in the soil treated sand. The amount of natural sand increases, the velocity difference in various lengths will be lower.