Experimental Determination of Dispersion Coefficient in Soil

Unsaturated soil dispersion experiment was done using three different soil samples which includes sand, clay and silt soil. The objective of the study is to determine experimentally dispersion coefficient in soil. Each sample was gradually introduced into a fabricated iron column, having a length of horizontal column 30cm and vertical part 60.96cm. A solution of silver nitrate was allowed to pass through the vertical column down to the horizontal part. Samples of soil were collected at a constant distance of 10cm and time interval of 5mins for 60mins .The concentration of nitrate was taken at a constant distance of 10cm .This was done three times for each of the soil sample. Thereafter, the dispersion coefficient was calculated, and a regression dispersion model developed as a function of permeability, average diameter of sieve, velocity of flow and time taken to flow. The model was calibrated and verified with experimental results and found to have a high correlation coefficient (r) of 0.943. Data obtained from the experiments were fitted into the two existing models and poor correlation coefficients of -0.529 and -0.524 were obtained respectively.


2-1-Dispersion in Soil
A model framework was developed to model the dispersion of contaminants in sandy, clay and silt soil.The framework utilizes the vertical column attached with a horizontal column to account for on ground nitrogen sources and corresponding loadings.The vertical column has 0.275 cm internal diameter, vertical column length of 60.96 cm, with 5 cm internal diameter of horizontal cross section area and a horizontal length of 30 cm.Moreover, 500 g of 2 mm sieved air dried soil of pre-determined textural class (sandy, silt and clay soils) were weighed out using a top loading mettler weighing balance.The sample was gradually introduced into the iron column.1000 ml of 100 mg/l AgNO3 was poured through the column and allowed to drain through the soil.Sample of the saturated soil was collected at 1, 5,10,15,20,25,30,35,40,45,50, 55 and 60 minutes interval at a constant distance of 10 cm.One (1g) of the sample was introduced into the extraction tube and 50 ml of 2.5 N KCl added and shook for about 2 hours.The sample was filtered through a No 42 What man filter paper into a 50 ml volumetric flask and made up to the mark with the extraction reagent.
A "Markham" distillation apparatus was set up and 10 ml of the extract introduced into the apparatus, 10 ml of 45% NaOH was gradually added to the sample and allowed to distill into a 10ml of 4% H3BO3, the distillate is then titrated with 0.02 N H2SO4 and the produced recorded as NH4-N, a pinch of Devader alloy was added to the residual sample and the reaction oxidizes the nitrogen form to NO3 -N about 50 ml distillate was also collected into another 10 ml of 4% H3BO3 and titrated with the same 0.02 N H2SO4 and the produced recorded as NO3 -N.Hence, the dispersion in soil were determined using the below equation [5].
Where  = is the amount of silver nitrate (AgNO3) That diffused into the soil (g /cm 2 )  = The concentration at a distance after dispersion over time (g /cm 3 ),  =    ℎ ,  = 3.142 t = Corressponding time interval (secs)

2-2-Statistical Regression Model of Dispersion in Soil
Statistical analyses were used to develop a regression model for dispersion coefficient.A regression dispersion model developed as a function of permeability, average diameter of sieve, velocity of flow and time taken to flow.Where D is the dispersion coefficient.In other to make D a function; Where D is the dispersion coefficient.
Taking the natural logarithm for both sides [6.7].

2-3-Model Calibration
The model calibration were necessary in other to determine the values of the parameters that can fit the model to the system [6].In essence of this, the calibration was deduced from the experimental data.In calibration, half of the experimental data is used and half used for verification.

2-4-Verification
Model evaluation involves determining how well the mathematical model describes the system [6].This was achieved by conducting a set of measurements from the system used in creating the model.The remaining data were used for verification.

2-5-Results and Discussion
The experimental results of sand, clay and silt soil generated from experiments were used to calculate dispersion using Equation 1.Thereafter, the dispersion coefficient was calibrated and verified.The reason for this is to ascertain if the developed model really satisfies the system being modeled.The Figure 1 and 2, below shows the calibration and the verification of the model.

2-6-Model Development
In the development of the regression model, the variables in Equation 2 were fixed as defined in Equation 3 to 11, and thereafter regressed, thus; D = e 6.353 × K 0.411 t 2.139 × V 1.143 × d 0.034 (12)

2-7-Calibration of Dispersion Model
The regression model given in Equation 12 was used to calibrate dispersion coefficient and thereafter verified using the measured dispersion.From the result, both has good agreement with R 2 value of 0.991 as shown in Figure 2.

2-8-Comparison of Experimental Data with the Existing Model
Results obtained from the experiment were fitted into Shao et al., [8] model and regressed.A low value of r = -0.529was obtained against r = 0.943 obtained in this research.Also, the same data were fitted into De Smeldt and Wierenga [9] model and regressed, in which a low value of r = -0.524was obtained against r =0.943 obtained in this research.The predicted curve of Shao et al. [8] and our predicted curve are shown in Figure 3. Looking at the Figure 3, It was observed that the measured values of the dispersion do not agree with values fitted in Shao et al., [8] model.It lacks consistency, since velocity decreased from the first 1.8 to 7.2 m distance which is good, but sharply increased to 3.96 cm 2 /s at a distance of 9 m.There was an observed decrease from 2.45 to 1.65 cm 2 /s at distances of 10.8 and 14.4 m.There was also a constant decrease to 0.0118 cm 2 /s at a distance of 16.2 m, which further decreased to 0.0079 and 0.0056 cm 2 /s at distances of 18 and 19.5 m respectively.
Similarly, predicted curve of [9] model in Figure 4 and the predicted curve of this work followed the same trend when plotted against distance from the borehole source.Therefore, we deduced that the model developed in this work is far much better than the two existing models compared, since four parameters were considered unlike Shao et al. [8] and De Smeldt and Wierenga [9] model that evaluated one and two parameters.Also, our regression value r has positive value of 0.945 against Shao, De Smeldt and Wierenga model that has r value as -0.529 and -0.524 respectively.

3-Conclusion
The understanding of solute dispersion in soil is the basis for protection of groundwater from been contaminated.Studies on the dispersion of solute transport through a fabricated column were conducted.The results obtained from the experiment was calibrated and verified.Thereafter, regression model of dispersion, a function of velocity, permeability, average diameter of sieves and time was developed as shown in Equation 1.The derived regression model was used to compare Shao et al. [8] model, De Smeldt and Wierenger [9] model.Data obtained from the experiment were fitted into Shao, et al [8] model and regressed in which a low value of r = -0.529was obtained against r= 0.943 obtained in the research.Also, the same data were fitted into De Smeldt and Wierenger model [9], and regressed, in which a low value of r = -0.524was obtained against r = 0.943 obtained in the research.Noteworthy of mention, this regression dispersion model was used in modifying a transport model of a nitrate after being linked to Notodarmojo et al. [10].

d=
Average diameter of sieves (cm), k= Permeability (cm/s), t = Time of flow, (s) v= Velocity of flow in (cm/s) Such that

Figure1.
Figure1.Calibration graph showing predicted and measured of dispersion in soil.

Figure 2 .
Figure 2. Verification of graph showing predicted and measured of dispersion in soil.