A Mathematical Model for Determining Diffusion of Nitrate Contaminant in Underground Water

Abstract

All living things, including human beings, need water to survive and to flourish. Unfortunately, many people in the world don’t have access to clean and safe water. Groundwater is a valuable source of drinking water in many societies through wells. However, throughout the world, aquifers are being contaminated by chemicals from many sources. The diffusion process of nitrate highly contaminates underground water and there is no known mathematical model to evaluate it’s effect. This study formulated a mathematical model for determining diffusion of nitrate contaminant in underground water. The mathematical model was used to determine the con- centration of nitrate as an agricultural chemical contaminant in the ground at a given time and point using the definition of the model given by Velten. The ana- lytical and numerical solutions of the diffusion equation characterizing the flow of chemicals in the ground were used to determine the concentration of nitrate. The numerical solutions of the model were implemented using MATLAB software and the analytical solutions were found using Fourier transform method. The results of the model were validated using the data of concentration of nitrate chemicals in wa- ter collected from eleven wells and tested using UV-Vis analysis. The tested water samples were collected from the wells around Moi University main Campus in Kesses Sub County. From the model formulated, one can easily predict the concentration of nitrate at given time and point in the soil using the diffusion equation and its solutions. The analytical solution of the model equation proved that nitrate contam- inant is normally distributed. The numerical simulation with MATLAB showed that the concentration keeps reducing at the disposal area as it spreads in all directions unless the source of contamination is permanent. In this case, the concentration will always remain high at the disposal point. The laboratory analysis to check for the concentration levels of Nitrate showed that some of the samples were polluted by nitrate since the results were between 6.71mg/l and 55.95mg/l. This is because the maximum allowed concentration level of nitrate is 10mg/l. The findings of this research can be used to advise farmers to build their wells at least a distance of 100m from the source of contamination. These findings can also be used by policy makers and chemical manufacturers to inform and advise the population on the side effect of the chemicals. The formulated model is also applicable to underground water flow and its contamination as well as other mathematical modeling.