Analytical solution for transport of pollutant from time-dependent locations along groundwater

The present work derives analytical solutions of advection-dispersion equation (ADE) with temporal coefficients, and a pollutant's point source moving linearly along the axis of a one-dimensional semi-infinite domain. The source is considered a varying and a uniform pulse source, respectively. The dispersion of pollutant originating from a varying pulse source may be supposed to occur along groundwater flow domain, and that from a uniform pulse source in an open medium like air or along a river flow. The location of the input concentration that is the pollutant's concentration emanating from the source in an open medium or that reaching the groundwater domain being infiltrated from its source on the ground, is considered moving linearly along the flow direction. The motion of the source is described through an asymptotically increasing temporal function. The illustration of the analytical solution clearly reflects this feature. It also renders that the concentration pattern of the proposed solution is proximal to that of an existing solution obtained with the stationary source. The pertinent existing solutions may also be derived from the proposed solutions. The proposed solutions are found approximate but it is also found that the error of approximation of one of them is too small to have any effect on the concentration pattern. To get these solutions, firstly, the moving source is reduced into a stationary source at the origin, then the governing equations including the ADE, are made free from the three temporal functions, one occurring in the time-dependent position of the source, and the other two as the coefficients of the ADE. In this process, three new position variables, and a new time variable are introduced using as many coordinate transformations. Then the Laplace Integral Transformation Technique (LITT) is used to get the final solutions. The solution in Laplacian domain with uniform pulse source is obtained as a special case of that with the varying pulse source.

Automated summary

This study presents analytical solutions of the advection-dispersion equation (ADE) with temporal coefficients in a one-dimensional semi-infinite domain. The solutions describe the dispersion of pollutants originating from a varying or uniform pulse source, considering both groundwater flow and open medium environments. The solutions also capture the linear motion of the source and show good agreement with existing solutions. The proposed solutions, obtained through coordinate transformations and Laplace Integral Transformation Technique (LITT), offer a simplified approach for analyzing pollutant dispersion.