Three‐dimensional analytical solutions to the advection‐dispersion equation in arbitrary Cartesian coordinates

Abstract

Analytical solutions for the advection‐dispersion equation (ADE) usually assume that boundary and initial conditions are orthogonal to the principal axes of the dispersion tensor. However, this is not always the case in field studies or modeling scenarios. Using the method of Green's functions, a generalized analytical solution of the three‐dimensional ADE set in an arbitrary Cartesian coordinate system (i.e., not orthogonal to the principal flow direction) is given for the solute resident concentration in a semi‐infinite porous medium with an arbitrary initial condition, surface boundary condition, and sink/source terms. Two particular solutions for a rectangular surface flux boundary condition and a buried parallelopiped, respectively, are derived from the general solution. The corresponding frequency domain solutions are also given which provide a more efficient method of computation for generating two‐and three‐dimensional grids via use of a two‐dimensional fast Fourier transform algorithm (as an alternative to the two‐dimensional numerical integration required to calculate the concentration using the real space solution). When the arbitrary coordinate system is orthogonal to the principal axes, it is shown that these particular solutions are the same as previously published results.