Analytical power series solution for contaminant transport with hyperbolic asymptotic distance-dependent dispersivity

A hyperbolic asymptotic function, which characterizes that the dispersivity initially increases with travel distance and eventually reaches an asymptotic value at long travel distance, is adopted and incorporated into the general advection–dispersion equation for describing scale-dependent solute transport in porous media in this study. An analytical technique for solving advection–dispersion equation with hyperbolic asymptotic distance-dependent dispersivity is presented. The analytical solution is derived by applying the extended power series method coupling with the Laplace transform. The developed analytical solution is compared with the corresponding numerical solution to evaluate its accuracy. Results demonstrate that the breakthrough curves at different locations obtained from the derived power series solution agree closely with those from the numerical solution. Moreover, breakthrough curves obtained from the hyperbolic asymptotic dispersivity model are compared with those obtained from the constant dispersivity model to scrutinize the relationship of the transport parameters derived by Mishra and Parker [Mishra, S., Parker, J.C., 1990. Analysis of solute transport with a hyperbolic scale dependent dispersion model. Hydrol. Proc. 4(1), 45–47]. The result reveals that the relationship postulated by Mishra and Parker [Mishra, S., Parker, J.C., 1990. Analysis of solute transport with a hyperbolic scale dependent dispersion model. Hydrol. Proc. 4(1), 45–47] is only valid under conditions with small dimensionless asymptotic dispersivity (aa) and large dimensionless characteristic half length (b).