An approximate analytical solution is derived for solute transport with monovalent-divalent ion exchange in saturated, steady, groundwater flow. The analytical solution is obtained by simplifying the complicated heterovalent ion exchange. Using a coordinate system that moves at the average solute advection velocity, a solution is obtained by direct integration with given boundary and initial conditions. The results agree well with a numerical solution using the original isotherm. The analytical solution presented is similar to that of monovalent-monovalent exchange, which suggests that a simple ion exchange model may be assumed for approximation.
In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.
An approximate analytical solution of depth to water table driven by periodical precipitation and evapotranspiration in shallow groundwater zones