Analytical solution for monovalent dival

An approximate analytical solution is derived for solute transport with monovalent–divalent ion exchange in saturated, steady, groundwater flow. The objective of this study is to simplify the complicated heterovalent ion exchange to derive a usable solution. A coordinate system moving at the average solute advection velocity is employed, and the solution is obtained through direct integration with specifically defined boundary and initial conditions. The results demonstrate good agreement with numerical solutions utilizing the original isotherm, indicating the reliability of the analytical approach. Furthermore, the analytical solution presented aligns closely with that of monovalent–monovalent exchange, suggesting that a simpler ion exchange model may be effectively assumed for future approximations. This work contributes valuable insights into groundwater solute transport modeling, potentially aiding in remediation efforts and enhancing understanding of ion exchange processes. Consequently, the findings underscore the viability of analytical methods in addressing complex environmental challenges.