An Approximate Analytical Solution for Multi-mono-valent Cation Exchange Reactive Transport in Groundwater

Abstract

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation-exchanging pollutants in groundwater is highly nonlinear. This makes difficult to derive analytical solutions for multication exchange reactive transport which are of interest for stochastic analyses of multicomponent reactive transport. Dou and Jin (1996) used the method of characteristics with a special treatment of shock waves and worked out a closed-form formulation for 1-D transport coupled to binary homovalent ion exchange. Jin and Ye (1999) extended this approach and developed an approximate analytical solution to binary monovalent-divalent ion exchange transport. Due to the complexity of the isotherms, most of the available analytical solutions are suitable only for 1-D transport with binary cation exchange. Here we present an approximate analytical solution for the general case of multi-monovalent cation exchange reactive transport accounting for any arbitrary number of monovalent cations. Time derivatives of concentrations of exchanged cations (those sorbed on the solid phase), β’, are related to time derivatives of concentrations of dissolved cations c’ through a Jacobian matrix J which is derived by taking time derivatives of the logarithmic version of the nonlinear cation exchange mass-action-law equations. The Jacobian matrix which in general depends on β and c is evaluated at selected values of β* and c*. Substitution of β’ into the reactive transport equations leads to a set of coupled partial differential equations (PDEs). Such coupled set of PDEs can be effectively decoupled by means of a similarity transformation which leads to a diagonal retardation matrix. By performing such transformation on boundary and initial concentrations, the set of linear uncoupled PDE’s can be solved in terms of transformed concentrations U by using standard available analytical solutions. Concentrations of the original problem c are obtained by performing the backwards transformation on U. Our analytical solution has been tested with numerical solutions computed with a general purpose reactive transport code (CORE2D)for several 1-D cases. Analytical solutions not only agree well with numerical results regardless of the choice of β* and c*, but provide also additional insight into the nature of the retardation factors caused by multicomponent mono-valent cation exchange. Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation-exchanging pollutants in groundwater is highly nonlinear. This makes difficult to derive analytical solutions for multication exchange reactive transport which are of interest for stochastic analyses of multicomponent reactive transport. Dou and Jin (1996) used the method of characteristics with a special treatment of shock waves and worked out a closed-form formulation for 1-D transport coupled to binary homovalent ion exchange. Jin and Ye (1999) extended this approach and developed an approximate analytical solution to binary monovalent-divalent ion exchange transport. Due to the complexity of the isotherms, most of the available analytical solutions are suitable only for 1-D transport with binary cation exchange. Here we present an approximate analytical solution for the general case of multi-monovalent cation exchange reactive transport accounting for any arbitrary number of monovalent cations. Time derivatives of concentrations of exchanged cations (those sorbed on the solid phase), β’, are related to time derivatives of concentrations of dissolved cations c’ through a Jacobian matrix J which is derived by taking time derivatives of the logarithmic version of the nonlinear cation exchange mass-action-law equations. The Jacobian matrix which in general depends on β and c is evaluated at selected values of β* and c*. Substitution of β’ into the reactive transport equations leads to a set of coupled partial differential equations (PDEs). Such coupled set of PDEs can be effectively decoupled by means of a similarity transformation which leads to a diagonal retardation matrix. By performing such transformation on boundary and initial concentrations, the set of linear uncoupled PDE’s can be solved in terms of transformed concentrations U by using standard available analytical solutions. Concentrations of the original problem c are obtained by performing the backwards transformation on U. Our analytical solution has been tested with numerical solutions computed with a general purpose reactive transport code (CORE2D)for several 1-D cases. Analytical solutions not only agree well with numerical results regardless of the choice of β* and c*, but provide also additional insight into the nature of the retardation factors caused by multicomponent mono-valent cation exchange.