An Approximate Analytical Solution for Multi-mono-valent Cation Exchange Reactive Transport in Groundwater
Creators: name: Samper, Javier
name: Yang, Changbing
name: Samper, Javier
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: 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.