10.4122/1.1000000590
Samper, Javier
Javier
Samper
jsamper@udc.es
Yang, Changbing
Changbing
Yang
cyang@udc.es
Samper, Javier
Javier
Samper
jsamper@udc.es
An Approximate Analytical Solution for Multi-mono-valent Cation Exchange Reactive Transport in Groundwater
XVI International Conference on Computational Methods in Water Resources
2006
2006
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.