On Diagonalization of Coupled Hydrologic Transport and Geochemical Reaction Equations

Abstract

Two basic ingredients present in modeling the transport of reactive multi-components: the transport is described by a set of advection-dispersion-reactive partial differential equations (PDEs) based on the principle of mass balance; the chemical reactions, under the assumptions of local equilibrium, are described by a set of highly nonlinear algebraic equations (AEs) base on the principles of mole balance and mass action. For a typical application, the complete set of nonlinear PDEs and AEs consist of more than one hundred simultaneous equations. Thus, it is impractical to solve this set of equations simultaneously. General practice is to divide this set of equations into two subsets: one is the primary governing equations (PGEs) consisting of mainly the transport equations and the other one is the secondary governing equations consisting of mainly the geochemical reaction equations. The PGEs are solved for the chosen primary dependent variables (PDVs) and the SGEs are used to compute for the secondary dependent variables (SDVs). The major difficulties in simulating the reactive transport is the numerical solution of PGEs. From the computational point of view, the solution of the set of highly nonlinear PDEs are solved either with the direct substitution approach (DSA) or with the sequential iteration approach (SIA). For DSA, geochemical equilibrium reaction equations are substituted into the hydrologic transport equations to results in a set of nonlinear partial differential equations. These nonlinear PDEs are solved simultaneously for the chosen PDVs. For SIA, the procedure consists of iterating between sequentially solving the hydrologic transport equations for the chosen PDVs and solving geochemical equilibrium equations for the SDVs. The most important consideration in choosing PDVs is: it should decouple the simultaneity of nonlinear PDEs as much as possible and it should yield nonlinear source/sink terms in any PDE as small as possible. This paper presents a discussion on the diagonalization of the simultaneous nonlinear PDEs.