Equilibrium Vs. Kinetic Formulation in Reactive Transport Models During a reactive transport simulation, the temporal scales of the various

transport and reaction processes control which formulation, should be used to best describe the chemical rate expressions (Mayer 1999). An equilibrium instead of a kinetic formulation for a reaction may be used as long as the local equilibrium assumption (LEA) is valid. LEA requires that the reaction time scale is sufficiently fast enough relative to the transport time scale, to establish a local equilibrium throughout a model domain (Yeh & Tripathi 1989; Novak &

Sevougian 1992; Walter et al. 1994; Viswanathan 1996; Gao et al. 2001). An equilibrium formulation, however, describes only the final composition of a geochemical system while a kinetically controlled description formulates the transient chemical evolution towards an equilibrium state. Therefore, the kinetic formulation is more general, and enables essentially both equilibrium and kinetic reactions to be modelled with the same general formulation (Steefel & Lasaga, 1994; Viswanathan 1999) as well as allows reactions to occur on any time scale (Novak & Sevougian 1992). Though, due to being dependent on the reaction path (Lasaga 1981), kinetically-controlled rate reactions are inherently more difficult to describe than an equilibrium one in reactive transport models. As a result, a kinetic formulation is computationally more intensive (Zysett et al. 1994) and causes generally more numerical difficulties to find a solution than equilibrium formulations. Mathematically expressed, for an equilibrium formulation by applying LEA to all reaction source/sink terms in the reactive

transport equation set, a set of mixed partial differential/algebraic (Zysett et al.

1994) is obtained instead of a set of a mixed partial/ordinary differential equation for a kinetic formulation (Viswanathan 1996). The resulting set of nonlinear algebraic and partial differential equations describing equilibrium concentrations and transport, respectively, can then be solved separately. By minimising the total Gibbs free energy of the chemical system (Novak & Sevougian 1992), solutions for each master or primary species or component can be found for the set of nonlinear algebraic equations. As equilibrium relationships are uniquely defined by the law of mass action (Novak & Sevougian 1992), once concentrations of the different components are known, concentrations of each complexed specie in (local) equilibrium with each component/master species can easily be computed from the concentrations of their components and their respective thermodynamic constants (K) (Viswanathan 1996). The concentration of the complexed species, calcium carbonate ([CaCO₃]), for instance, is calculated from the concentrations of its components [Ca²⁺] and [CO32-] and from the K value of 10^-3.224. Moreover, by replacing the kinetic by the equilibrium relationships, the number of unknowns describing a reaction network can greatly be reduced. This is because for each equilibrium relationship that replaces a kinetic relationship, one primary unknown can be eliminated. Numerically, the number of primary unknown (N_p) per spatial discretisation point can be decreased to the number of components (Nc) instead of Nc + Nx + Ng + Ns + Nm

(N_x = number of aqueous complexes, N_g = number of gaseous phases, N_s = number of sorbed species to the mineral surfaces, Nm = number of mineral phases), or even less, if mineral phases are at equilibrium with the solution (Mayer 1999). With regards to the convergence, the replacement of kinetic controlled rate expressions by equilibrium relationships decreases the stiffness of the set of equations associated with an improvement of its convergence properties. Thus, an equilibrium description may also provide a solution algorithm that converges better than a kinetic description.

Despite of being computationally more intensive and having more numerical problems, kinetic formulations are often demanded to describe mineral rock interactions such as precipitation/dissolution, which often display kinetic limitations in many groundwater systems (Steefel & Lasaga 1994). For such problems, the local equilibrium assumption might not be applicable (Viswanathan 1996). The LEA assumption may also not be applicable to some very slow reactions such as the acid buffering reactions with non-carbonate minerals and redox reactions without acceleration, e.g., Fe²⁺/Fe³⁺ transfer without bacterial acceleration (Gao et al. 2001). In fractured system, since solid phase reactions can potentially plug pores or open fractures reducing matrix diffusion and promoting rapid flow through fractures (Yeh et al. 2001), a kinetic formulation may be required to describe such effects more appropriately. Further, amorphous phases that are commonly encountered in ashes and slags are limited by a finite dissolution rate or non-existence of equilibrium and need therefore a kinetic formulation (Bäverman et al. 1999). Nevertheless, an equilibrium formulation, although a kinetic one is required, can still be helpful in finding quicker solutions, where a chemical system will proceed with the given boundary conditions, such as those of redox systems.

Another criterion to decide whether an equilibrium formulation may be used is to

rely upon the differentiation between homogeneous and heterogeneous reactions. Geochemical processes of interest in groundwater systems can be divided into homogeneous and heterogeneous reactions. Homogeneous (or also intra-aqueous, Novak & Sevougian 1992) reactions take place in a single phase (e.g., aqueous complexation between dissolved constituents) while heterogeneous reactions involve mass exchange between two or more phases such as dissolution and precipitation of minerals (Mayer 1999). As a rule of thumb, homogeneous reactions may be described by equilibrium formulations whereas heterogeneous reactions may need kinetic formulations (Novak &

Sevougian 1992).

Due to computational limitations and numerical problems as well as the virtually nonexistent database of kinetically controlled rates (Pfingsten & Carnahan 1995), the earlier simulation codes that coupled chemistry with transport typically assumed that all geochemical components present in a system were in local equilibrium (Cederberg et al. 1985; Engesgaard & Kipp, 1992; Rubin, 1983; Yeh

& Tripathi 1989). In present days also, equilibrium approaches in reactive transport models are popular due to the accessibility of large geochemical databases and sophisticated software packages (Viswanathan 1999). However, to allow kinetic controlled rate expressions for certain types of reactions, the current trend in reactive transport simulation is to develop combined kinetic and equilibrium transport, mixed equilibrium/kinetic models (Viswanathan 1999) or partial equilibrium transport models (Mayer 1999). This type of approach is not as computationally intense as that of fully kinetic models. Several options are available to add a mixed kinetic/equilibrium reaction capability to reactive transport models. A common technique adopted by mixed equilibrium-kinetic reactive transport codes is the reduction of the number of independent variables by using the idea of chemical components. A set of chemical components or master or primary species is defined as the minimum number of species that uniquely describe a chemical system (Mangold & Tsang 1991). By transporting components rather than species, the number of coupled partial differential equations (PDEs) is reduced. Due to the reduction of mixed partial and ordinary differential equations, the mixed formulation solves for fewer coupled PDEs and will therefore also be computationally more efficient than a fully kinetic formulation (Viswanathan 1996). An alternative to a mixed equilibrium/kinetic approach is to make use of so-called pseudo-kinetic rate expressions like the CHEMFRONTS model developed by Bäverman et al. (1999) does. This also allows relaxing the stringent conditions of local equilibrium.

In sum, consideration of not only equilibrium but also kinetic chemistry, along with the hydrologic transport and the interaction between fluid flow and reactive transport is necessary to be able to reflect the complexity of many real systems (Yeh et al. 2001). Due to the current existence of a considerable body of knowledge of mineral kinetic rate constants, the increase in relatively inexpensive computational resources as well as developments in formulating chemical rate expressions, in recent years, a number of kinetic controlled reactions are possible to include in a simulation along with equilibrium reactions while achieving results in a reasonable time.