Abstract. We present a concise and general mathematical formulation for reactive transport in groundwater for general applications. By means of linear algebraic manipulations of the stoichiometric coefficients of the chemical reactions we are able to reduce the number of unknowns of the equations to be solved to the number of degrees of freedom according to thermodynamic rules. We present six formulations that differ from each other by number and type of unknowns and discuss their advantages and disadvantages with respect to the two most important numerical solution methods, the Sequential Iteration Approach (SIA) and the Direct Substitution Approach (DSA). Our conclusion is that the proposed reduction of the number of variables is of special interest for the DSA. We have applied one of these formulations to an example of the flushing of saline water by fresh water.
[1] Chemical reactions are driven by disequilibrium, which is often caused by mixing. Therefore quantification of the mixing rate is essential for evaluating the fate of solutes in natural systems, such as rivers, lakes, and aquifers. We propose a novel mixing ratios-based formulation to evaluate solute concentrations and reaction rates when equilibrium aqueous reactions and precipitation/dissolution of minerals are driven by mixing of different end-members. Each end-member corresponds to a water from a given source with a specific chemical signature. The approach decouples the solute transport and chemical speciation problems, so that mixing ratios can be first obtained from the solution of conservative transport and then be used in general speciation codes to obtain the concentration of reacting species. One key finding is a general expression for reaction rates which demonstrates that the amount of reactants evolving into products depends on the rate at which solutions mix. Our formulation constitutes a general framework according to which one can design and interpret experimental analyses devoted to study mixing-driven reactive processes and obtain transverse dispersion coefficients. The formulation is also proposed as a useful tool to derive analytical solutions of reactive transport problems and may result computationally advantageous when compared to previous approaches to reactive transport modeling. We apply the developed formulation to provide an analytical solution of the reactive transport process resulting from mixing different CaCO 3 -saturated waters in a two-dimensional setup.
Reactive transport equations may become cumbersome to solve when a large number of species are coupled through fast (equilibrium) and slow (kinetic) reactions. Solution is further encumbered when both mobile (solutes) and immobile (minerals) species are involved. In these cases, sequential iteration approaches (SIA) may become extremely slow to converge. Direct substitution approaches (DSA), which solve all transport equations together, may become extremely large. Here we propose a formulation for optimally decoupling the reactive transport equations. The procedure is described sequentially using a paradigm system. We start by a tank paradigm in which all species are mobile and undergo equilibrium reactions. In this case, all components are fully decoupled. If some of the reactions are kinetic (canal paradigm), then we can still decouple components corresponding to equilibrium reactions. The same can be said regarding a system in which immobile species only react kinetically (river). The number of components can be reduced in the case that both types of reactions and species are present (aquifer). In short, the number of coupled equations that need to be solved simultaneously is, at most, equal to the number of kinetic reactions. This benefits both SIA and DSA solution methods. SIA should improve convergence because most components are linear and effectively decoupled, thus reducing nonlinearity to kinetic terms only in transport equations for kinetic components. DSA systems become reduced as the number of components that need to be solved together is, at most, equal to the number of independent kinetic reactions.
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