A formulation for decoupling components in reactive transport problems

Abstract

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.

Toward efficiency in heterogeneous multispecies reactive transport modeling: A particle‐tracking solution for first‐order network reactions

“…For instance, first-order degradation reactions of a single species can be included into PTMs by assigning to every particle a variable mass, which develops in time according to first-order kinetics [Kinzelbach, 1987;Wen and Gomez-Hernandez, 1996]. When all species share the same transport operator, certain reactions in chemical equilibrium can be easily simulated with particle tracking by using conservative components [Molins et al, 2004;Kr€ autle and Knabner, 2005;De Simoni et al, 2005;FernandezGarcia et al, 2008;Fernandez-Garcia and Sanchez-Vila, 2011], i.e., a linear combination of the species concentrations that can be used to decouple the system of equations into simpler problems. Fast kinetic reactions have been properly simulated by applying simple proximity relationships between nearby particles [Edery et al, 2009[Edery et al, , 2010.…”

Conditional Probability Density Functions of Concentrations for Mixing-Controlled Reactive Transport in Heterogeneous Aquifers

“…This approach can be applied when the concentrations of the reacting species stand in algebraic relationships, and the coefficients describing physical mixing in the system coincide for all compounds. The former requirements are met by systems either in local chemical equilibrium or instantaneous, complete, irreversible reactions (Ham et al 2004;Liedl et al 2005) and can also be used for specific cases of kinetic reactions (Molins et al 2004;Cirpka and Valocchi 2007). A methodology to compute directly homogeneous and heterogeneous reaction rates under instantaneous equilibrium has been presented recently by De Simoni et al (2005.…”

“…A common method to estimate spatial distributions of chemical species concentrations and associated reaction rates is to employ numerical modeling. A series of mathematical formulations are available in the literature, which are included in a variety of codes (Rubin 1990;Yeh and Tripathi 1991;Friedly and Rubin 1992;Lichtner 1996;Steefel and MacQuarrie 1996;Tebes-Stevens et al 1998;Clement et al 1998;Saaltink et al 1998;Parkhurst and Appelo 1999;Robinson et al 2000;Molins et al 2004). All these methodologies are based on the idea that reactive transport problems can be reformulated mathematically in terms of chemical components, defined as linear combinations of reactive species concentrations.…”

Computational Efficiency of Decoupling Approach in Solving Reactive Transport Model: A Case Study of Pyrite Oxidative Dissolution

“…More detailed solution steps of F can be seen in Molins et al [32]. (4) The fourth is aquifer system, where some heterogeneous reactions are fast enough to be considered in equilibrium.…”

“…According to Molins et al [32], four types of reactive transport systems are classified by the types of reactions, as shown in Table 1.…”