Using a Reaction-Based Differential-Algebraic Equation Approach for Reactive Transport Modeling in Hydrologic Systems

Abstract

We discuss how a Reaction-Based, DifferentialAlgebraic Equation (RB-DAE) approach can provide robust numerical solutions for biogeochemical systems that are composed of fast, slow, reversible, and irreversible reactions. We also discuss several numerical strategies for solving reactive transport equations using high performance computing (HPC). Problem SpecificationComputer simulation of flow and transport is an essential component in the rigorous analysis of water supply, contamination, environmental cleanup, and ecosystem restoration. Simulation of coupled equations for transport and biogeochemical reactions based on the principle of conservation is the only quantitative approach to date for integrating multiple, complex, environmental processes into an internally consistent conceptual model with which to assess water quality and to design engineered solutions for remedial alternatives. A reactive transport (RT) model in a more general sense treats a multi-component, multi-species system in which a number of equilibrium-controlled (fast reversible) and perhaps kinetic (slow) and instantaneous (fast irreversible) reactions occur simultaneously. In a typical hydrologic cycle, water moves on, above, or below the surface of the earth, and its speed can vary over many orders of magnitude. Similarly, the rate of a biogeochemical reaction, whether natural or man-induced, can change drastically over time and space. The combination of these two facts makes RT modeling in hydrologic systems an extremely difficult task. RT Equations in Primitive Form.A typical set of RT equations include transport processes (e.g., advection, diffusion, dispersion), biogeochemical reactions, and sources/sinks. They can be written in the so-called primitive form as where C is the species concentration vector; L() denotes the linear transport operator that accounts for advection, diffusion, and dispersion; SS R represents nonlinear sources/sinks due to biogeochemical reactions; and SS C represents linear sources/sinks due to other activities, such as injection or extraction. For immobile species, the transport and source/sink terms in Eq. (1.1) may be neglected.As previously mentioned, L(C), SS R , and SS C in Eq. (1.1) may vary over wide ranges. While L(C) and SS C fall in specific ranges as defined by the associated hydrologic system, the range of SS R depends on the characteristics of the reactions taken into account. The relative importance of reaction and transport at a specific distance scale L can be described by the non-dimensional . R SS C = dt dNo matter which solution technique is used in solving RT equations, it is vital to model biogeochemical reactions and solve for concentration distributions among species accurately and efficiently. Bethke [2] discussed the conceptual model, mathematical formulations, and numerical solutions for reaction processes and systems of various kinds. His discussion demonstrates the capabilities of a reaction model and determines the model's effect on the performance of reactive transport mo...