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

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). 1 Problem Specification Computer 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. 1.1 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 (1.1) ( ) , C R SS SS C C + + = ∂ ∂ L t * Supported by the System-Wide Water Resources Program (SWWRP) of U.S. Army Corps of Engineers; Allowed to present by the chief of US Army Corps of Engineers. Coastal and Hydraulics Laboratory, US Army Engineer Research and Development Center, Vicksburg, MS. where C is the species concentration vector; L() denotes the linear transport operator that accounts for advection, diffusion, and dispersion; SSR represents nonlinear sources/sinks due to biogeochemical reactions; and SSC 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), SSR, and SSC in Eq. (1.1) may vary over wide ranges. While L(C) and SSC fall in specific ranges as defined by the associated hydrologic system, the range of SSR 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 DamKöhler number [1], Da, which is defined as