An analysis of complex reaction networks in groundwater modeling

Abstract

Abstract. The complex chemistry describing the biogeochemical dynamics in the natural subsurface environments gives rise to heterogeneous reaction networks, the individual segments of which can feature a wide range of timescales. This paper presents a formulation of the mass balance equations for the batch chemistry and the transport of groundwater contaminants participating in such arbitrarily complex networks of reactions. We formulate the batch problem as an initial-value differential algebraic equation (DAE) system and compute its "index" so that the ease of solvability of the system is determined. We show that when the equilibrium reactions obey the law of mass action, the index of this initial-value DAE system is always unity (thus solvable with well-developed techniques) and that the system can be decoupled into a set of linearly implicit ordinary differential equations and a set of explicit algebraic equations. The formulations for the transport of these reaction networks can take advantage of their solvability properties under batch conditions. To avoid the error associated with time splitting fast reactions from transport, we present a split-kinetics approach where the fast equilibrium reactions are combined with transport equations while only the slower kinetic reactions are time split. These results are used to formulate and solve a simplified reaction network for the biogeochemical transformation of Co(II) ethylenediaminetetraacetic acid (EDTA) in the presence of iron-coated sediments.

A step-by-step procedure for pH model construction in aquatic systems

“…Effectively, the process with the smallest characteristic time scale will set the pace of how the integration procedure progresses with time. Given the small time scales of the acid base reactions, pH models are very impractical or virtually impossible to solve with the FKA, even with integration methods that are specifically geared towards stiff problems (Chilakapati et al, 1998). A runtime comparison of all our presented approaches, including the FKA (solution method 1b), is given at the end of the paper.…”

“…The system can be brought into a DASSL-solvable form by means of a transformation into the canonical form as discussed in, e.g., DiToro (1976), Steefel and MacQuarrie (1996), Lichtner (1996), Saaltink et al (1998), Chilakapati et al (1998) and Meysman (2001), based on an idea put forward by Aris and Mah (1963). During this transformation, the unknown equilibrium reaction rates are eliminated from the system.…”

“…Regnier et al (1997) used the OSA to model pH along an estuarine gradient, Marinelli and Boudreau (1996) used it to study the pH in irrigated anoxic coastal sediments, and Follows et al (1996) used the OSA to investigate the carbonate system in the water column of the North Atlantic. Besides pointing out different varieties of the FNA, Chilakapati et al (1998) also applied the OSA to simple groundwater problems. While not explicitly reformulating the system, Boudreau (1987), Boudreau and Canfield (1988), Boudreau (1991), Boudreau and Canfield (1993), and Boudreau (1996a) (the CANDI model) used the notion of dividing the reaction set into kinetic reactions and equilibrium reactions.…”

“…The most basic approach, the Full Kinetic Approach (FKA) has only been implemented sporadically (Steefel and MacQuarrie, 1996;Zeebe, 2007), because of the numerical stiffness of the obtained equation systems (solution method 1a and 1b), and the need to obtain parameters that are not very well constrained (forward and backward rates of acid-base reactions in solution method 1a). After one reformulation step termed the transformation into canonical form (DiToro, 1976;Lichtner, 1996;Steefel and MacQuarrie, 1996;Chilakapati et al, 1998;Saaltink et al, 1998;Meysman, 2001) based on an idea first put forward by Aris and Mah (1963), one can implement the Full Numerical Approach (FNA), which involves a direct numerical solution of the resulting differential algebraic equation system. Steefel and MacQuarrie (1996) list a number of packages, including DASSL (Petzold, 1982), capable of solving a system according to the FNA.…”

Uranium Mobility During In Situ Redox Manipulation of the 100 Areas of the Hanford Site

“…The combined method is a direct numerical approximation of the Reynolds transport theorem in that fluid packets are followed along volume-preserving streamlines and the dispersive, reactive contributions are computed. The code has been tested extensively and the convergence of the numerical methods established (Chilakapati et al 1998). The accuracy of the multireaction reactive transport system in this study was compared with an analytical solution of transport with a single kinetic reaction (Parker and van Genuchten 1984) and a numerical code transport with multiple kinetic/equilibrium reactions (Salvage et al 1995;Yeh et al 1998).…”