Comparison between global and operator splitting approaches for modeling multicomponent reactive transport in porous media

Multicomponent reactive transport with chemical equilibrium reactions, involving advective and dispersive solute transport coupled with the nonlinear reaction, is fundamental feature of subsurface environments. At the equilibrium, the governing system of equations is formed by the partial differential equations (PDEs) of the transport operator and the nonlinear algebraic equations (AEs) describing the chemical reactions. These two sets of equations are coupled and needed to be solved using the operator splitting (OS) approach or the global approach. With (OS), the transport and reaction equations are separated and solved sequentially for each time step. The OS includes the Sequential non-iterative approach (SNIA) and the Sequential iterative approach (SIA), which iterates between transport and chemistry until convergence for each time step. With the global approach, the governing equations of transport and reaction are solved simultaneously. We distinguish two methods in the global approach. (i) The Differential Algebraic Equations Approach (DAE) solves a large system formed by both transport and chemistry equations (Miller, 1983). (ii) The direct substitution approach (DSA) solves a reduced system obtained after substituting the chemistry algebraic equations in the transport partial differential equations (Shen and Nikolaidis, 1997; Jenning et al., 1982). Since the reference work of Yeh and Tripathi (1989), the (OS) approach is widely used to simulate reactive transport problems since it allows different numerical methods to be used for the reactive and transport components. However, the (OS) approach can introduce operator-splitting errors which are avoided with the global approach. In this work, we show that DSA and DAE have the same numerical behavior. For a fine discretization and/or for a large number of chemical species, DSA is shown to be more efficient then DAE. Both DSA and SIA give accurate results. Contrarily to DSA, SIA solves sequentially two small systems (transport and chemistry). It was shown in Saaltink (2001) that DSA runs faster than SIA in chemically difficult cases and the SIA may become faster than the DSA for very large, chemically simple problems. In this work, we combine DSA with a very efficient linear solver UMFPACK. Our numerical experiments suggest that for all cases, DSA is shown to be more efficient than SIA. DSA requires less iteration to reach the convergence and allows large time steps contrarily to SIA. Comparisons between DSA and SNIA show that SNIA spends less CPU time than DSA. However OS errors introduced with SNIA are proportional to the time step size and can therefore be significant. When SNIA is combined with an adaptative time stepping procedure based on a posteriori control error, DSA can be more efficient then SNIA.