Although multicomponent reactive transport modeling is gaining wider application in various geoscience fields, it continues to present significant mathematical and computational challenges. There is a need to solve and compare the solutions to complex benchmark problems, using a variety of codes, because such intercomparisons can reveal promising numerical solution approaches and increase confidence in the application of reactive transport codes. In this contribution, the results and performance of five current reactive transport codes are compared for the 1D and 2D subproblems of the so-called easy test case of the MoMaS benchmark (Carrayrou et al., Comput Geosci, 2009, this issue). This benchmark presents a simple fictitious reactive transport problem that highlights the main numerical difficulties encountered in real reactive transport problems. As a group, the codes include iterative and noniterative operator splitting and global implicit solution approaches. The 1D easy advective and 1D easy diffusive scenarios were solved using all codes, and, in general, there was a good agreement, with solution discrepancies limited to regions with rapid concentration changes. Computational demands were typically consistent with what was expected for the various solution approaches. The differences between solutions given by the three codes solving the 2D problem are more important. The very high computing effort required by the 2D problem illustrates the importance of parallel computations. The most important outcome of the benchmark exercise is that all codes are able to generate comparable results for problems of significant complexity and computational difficulty.Keywords MoMaS · Benchmark · Code intercomparison · Numerical methods for reactive transport · Direct substitution approach (DSA) · Differential and algebraic equations (DAE) · Sequential iterative approach (SIA) · Sequential noniterative approach (SNIA) 484 Comput Geosci (2010) 14:483-502

1] In this article a systematic approach for the efficient computation of the transport and reaction of a multispecies, multireaction system is proposed. The objective of this approach is to reformulate the given system of differential or differential-algebraic equations in such a way that the couplings and the nonlinearities are concentrated in a reduced number of equations (if compared to the original formulation), while some linear equations decouple from the system. The resulting system is handled in the spirit of a global implicit approach (''one step method'') avoiding operator splitting techniques. The reduction of the problem size proposed in this article helps to limit the large computational costs of numerical simulations for such problems. The reduction mechanism is a generalization of the method proposed in a previous paper. Now, problems with mixed mobile/immobile species, homogeneous/heterogeneous kinetic/equilibrium reactions are considered, while the previous publication was restricted to problems without heterogeneous equilibrium reactions (such as equilibrium sorption). An application of the reduction mechanism to an example problem is given in order to investigate the reduction of the number of coupled nonlinear equations and to compare it to other methods.Citation: Kräutle, S., and P. Knabner (2007), A reduction scheme for coupled multicomponent transport-reaction problems in porous media: Generalization to problems with heterogeneous equilibrium reactions, Water Resour. Res., 43, W03429,

[1] A new systematic approach for the efficient computation of the transport and reaction of a multispecies multireaction system is developed. The objective of this approach is to reduce the number of coupled nonlinear differential equations drastically, while splitting errors are avoided. The reduction mechanism is able to handle both kinetic reactions and heterogeneous equilibrium reactions and mobile and immobile species. It leads to a formulation of the nonlinear system with a Jacobian that has very few nonzero entries. Applications of the reduction mechanism to reaction networks, including a biodegradation problem which is modeled by the Monod approach, are given. Two numerical examples demonstrate the speed up of the presented reduction mechanism.Citation: Kräutle, S., and P. Knabner (2005), A new numerical reduction scheme for fully coupled multicomponent transportreaction problems in porous media, Water Resour. Res., 41, W09414,

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In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton's method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method.