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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

International audienc

International audienceNumerical benchmark can be an efficient way to validate reactive transport codes. The reactive transport benchmark of GNR MoMaS is here presented and solved on its easy 1D version. The reactive transport code SPECY is presented with a brief description of its main numerical methods: discontinuous finite elements for solving advection, mixed hybrid finite elements for solving dispersion and Newton–Raphson method to linearise the equilibrium chemistry and respect of the chemically allowed interval and positive continuous fractions methods to increase the robustness of the chemistry resolution. By successive mesh and time step refinement, we use the reactive transport code SPECY to look for a reference solution to this problem

Nonlinear reactive transport problems can be solved using the Operator Splitting (OS) approach, where transport and reaction processes are separated or the Direct Substitution Approach (DSA) where chemical and transport equations are solved simultaneously. The OS techniques can be very attractive, but are known to introduce splitting errors with SNIA (Non Iterative OS) and have low convergence rate with SIA (Iterative OS). These problems are avoided with DSA which is more robust than OS schemes. On the other hand, DSA is more complicated and very demanding in terms of computing time and memory requirements. This can make DSA less efficient than OS schemes especially for fine discretizations and chemically simple problems. In this work, DSA, SIA and SNIA are combined with a new sparse direct (unifrontal/multifrontal) solver. The efficiency of this solver is not dependent on the matrix conditioning. The performance of the three approaches is studied for two transport problems with simple and difficult chemical reactions and for different number of unknowns. Results show that when combined with an efficient sparse direct solver, DSA is more efficient than SIA and SNIA even for chemically simple problems and large number of unknowns.

In numerical codes for reactive transport modeling, systems of nonlinear chemical equations are often solved through the Newton Raphson method (NR). NR is an iterative procedure that results in a sequential solution of linear systems. The algorithm is known for its effectiveness in the vicinity of the solution but also for its lack of robustness otherwise. Therefore, inaccurate initial conditions can lead to non‐convergence or excessive numbers of iterations, which significantly increase the computational cost. In this work, we show that inaccurate initial conditions can lead to very ill‐conditioned system matrices, which makes NR inefficient. This efficiency is improved by preconditioning techniques and/or by coupling the NR method with a zero‐order method called the positive continuous fraction method. Numerical experiments that are based on seven different test cases show that the ill‐conditioned linear systems within NR represent a problem and that coupling NR with a method that bypasses the computation of the Jacobian matrix significantly improves the robustness and efficiency of the algorithm. © 2016 American Institute of Chemical Engineers AIChE J, 63: 1246–1262, 2017

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