A parallel global-implicit 2-D solver for reactive transport problems in porous media based on a reduction scheme and its application to the MoMaS benchmark problem

Abstract

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.

A global approach to reactive transport: application to the MoMas benchmark

“…We check convergence in our numerical experiments and focus on specific behaviour such as numerical oscillations. This benchmark is also studied by other participants [4,16,19,22]. Our results are in good agreement with theirs.…”

Looking for some reference solutions for the reactive transport benchmark of MoMaS with SPECY

“…5). As presented by Hoffmann et al [11], these oscillations are dependent Figure 4 shows that a good convergence is achieved for a number of cells greater than 2,050. Nevertheless, it can be seen in Fig.…”

“…For this reason, we think that further developments about the numerical methods used for reactive transport should deal with time step and mesh adaptation. The actual way used for time step adaptation is a heuristic one [11,14]: if the iterative procedure (between transport and chemistry for sequential iterative approaches or for solving the reactive transport system for global approaches) converges father (resp. lower) than a prescribed value, the time step is increased (resp.…”

Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems—the MoMaS benchmark case

“…Application of (a) and (b) leads to a decoupling of the linear PDEs; this decoupling in combination with (c) leads to a reduction of the size of the nonlinear system (see Kräutle and Knaber [22], Hoffmann et al [19], and the references therein for details). The system of equations is handled in the spirit of a global implicit approach (one-step method) and avoids operator splitting.…”

“…We use four test cases, from the so-called Easy test case collection of the MoMaS reactive transport benchmark. Additional simulation results for these test cases and other test cases [10] are documented in the contributions by the individual participants [6,12,19,23,26].…”