On the Efficiency of the Direct Substitution Approach for Reactive Transport Problems in Porous Media

Abstract

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.

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

“…In addition, there are very few comparisons that provide information about the performance of the numerical methods used. The literature devoted to the comparison of sequential and global approaches for T-C coupling [13,16,35,36,39,42,43] provides some discussion that is mostly qualitative in nature. Reeves and Kirkner [34] provide the computing times required for the solution of a 1D problem with sorption of one, two, or three components for a number of methods.…”

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

“…Since the review article of Yeh and Tripathi [28], three ways are well known for reactive transport modelling: the global approach where both transport and chemical operators are solved simultaneously [9,10,22,24]; the iterative operator splitting approach where both operators are solved separately but coupled by Picard-like iterations [5,6,14,18,26]; and the non-iterative operator splitting approach where transport and chemistry operators are separately solved only one time per time step [1,2,27]. In this work, we present the non-iterative operator splitting approach.…”

An Efficient Implementation of the Method of Lines for Multicomponent Reactive Transport Equations

“…1 is discretized using centered finite volumes for the spatial derivative and first-order backward Euler scheme for the time derivative, (2) The chemical Eqs. 3-8 are substituted into the discretized transport equation, and (3) The resulting system of nonlinear algebraic equations is solved iteratively with the Newton-Raphson method (Fahs et al 2008). The tolerance criterion for the convergence of the iterative process is ε STD =10 −5 .…”

“…However, the GIA is known to be difficult to implement and requires the solution of a larger system than with the OS approach (Saaltink et al 2001). The comparison between GIA and OS approach has been largely discussed in the reactive transport literature (Yeh and Tripathi 1989;Steefel and Yabusaki 1995;Steefel and MacQuarrie 1996;Shen and Nikolaidis 1997;Saaltink et al 2001;Hammond et al 2005;Fahs et al 2008). Several recent works have shown that the GIA can be more efficient than the OS approach (Steefel and MacQuarrie 1996;Shen and Nikolaidis 1997;Saaltink et al 2001;Fahs et al 2008).…”

“…The comparison between GIA and OS approach has been largely discussed in the reactive transport literature (Yeh and Tripathi 1989;Steefel and Yabusaki 1995;Steefel and MacQuarrie 1996;Shen and Nikolaidis 1997;Saaltink et al 2001;Hammond et al 2005;Fahs et al 2008). Several recent works have shown that the GIA can be more efficient than the OS approach (Steefel and MacQuarrie 1996;Shen and Nikolaidis 1997;Saaltink et al 2001;Fahs et al 2008). In the last decade, the GIA has been increasingly implemented in numerical models for MRT simulations (for example, TOUGH2 (White 1995), GIMRT (Steefel and Yabusaki 1995), and ARASE (Grindrod and Takase 1996).…”