## 1. Introduction

[2] This article is concerned with the efficient numerical solution of a large system of partial differential equations (PDEs), ordinary differential equations (ODEs), and algebraic equations (AEs), as they occur in the modeling of transport, chemical reactions and biodegradation below the Earth's surface. The topic of this article is the proposition of a method to reformulate the given system of equations describing the problem, in such a way that the resulting system is easier to solve. A rather general approach for a wide class of such biogeochemical problems is proposed in this article. In fact, this article is a generalization of the work by *Kräutle and Knabner* [2005], where equilibrium reactions between mobile and immobile species were excluded.

[3] The chemical species which are considered are divided into two classes: Mobile (dissolved) species and immobile (sorbed chemical species, minerals, immobile bacteria) ones. The system of equations for the concentrations consists of PDEs for the mobile species and ODEs for the immobile ones, all of them coupled through the reaction terms. The characteristic timescale of the different reactions may cover a large range making it desirable to model some reactions as equilibrium-controlled and others as kinetically controlled, leading to a differential algebraic system of equations (DAE).

[4] Basically there are two different concepts to treat these kinds of problems numerically: By global implicit approaches (GIA) and by sequential iterative and sequential noniterative approaches (SIA/SNIA). The GIA requires most resources per time step, but is usually considered to be the most stable solution method. SIA and SNIA, depending on the specific problem to solve, may suffer from heavy restrictions on the time step size to gain convergence, or from the introduction of large splitting errors, respectively [see, e.g., *Steefel and MacQuarrie*, 1996; *Valocchi and Malmstead*, 1992; *Saaltink et al.*, 2000; *Berkvens et al.*, 2002].

[5] In this article, our goal is to avoid such problems by focussing on the GIA. In order to keep the computational effort with respect to memory requirements and cpu time limited, we want to reformulate the given system of PDEs/ODEs or DAE in such a way that some of the equations decouple, leading to a smaller nonlinear system to which we apply the GIA. The reformulation of the given system is performed by (1) taking linear combinations of the given equations and (2) by introducing new variables which are linear combinations of the unknown concentrations. This leads to a decoupling of some scalar linear transport equations and a smaller remaining nonlinear system of PDEs, ODEs, and AEs. In another step, the local equations, i.e., the ODEs and the AEs, are solved for certain variables, and these variables are eliminated in the remaining PDEs, which reduces the size of the coupled system again and which resembles the so-called direct substitutional approach (DSA).

[6] There are many papers dealing with the efficient solution of transport-reaction problems in porous media [e.g., *Lichtner*, 1985; *Yeh and Tripathi*, 1989; *Friedly*, 1991; *Friedly and Rubin*, 1992; *Saaltink et al.*, 1998; *Chilakapati et al.*, 1998, 2000; *Robinson et al.*, 2000; *Holstad*, 2000; *Fang et al.*, 2003] (also, recently, the very advanced paradigm system by *Molins et al.* [2004]). A main difference of the method proposed in this article to other reformulations [e.g., *Saaltink et al.*, 1998, 2000; *Molins et al.*, 2004] is that when we introduce linear combinations of concentrations or equations, we lay special emphasis on the distinction between mobile species and immobile species, not mixing up mobile and immobile species during the transformation. A benefit of this proceeding compared to other methods is that it enables a decoupling of some equations without posing additional assumptions on the stoichiometry of the problem and without “enforcing” a decoupling by splitting techniques. Another advantage can be seen in the fact that the DSA-like treatment of the local equations in our resulting system preserves a very sparse population of the Jacobian [see *Kräutle and Knabner*, 2005, section 3.4], which can be exploited if the linear solver is an iterative method.

[7] The article is structured as follows: In section 2, the equations for the coupled reactive transport are given. In section 3, which is the main part of this article, the general reduction algorithm including the case of heterogeneous equilibrium reactions is derived. The algorithm presented in section 3 requires a certain condition on the stoichiometric matrix. In section 4 it is demonstrated that every stoichiometric system can be written in such a form that the required condition is met. Section 5 demonstrates the application of the method to an example problem. A comparison of the proceeding to other methods is included in order to motivate our method.

[8] A mathematical proof that the local equations can always be solved for certain variables, if mass action law is assumed for the equilibrium reactions, is given in Appendix A.