A domain decomposition method using efficient interface-acting preconditioners - Kräutle, Serge (Mathematics of Computation, 2005)
The conjugate gradient boundary iteration (CGBI) is a domain decomposition method for symmetric elliptic problems on domains with large aspect ratio. High efficiency is reached by the construction of preconditioners that are acting only on the subdomain interfaces. The theoretical derivation of the method and some numerical results revealing a convergence rate of 0.04-0.1 per iteration step are given in this article. For the solution of the local subdomain problems, both finite element (FE) and spectral Chebyshev methods are considered. © 2004 American Mathematical Societ
Keywords: Domain decomposition; FETI; Parallelization; Preconditioning

A general reduction scheme for reactive transport in porous media - Hoffmann, Joachim; Kräutle, Serge; Knabner, Peter (Computational Geosciences, 2012)
We present a method to transform the governing equations of multispecies reactive transport in porous media. The reformulation leads to a smaller problem size by decoupling of equations and by elimination of unknowns, which increases the efficiency of numerical simulations. The reformulation presented here is a generalization of earlier works. In fact, a whole class of transformations is now presented. This class is parametrized by the choice of certain transformation matrices. For specific choices, some known formulations of reactive transport can be retrieved. Hence, the software based on the presented transformation can be used to obtain efficiency comparisons of different solution approaches. For our efficiency tests, we use the MoMaS benchmark problem on reactive transport. © 2012 Springer Science+Business Media B.
Keywords: Global implicit approach; Numerical simulation; Porous media; Reactive transport; Reduction of problem size

A Navier-Stokes solver based on CGBI and the method of characteristics - Kräutle, Serge (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2001)

A new numerical reduction scheme for fully coupled multicomponent transport-reaction problems in porous media - Kräutle, Serge; Knabner, Peter (Water Resources Research, 2005)
[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. Copyright 2005 by the American Geophysical Unio

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 - Hoffmann, Joachim; Kräutle, Serge; Knabner, Peter (Computational Geosciences, 2010)
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. © Springer Science + Business Media B.V. 200
Keywords: Complementarity problems; Numerical simulation; Porous media; Reactive transport; Size reduction

A Parallel Hybrid Highly Accurate Elliptic Solver for Viscous Flow Problems - Borchers, Wolfgang; Forestier, Yvon; Kräutle, Serge; Pasquetti, Richard; Peyret, Roger; Rautmann, Reimund; Roß, N.; Sabbah, Claude (Numerical Flow Simulation I, 1998)
We present a new parallel hybrid method to solve numerically elliptic equations on a channel-like domain. The method combines the highly accurate Chebyshev — spectral method with a standard finite difference one, via the CGBI — domain decomposition procedure. By this approach the solution of linear elliptic boundary value problems is reduced to a minimization principle for the unknown Neumann boundary data distributed on the subdomain interfaces. The subdomain solvers are based on Chebyshev spectral / finite difference methods, but finite elements, instead of finite differences, could be used to deal with more complicated geometries.

Approximationen der Navier-Stokes-Gleichungen mit Finiten Differenzen - Kräutle, Serge (Friedrich-Alexander-Universität Erlangen-Nürnberg, 1996)

A reduction scheme for coupled multicomponent transport-reaction problems in porous media: Generalization to problems with heterogeneous equilibrium reactions - Kräutle, Serge; Knabner, Peter (Water Resources Research, 2007)
[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. Copyright 2007 by the American Geophysical Unio

Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems-the MoMaS benchmark case - Carrayrou, Jérôme; Hoffmann, Joachim; Knabner, Peter; Kräutle, Serge; de Dieuleveult, Caroline; Erhel, Jocelyne; Van der Lee, Jan; Lagneau, Vincent; Mayer, K. Ulrich; MacQuarrie, Kerry T. B. (Computational Geosciences, 2010)
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. © Springer Science+Business Media B.V. 201
Keywords: Benchmark; Code intercomparison; Differential and algebraic equations (DAE); Direct substitution approach (DSA); MoMaS; Numerical methods for reactive transport; Sequential iterative approach (SIA); Sequential noniterative approach (SNIA)

Computation of the MoMaS Benchmark-Problem with a Parallel Global-Implicit 2-D solver based on a Reformulation of the PDE-ODE System - Hoffmann, Joachim; Kräutle, Serge; Knabner, Peter (International Workshop on Modelling Reactive Transport in Porous Media, 2008)

Efficient and Accurate Simulation of Large General Reactive Multicomponent Transport Processes in Porous Media by Model-Preserving Decoupling Techniques - Kräutle, Serge; Knabner, Peter; Hoffmann, Joachim (International Conference on Computational Methods in Water Resources, 2006)
Detailed modelling of reactive transport processes in the underground often requires the consideration of a wide range of reactive species. A prominent example is natural attenuation, that is the assessment and monitoring of microbially catalysed degradation processes of organic contaminants in the subsoil or aquifer with full geochemistry. Often the reactions exhibit a wide range of relaxation times, which advises to model those reactions being much faster than the time scale of the transport processes in a quasistationary manner, e.g. as (algebraicly described) equilibrium processes. Additionally not only mobile species (in solution) appear, but also immobile ones (attached to the porous skeleton). In summary, the resulting system is not semilinear and parabolic, but rather quasilinear and couples partial differential equations (pde), ordinary differential equations and algebraic equations. An often used approach is operator splitting, in which transport and reaction becomes (iteratively) decoupled. This procedure either introduces a further consistency error (in the non-interative version) which can only be controlled by the time stepping, or applies a fixed point type iteration of unclear convergence properties. We rather propose, after appropriate (mixed) finite element discretization, to deal with the full discrete nonlinear system (the Global Implicit Approach,in principal by a damped Newton's method). To make the problem still feasible we advise two means: The first is concerned with the continuous model and aims at a transformation of the dependent variables such that as many as possible are determined by decoupled linear pde's or by local algebraic relations, leading to a smaller coupled system. The problem lies here in the combined appearance of kinetics and equilibrium and mobile and immobile species. Alternatively to this exact a priori decoupling we use an a posteriori decoupling on the level of the linear system of equation in the Newton's method by sparsening, i.e. ignoring weak couplings in the Jacobian matrix. The resulting benefit in the solution of the linear system should supersede a possible deterioration in the convergence of the iterative method, being now only an approximate Newtons's method. Iterative operator splitting can be viewed as of this type with two half-steps, first ignoring e.g. the reactive couplings (the transport step) and then ignoring the transport couplings (the reaction step). The introduced larger flexibility of the method helps to find selective couplings where there is still a good (Newton-like) convergence in cases where operator splitting fails. The benefit of the two approaches, which in principle are also combinable, will be elucidated for several large problems from the hydrological literatur

Entwicklung, Zuverlässigkeit und Effizienz reaktiver Mehrkomponententransportmodelle - Knabner, Peter; Frank, Florian; Hoffmann, Joachim; Kräutle, Serge; Oßmann, Stephan; Prechtel, Alexander (KORA - Synopse "Systemanalyse, Modellierung und Prognose der Wirkungen natürlicher Schadstoffminderungsprozesse - eine rezente Synopse", 2008)

Existence and uniqueness of a global solution for reactive transport with mineral precipitation-dissolution and aquatic reactions in porous media - Hoffmann, Joachim; Kräutle, Serge; Knabner, Peter (SIAM Journal on Mathematical Analysis, 2017)
We consider a macroscopic (averaged) model of transport and reaction in the porous subsurface. The model consists of PDEs for the concentrations of the mobile (dissolved) species and of ODEs for the immobile (mineral) species. For the reactions, we assume the kinetic mass action law. The constant activity of the mineral species leads to set-valued rate functions or complementarity conditions coupled to the PDEs and ODEs. In this paper we first prove the equivalence of several formulations in a weak sense. Then we prove the existence and the uniqueness of a global solution for a multispecies multireaction setting with the method of a priori estimates. In addition to mineral precipitation-dissolution reactions, the model also allows for aquatic reactions, i.e., reactions among the mobile species. In both the mineral precipitation-dissolution rates and the aquatic reaction rates we consider polynomial nonlinearities of arbitrarily high order.
Keywords: existence of global solution; reactive transport in porous media; kinetic mineral precipitation-dissolution; complementarity problems; law of mass action

Existence of global solutions of multicomponent reactive transport problems with mass action kinetics in porous media - Kräutle, Serge (JOURNAL OF APPLIED ANALYSIS AND COMPUTATION, 2011)
We prove the existence and uniqueness of time-global solutions for multi-species multi-reaction advection-diffusion-dispersion problems with mass action kinetics in the space W-p(2,1)([0, T] x Omega). The reaction terms of mass action kinetics may contain polynomial expressions of arbitrarily high order. The difficulty to obtain an a priori estimate for the semilinar system of PDEs is tackled with a special Lyapunov function.
Keywords: Existence of solutions; A priori estimate; Parabolic equations; Reactive transport; Mass action kinetics

General multi-species reactive transport problems in porous media: Efficient numerical approaches and existence of global solutions - Kräutle, Serge (2008)

Global implicit solver for multiphase multicomponent flow in porous media with multiple gas components and general reactions: Global implicit solver for multiple gas components - Knodel, Markus M.; Kräutle, Serge; Knabner, Peter (Computational Geosciences, 2022)
In order to study the efficiency of the various forms of trapping including mineral trapping scenarios for CO2 storage behavior in deep layers of porous media, highly nonlinear coupled diffusion-advection-reaction partial differential equations (PDEs) including kinetic and equilibrium reactions modeling the miscible multiphase multicomponent flow have to be solved. We apply the globally fully implicit PDE reduction method (PRM) developed 2007 by Kräutle and Knabner for one-phase flow, which was extended 2019 to the case of two-phase flow with a pure gas in the study of Brunner and Knabner. We extend the method to the case of an arbitrary number of gases in gaseous phase, because CO2 is not the only gas that threats the climate, and usually is accompanied by other climate killing gases. The application of the PRM leads to an equation system consisting of PDEs, ordinary differential equations, and algebraic equations. The Finite Element discretized / Finite Volume stabilized equations are separated into a local and a global system but nevertheless coupled by the resolution function and evaluated with the aid of a nested Newton solver, so our solver is fully global implicit. For the phase disappearance, we use persistent variables which lead to a semismooth formulation that is solved with a semismooth Newton method. We present scenarios of the injection of a mixture of various gases into deep layers, we investigate phase change effects in the context of various gases, and study the mineral trapping effects of the storage technique. The technical framework also applies to other fields such as nuclear waste storage or oil recovery.
CRIS-Team Scopus Importer:2022-04-29
Keywords: CO; Equilibrium reactions; Globally implicit solver; Injection of various gases; Nested Newton; PDE reduction method; Porous media

Global implicit solver for multiphase multicomponent flow in porous media with multiple gas phases and general reactions - Knodel, Markus; Kräutle, Serge; Knabner, Peter (Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples, 2020)
Multiphase multicomponent flow processes in porous media have to be consid- ered to study the efficiency of mineral trapping mechanisms for climate killing gas storage in deep layers. Robust predictions ask for the solution of large nonlinear cou- pled systems of diffusion-advection-reaction (partial differential) equations contain- ing equilibrium reactions. In that we elaborate the fully globally implicit Kräutle- Knabner PDE reduction method (cf. a former paper [8]) for the case of multiple gas phases, we solve the arising Finite Element discretized / Finite Volume stabilized equations by means of a semismooth nested Newton solver. We present prelimi- nary simulation results for the case of mutual injection of CO2, CH4 and H2S into deep layers and investigate the arising mineral trapping scenario. Our methods are applicable also to other fields such as nuclear waste storage or oil recovery.<br
Keywords: PDE reduction method; nested Newton; equilibrium reactions; CO2

Modellierung des reaktiven Transports von Schadstoffen in der (un-)gesättigten Bodenzone zur Prognose der natürlichen Selbstreinigung - Komplexe Probleme berechenbar machen - Knabner, Peter; Kräutle, Serge; Oßmann, Stephan; Prechtel, Alexander (Statusseminar Forschungsverbund KORA - Kontrollierter natürlicher Rückhalt und Abbau von Schadstoffen bei der Sanierung kontaminierter Grundwässer und Böden, 2005)

Multi-domain finite element - Spectral Chebyshev parallel Navier-Stokes solver for viscous flow problems - Borchers, Wolfgang; Kräutle, Serge; Pasquetti, Richard; Peyret, Roger; Rautmann, Reimund (Numerical Flow Simulation III, 2003)
The paper is concerned with a hybrid finite element - spectral Chebyshev parallel solver of the incompressible Navier-Stokes equations. A domain decomposition, well adapted to the computation of wake or jet type flow, is assumed. Subdomains with complex geometry are handled with finite elements and the other ones with the highly accurate spectral method. The iterative resolution of the multi-domain problem is carried out with the "Conjugate Gradient Boundary Iteration" method. Here we focus on its optimal preconditioning when Gauss-Lobatto type grids axe involved. The resulting algorithm is then applied to the flow past a cylinder and over a backward facing step at higher Reynolds numbers. For the latter case the numerical results display some transitional laminar-turbulent behaviour of the flow arising from unstable steady states. Numerical results are presented for both convective and absolute instability regions.

Numerical results for the CGBI method to viscous channel flow - Kräutle, Serge; Wielage, Kerstin (The Navier-Stokes Equations: Theory and Numerical Methods, 2001)
The conjugate gradient boundary iteration (CGBI) is a domain decom- position method for elliptic partial differential equations [1] [2] [3] [6 ] [7]. In the context of fractional step methods [10], a Navier-Stokes solver based on CGBI and a characteristic’s method was constructed. In this article, numerical results for the incompressible flow past a backward facing step and past a cylinder (both modelled in 2d) are presented.

ParRichy: Parallel Simulation of Bioreactive Multicomponent Transport Processes in Porous Media - Kräutle, Serge; Bause, Markus; Prechtel, Alexander; Radu, Adrian Florin; Knabner, Peter (High Performance Computing in Science and Engineering, Garching 2004, 2005)
Numerical simulations have become an important tool to predict the evolution of groundwater and subsurface contamination by organic compounds. Due to the observation of sharp interfaces on which biochemical degradation of the contaminants takes place, a reliable and accurate prognosis requires high spatial resolution of the considered subsurface domain and/or the application of higher order discretization methods. A successfull realization of these simulations can only be provided by parallel computer systems with high computing power and memory availability. In this article, our concepts and simulation results of “real world” scenarios of organic contamination are presente

Reaktive Mehrkomponentenprobleme: Sicherung von Effizienz und Zuverlässigkeit - Prechtel, Alexander; Hoffmann, Joachim; Kräutle, Serge; Knabner, Peter (Modellierung und Prognose von Natural Attenuation-Prozessen im Untergrund, 2006)

Revisiting the Analytical Solution Approach to Mixing-Limited Equilibrium Multicomponent Reactive Transport Using Mixing Ratios: Identification of Basis, Fixing an Error, and Dealing With Multiple Minerals - Ginn, Timothy; Schreyer, Lynn; Sanchez-Vila, Xavier; Nassar, Mohamed K.; Ali, A. A.; Kräutle, Serge (Water Resources Research, 2017)
Multicomponent reactive transport involves the solution of a system of nonlinear coupled partial differential equations. A number of methods have been developed to simplify the problem. In the case where all reactions are in instantaneous equilibrium and the mineral assemblage is constant in both space and time, de Simoni et al. (2007) provide an analytical solution that separates transport of aqueous components and minerals using scalar dissipation of mixing ratios between a number of boundary/initial solutions. In this approach, aqueous speciation is solved in conventional terms of primary and secondary species, and the mineral dissolution/precipitation rate is given in terms of the scalar dissipation and a chemical transformation term, both involving the secondary species associated with the mineral reaction. However, the identification of the secondary species is nonunique, and so it is not clear how to use the approach in general, a problem that is keenly manifest in the case of multiple minerals which may share aqueous ions. We address this problem by developing an approach to identify the secondary species required in the presence of one or multiple minerals. We also remedy a significant error in the de Simoni et al. (2007) approach. The result is a fixed and extended de Simoni et al. (2007) approach that allows construction of analytical solutions to multicomponent equilibrium reactive transport problems in which the mineral assemblage does not change in space or time and where the transport is described by closed-form solutions of the mixing ratios.
Keywords: reactive transport; equilibria; upscaling; mixing

Robust simulation of mineral precipitation-dissolution problems with variable mineral surface area - Kräutle, Serge; Hodai, Jan; Knabner, Peter (Journal of Engineering Mathematics, 2021)
We consider a macroscale model of transport and reaction of chemical species in a porous medium with a special focus on mineral precipitation–dissolution processes. In the literature, it is frequently proposed that the reaction rate should depend on the reactive mineral surface area, and so on the amount of mineral. We point out that a frequently used model is ill posed in the sense that it admits non-unique solutions. We investigate what consequences this non-uniqueness has on the numerical solution of the model. The main novelty in this article is our proposal of a certain substitution which removes the ill-posedness from the system and which leads to better numerical results than some “ad hoc methods.” We think that the proposed substitution is a rather elegant way to get rid of the non-uniqueness and the numerical difficulties and is much less technical than other ideas. As a proof of concept, we present some numerical tests and simulations for the new mode
Keywords: Complementarity problem; Dissolution; Mineral reactions; Newton’s method; Non-Lipschitz; Precipitation; Reactive transport; Uniqueness

The CGBI method for viscous channel flows and its preconditioning - Kräutle, Serge; Wielage, Kerstin (Nonlinear Analysis - Theory Methods & Applications, 2001)
The conjugate gradient boundary iteration (CGBI) is a domain decomposition method for elliptic partial differential equations. In the context of fractional step methods [7], a Navier-Stokes solver based on CGBI [1],[2],[3],[5] and a characteristic's method [5] is constructed. Numerical results for the flow around a cylinder (modelled in 2d) are presented.

The semismooth Newton method for multicomponent reactive transport with minerals - Kräutle, Serge (Advances in Water Resources, 2011)
In this article a method is proposed for the efficient simulation of reactive systems or reactive transport problems including (equilibrium) mineral precipitation-dissolution. The difficulty lies in the fact that for larger systems it is usually a priorily not known for which mineral the fluid is saturated and for which it is undersaturated. Currently wide-spread algorithms use some trial-and-error strategy requiring repeated computations, or an approximation of equilibrium precipitation-dissolution by a kinetic description. In this article we propose to formulate the problem as a so-called complementarity problem (CP) and to solve it with the semismooth Newton method, a solution strategy well known in the field of optimization theory. The CP formulation of the mineral reactions is then combined with a reformulation for the full multicomponent reactive transport problem which leads to a reduction of the number of unknowns. The reactive transport problem is tackled in the sense of a one step (global implicit) method. © 2010 Elsevier Ltd.
Keywords: Reactive transport; Minerals; Complementarity problems; Semismooth Newton method

The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions - Buchholzer, Hannes; Kanzow, Christian; Knabner, Peter; Kräutle, Serge (Computational Optimization and Applications, 2011)
The semismooth Newton method was introduced in a paper by Qi and Sun (Math. Program. 58:353-367, 1993) and the subsequent work by Qi (Math. Oper. Res. 18:227-244, 1993). This method became the basis of many solvers for certain classes of nonlinear systems of equations defined by a nonsmooth mapping. Here we consider a particular system of equations that arises from the discretization of a reactive transport model in the subsurface including mineral precipitation-dissolution reactions. The model is highly complicated and uses a coupling of PDEs, ODEs, and algebraic equations, together with some complementarity conditions arising from the equilibrium conditions of the minerals. The aim is to show that this system, though quite complicated, usually satisfies the convergence criteria for the semismooth Newton method, and can therefore be solved by a locally quadratically convergent method. This gives a theoretical sound approach for the solution of this kind of applications, whereas the geoscientists community most frequently applies algorithms involving some kind of trial-and-error strategies. © 2010 Springer Science+Business Media, LL
Keywords: Complementarity problems; Mineral precipitation-dissolution; Quadratic convergence; Reactive transport; Semismooth Newton method

Towards a parallel hybrid highly accurate Navier-Stokes solver - Borchers, Wolfgang; Kräutle, Serge; Pasquetti, Richard; Rautmann, Reimund; Roß, N.; Wielage, Kerstin; Xu, Chuanju (Numerical Flow Simulation II, 2001)
This paper presents the last progresses obtained in the development of a parallel multidomain/multi-method Navier-Stokes solver. Three points are addressed: (i) the preconditioning of the iterative procedure which results from using a multi-domain approach, (ii) the semi-Lagrangian method to be used for the transport step and (iii) the first results obtained with finite difference/finite element - spectral Chebyshev couplings.

Upscaling of a system of semilinear parabolic partial differential equations coupled with a system of nonlinear ordinary differential equations originating in the context of crystal dissolution and precipitation inside a porous medium: existence theory and periodic homogenization - Mahato, Hari Shankar; Kräutle, Serge; Böhm, Michael; Knabner, Peter (Advances in Mathematical Sciences and Applications, 2017)
In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and precipitation of immobile species present inside a porous medium. The transport of mobile species in the pores is modeled by a system of semilinear parabolic partial differential equations. The reactions amongst the mobile species are assumed to be reversible, i.e. both forward and backward reactions are considered. These reversible reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial differential equations. This system of equations for the mobile species is complemented by flux boundary conditions at the outer boundary. Furthermore, the dissolution and precipitation of immobile species on the surface of the solid parts are modeled by mass action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution term. The model is posed at the pore (micro) scale. The contribution of this paper is two- fold: first we show the existence of a unique positive global weak solution for the coupled systems and then we upscale (homogenize) the model from the micro scale to the macro scale. For the existence of solution, some regularization techniques, Schaefer’s fixed point theorem and Lyapunov type arguments have been used whereas the concepts of two-scale convergence and periodic unfolding are used for the homogenization.<br
Keywords: semilinear parabolic partial differential equations; nonlinear ordinary differential equations; reversible reactions; fixed point theorems; Lyapunov functionals; maximal regularity; periodic homogenization; two-scale convergence; periodic unfolding