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The problem of obtaining suitable global descriptions for some complex reactive flows in porous media was addressed in the literature by

using various upscaling methods: heuristic and empirically based methods, variational methods, stochastic methods, methods based on homogenization,

mixture theories, or volume averaging techniques. Also, the use of numerical models for studying single-phase or multi-phase flows in heterogeneous porous media has received considerable attention in the last

decades. However, even with the increases in the power of computers, the complex multiscale structure of the analyzed media constitutes a critical problem

in the numerical treatment of such models and there is a considerable interest in the development of upscaled or homogenized models in

which the effective properties of the medium vary on a coarse scale which proves to be suitable for efficient computation, but enough accurately to capture

the influence of the fine-scale structure on the coarse-scale properties of the solution.

Porous media play an important role in many areas, such as hydrology

(groundwater flow, salt water intrusion into coastal aquifers), geology (petroleum reservoir engineering, geothermal energy), chemical

engineering (packed bed rectors, drying of granular materials), mechanical engineering (heat exchangers, porous gas burners),  the study of industrial materials

(glass fiber materials, brick manufacturing). There is an extensive literature on the determination of the

effective properties of heterogeneous porous media (see, e.g., \\cite{Hornung}, \\cite{Bear}, and the references therein).

Transport  processes  in  porous  media  have  been  extensively studied in last decades by engineers, geologists, hydrologists, mathematicians, physicists.

In particular, mathematical modeling of chemical reactive flows through porous media is a topic of huge practical importance in many engineering,

physical, chemical, and biological applications. Obtaining suitable macroscopic laws for the processes in geometrically complex porous media (such as soil,

concrete, rock, or pellets) involving flow, diffusion, convection, and chemical reactions is a difficult task. The homogenization theory proves to be a

very efficient tool by providing suitable techniques allowing us to pass from the microscopic scale to the macroscopic one and to obtain suitable macroscale

models. Since the seminal work of G.I. Taylor \\cite{Taylor}, dispersion phenomena in porous media have attracted a lot of attention. There are many formal or

rigorous methods in the literature. We refer to \\cite{Hor-Jag} and \\cite{Jager} as one of the first works containing rigorous homogenization results for

reactive flows in porous media. By using the two-scale convergence method, coupled with monotonicity methods and compensated compactness, the convergence of the

homogenization procedure was proven for problems with nonlinear reactive terms and nonlinear transmission conditions. Since then, many works have been devoted to

the homogenization of reactive transport in porous media (see  \\cite{All-Bri-Mik-Pia}, \\cite{Aur-Adl}, \\cite{Bear}, \\cite{Mei}, \\cite{Mik-Pri},

\\cite{Kumar}, \\cite{Hor-Jag-Mik}, \\cite{Mauri}, \\cite{Pop}, \\cite{Pop1} and the references therein). For instance, rigorous homogenization results for reactive

flows with adsorption and desorption at the boundaries of the perforations,

for dominant P\\\'{e}clet numbers and Damkohler  numbers, are obtained in \\cite{Allaire2}, \\cite{All-Mik-Pia}, and \\cite{Mik-Dev-Dui}. For reactive flows combined with

the mechanics of cells, we refer to \\cite{Jag-Mik-Neu}. Rigorous homogenization techniques for obtaining the effective model for dissolution and precipitation in

a complex porous medium were successfully applied in \\cite{Pop}. Solute transport in porous media is also a topic of interest for chemists, geologists and

environmental scientists (see, e.g., \\cite{All-Hut} and \\cite{Dui-Kna}). Related problems, such that single or two-phase flow or miscible

displacement problems were addressed in various papers (see, for instance,  \\cite{Ama-Pan}, \\cite{Arb-Dou-Hor}, \\cite{Arbogast2}, \\cite{Mikelic}).

For an interesting survey on homogenization techniques applied to problems involving flow, diffusion, convection, and reactions in porous media,

we refer to \\cite{Hornung}.

In this chapter, some applications of the {\\em  homogenization method} to the study of reactive flows in periodic porous media will be presented.

The chapter represents a summary of the results I obtained in this area, alone or in collaboration, and is based on the papers \\cite{Con-Dia-Tim}, \\cite{Con-Dia-Lin-Tim}, \\cite{Tim-Acta},

and  \\cite{Tim-Ann-Ferr}.

\\section{Upscaling in stationary reactive flows in porous media}

We shall discuss now, following \\cite{Con-Dia-Lin-Tim} and \\cite{Tim-Acta}, some homogenization results for chemical reactive flows through porous media. For more details

about the chemical aspects involved in this kind of problems and, also, for some mathematical and historical backgrounds, we refer to S. N. Antontsev et al.

\\cite{Antontsev}, J. Bear \\cite{Bear}, J. I. D\\\'{\\i}az \\cite{Diaz1}, \\cite{Diaz2}, \\cite{Diaz3}, and U. Hornung \\cite{Hornung} and the references therein.

We shall be concerned with a problem modeling the stationary reactive flow of a fluid confined in the exterior of some periodically distributed obstacles,

reacting on the boundaries of the obstacles. More precisely, the challenge in our first paper dedicated to this subject, namely \\cite{Con-Dia-Lin-Tim}, consists

in dealing with Lipschitz or even non-Lipschitz continuous reaction rates such as Langmuir

or Freundlich kinetics, which, at that time, were open cases in the literature. Our results represent a generalization of some of the results in \\cite{Hornung}.

Using rigorous multiscale techniques, we derive a macroscopic model system for such elliptic problems modeling chemical reactions on the grains of a porous medium.

The effective model preserves all the relevant information from the microscopic level. The case in which chemical reactions arise inside the grains

of a porous medium will be also discussed. Also, we shall present some results obtained in \\cite{Tim-Acta}, where we have analyzed  the effective behavior of

the solution of a nonlinear

problem arising in the modeling of enzyme catalyzed reactions through the exterior of a domain containing periodically distributed reactive solid obstacles.

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