Abstract. We investigate a two-dimensional micro-scale model for crystal dissolution and precipitation in a porous medium. The model contains a free boundary and allows for changes in the pore volume. Using a level-set formulation of the free boundary, we apply a formal homogenization procedure to obtain upscaled equations. For general micro-scale geometries, the homogenized model that we obtain falls in the class of distributed microstructure models. For circular initial inclusions the distributed microstructure model reduces to system of partial differential equations coupled with an ordinary differential equation. In order to investigate how well the upscaled equations describe the behavior of the micro-scale model, we perform numerical computations for a test problem. The numerical simulations show that for the test problem the solution of the homogenized equations agrees very well with the averaged solution of the micro-scale model. Introduction.In this paper we use a formal limiting procedure with asymptotic expansions to derive a macroscopic law for crystal dissolution and precipitation in a porous medium. The microscopic model that serves as the staring point for the limiting process, incorporates the change in volume of the pore space as a result of the precipitation/dissolution process. In [17], the same microscopic model is considered in a thin strip. In the present paper we investigate the model on a perforated domain.Macroscopic laws for reactive transport in porous media, which include the present case of crystal dissolution and precipitation, are of practical importance in many physical, biological and chemical applications. Macroscopic laws for reactive transport in porous media are derived rigorously in, e.g., [6]. For the more specific case of crystal dissolution and precipitation, macroscopic models are given in [3,4,7,8]. In these papers the presented macroscopic models are analysed, but are not supported by a rigorous derivation. In most of these papers also the numerical solution of the proposed model equations is studied. Related work, in which the transport of dissolved material is analysed, can be found in [14,16].The main difficulty in performing the formal homogenization for the crystal dissolution and precipitation reaction is that the equations that describe the micro-scale processes contain a free boundary. This free boundary describes the interface between the layer of crystalline solid attached to the grains and the fluid occupying the pores. The location of this free boundary is an unknown in the model and moves with speed proportional to the local dissolution/precipitation rate. The micro-scale model with the free boundary has been studied in [18] in a one dimensional setting and without flow. Other works that study crystal dissolution and precipitation on the micro-scale are [12] and [13].The crystal dissolution and precipitation problem has been studied also in [7] and [15,19]. The main difference between the cited papers and the present paper is that in

[1] The focus of this paper is the derivation of an effective model for biofilm growth in a porous medium and its effect on fluid flow. The starting point is a pore-scale model in which the local geometry of the pore is represented as a thin strip. The model accounts for changes in pore volume due to biomass accumulation. As the ratio of the width of the strip to its length approaches zero, we apply a formal limiting argument to derive a onedimensional upscaled (effective) model. For a better understanding of the terms and parameters involved in the equations derived here, we compare these equations to a wellknown core-scale model from the literature.

We present a two-dimensional micro-scale model for crystal dissolution and precipitation in a porous medium. The local geometry of the pore is represented as a thin strip and the model allows for changes in the pore volume. A formal limiting argument, for the limit of the width of the strip going to zero, leads to a system of one-dimensional effective upscaled equations. We show that the effective equations allow for travelling-wave solutions and prove the existence and uniqueness of these. Numerical solutions of the effective equations are compared with numerical solutions of the original equations on the thin strip and with analytical results. We also show that a model from the literature that does not allow changes in the pore volume can be obtained from the present model as a formal limit.

We aim at understanding transport in porous materials consisting of regions with both high and low diffusivities. We apply a formal homogenisation procedure to the case wherethe heterogeneities are not arranged in a strictly periodic manner. The result is a two-scale model formulated inx-dependent Bochner spaces. We prove the weak solvability of the limit two-scale model for a prototypical advection–diffusion system of minimal size. A special feature of our analysis is that most of the basic estimates (positivity,L∞-bounds, uniqueness, energy inequality) are obtained in thex-dependent Bochner spaces.

We consider a pore-scale model for reactive flow in a thin 2-D strip, where the convective transport dominates the diffusion. Reactions take place at the lateral boundaries of the strip (the walls), where the reaction product can deposit in a layer with a non-negligible thickness compared to the width of the strip. This leads to a free boundary problem, in which the moving interface between the fluid and the deposited (solid) layer is explicitly taken into account. Using asymptotic expansion methods, we derive an upscaled, one-dimensional model by averaging in the transversal direction. The result is consistent with (Taylor dispersion) models obtained previously for a constant geometry. Finally, numerical computations are presented to compare the outcome of the effective (upscaled) model with the transversally averaged, two dimensional solution.