We study nonlinear reactive transport in a layered porous medium separated by an $\varepsilon$-thin, highly heterogeneous fracture whose aperture and obstacle pattern vary periodically. Species transport in the bulk is governed by parabolic reaction--diffusion equations, coupled to a convection-diffusion-reaction problem in the fracture with nonlinear wall and obstacle reactions and Peclet number of order $O(\varepsilon^{-1})$. Via multiscale analysis as $\varepsilon \to 0$, when the fracture collapses to a flat interface, we derive a new type of homogenized model consisting of bulk diffusion--reaction equations coupled through nonlinear interface conditions and a first-order semilinear hyperbolic system on the interface. We prove well-posedness and regularity of the limit system, construct a multiscale approximation with boundary-layer correctors, and derive quantitative error estimates in suitable energy norms.
We consider reactive species transport in fractured porous media. As a fracture we understand structures that have a small aperture relative to their length. Fractures are ubiquitous in natural and engineered porous media and strongly influence flow and transport of dissolved species. In many applications, such as hydrogeology, contaminant remediation, and subsurface reactive flows, fractures act as preferential pathways with much larger convective transport than the surrounding matrix. In reality, fractures are often geometrically complex; for example, their aperture may vary rapidly and they may contain embedded obstacles, possibly inducing several spatial scales.
These microscale heterogeneities complicate direct numerical simulation and motivate rigorous upscaling of the microscale models to derive effective macroscopic models, in which the thin fracture is reduced to a lower-dimensional interface.
In this contribution we are interested in chemically reactive transport of species in a two-dimensional porous medium that is separated by a horizontal channel-like fracture into two bulk domains. For some small parameter ε, the fracture has O(ε)-aperture. As geometrical heterogeneities we account for the fracture’s wall roughness (differently on the fracture’s upper and lower side) and some internal obstacle pattern which are assumed to vary periodically on the same microscale. For the sake of illustration we refer to Fig. 1.
. Sketch of the two bulk domains and the thin fracture.
The reactive transport of the species in the two bulk domains is modelled by a system of parabolic reactiondiffusion equations which couples to a convection-diffusion-reaction problem in the thin fracture accounting for nonlinear chemical reactions on the fracture’s boundaries. We assume that the species transport in the fracture is convection-dominated, i.e., the Péclet number is proportional to the inverse of ε . The coupling between the bulk domains and the fracture is done by nonlinear side-dependent flux conditions on the rough boundaries whereas nonlinear Robin conditions are employed on the obstacle boundaries. The complete microscale model is given in (2.2) below. Up to our knowledge such a setting has not been analyzed before with respect to the limit ε → 0 .
Due to the nonlinear interface laws and convection-dominated transport in the fracture, the asymptotic analysis leads to qualitatively different homogenized behaviour compared to previous works. The homogenized limit (see (4.1)) consists of nonlinear diffusion-reaction equations in the bulk domains, coupled through homogenized nonlinear interface conditions, together with a first-order semilinear hyperbolic system posed on the flat interface that captures the effective fracture dynamics generated by strong advection and the microscale geometry. The well-posedness of this homogenized system is established in Theorem 4.1, and the C 3 -regularity of its solution, required for proving the error estimates, is derived in Lemma 4. 3.
Another novel contribution is the complete approximation for the microscopic solution, i.e., we construct an approximation (see (5.1)) that incorporates boundary-layer correctors near the oscillating boundaries in the bulk domains ( § 3.2) as well as additional correctors near the vertical boundaries of the fracture ( § 4.3) to satisfy the prescribed Dirichlet condition.
Finally, we obtain error estimates of order O( √ ε) in the energy norm (see Theorem 5.1), thereby rigorously justifying the adequacy of the homogenized model. These estimates clearly reveal the influence of the interface microstructure through the second-order terms of the asymptotic approximation, including the boundary-layer correctors, and thus provide deeper insight into how microscopic features and convection-dominated transport shape the macroscopic behavior.
We next review the existing literature on homogenization in fractured porous media. The literature pertaining to such issues has expanded considerably over the past two decades such that we restrict ourselves to the most closely related works and the differences between them and our own. For broader context, we refer to works on related classes of multiscale problems, including thin-domain models [3,[37][38][39], models in junctions of thin structures [5,18,35,36], and homogenization problems in porous media [2,4,6,13,15,31,51]. Among those, the papers [2,[35][36][37][38]51] address convection-dominated transport, but not in a fractured porousmedia geometry like in Fig. 1. It is worth noting that only the results in [35][36][37][38] yield hyperbolic dispersion equations in the limit.
A significant body of research has been dedicated to the study of problems involving the separation of two bulk domains by an ε-thin fracture. These problems have been examined from the perspective of effective interface laws for the limit ε → 0. We restrict ourselves to the most recent contributions [8,9,11,14,16,17,28,50] that
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