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Multiscale Hyperbolic-Parabolic Models for Nonlinear Reactive Transport in Heterogeneously Fractured Porous Media

Taras Mel'nyk, Sorin Pop, Christian Rohde

Abstract

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.

Multiscale Hyperbolic-Parabolic Models for Nonlinear Reactive Transport in Heterogeneously Fractured Porous Media

Abstract

We study nonlinear reactive transport in a layered porous medium separated by an -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 . Via multiscale analysis as , 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.
Paper Structure (13 sections, 12 theorems, 217 equations, 4 figures)

This paper contains 13 sections, 12 theorems, 217 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Formal asymptotic analysis
  4. Analysis in the thin perforated domain
  5. Model cell problem in $Y_0$
  6. Analysis in the bulk domains $\Omega^+_\varepsilon$ and $\Omega^-_\varepsilon$
  7. Model cell problem in $\Pi^+_0$
  8. The homogenized problem
  9. Recursive inequality
  10. Well-posedness of the homogenized problem
  11. Boundary-layer asymptotics
  12. Asymptotic Approximation and Asymptotic Estimates
  13. Conclusions

Key Result

Proposition 3.1

Problem cell-problem has a unique weak solution if and only if

Figures (4)

  • Figure 1: Sketch of the two bulk domains and the thin fracture.
  • Figure 2: Schematic illustration of the thin fracture $\Omega^f_\varepsilon$
  • Figure 3: Sketch of the domain $\Pi^+_0$
  • Figure 4: Semi-strip $\mathfrak{C}$ with perforations and upper/lower boundaries all varying with period$1$

Theorems & Definitions (38)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.1
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 3.1
  • Definition 3.1
  • ...and 28 more