[Paper Review] Homogenization of reaction-diffusion equations in fractured porous media

Hermann Douanla, Jean Louis Woukeng|arXiv (Cornell University)|Jun 27, 2015

Advanced Mathematical Modeling in Engineering9 references3 citations

TL;DR

This paper develops a homogenization framework for reaction-diffusion equations with large reaction terms in fractured porous media using multi-scale convergence. It derives a macroscopic convection-diffusion-reaction equation on a fixed domain, capturing effective transport behavior through periodic microstructures of pores and fractures, with convergence of solutions as the scale parameter ε → 0.

ABSTRACT

The paper deals with the homogenization of reaction-diffusion equations with large reaction terms in a multi-scale porous medium. We assume that the fractures and pores are equidistributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of convection-diffusion-reaction type.

Motivation & Objective

• To model diffusion and reaction processes in multi-scale fractured porous media with equidistributed pores and fractures.
• To address the challenge of numerical simulation in complex multi-scale geometries by deriving an effective macroscopic model.
• To analyze the asymptotic behavior of reaction-diffusion equations with large reaction terms as the microscale parameter ε → 0.
• To establish convergence of the ε-solution to a limit problem defined on a fixed domain.

Proposed method

• Utilizes the multi-scale convergence method to analyze sequences of solutions in a periodic, fractured porous medium with distinct matrix, pore, and fracture phases.
• Models the medium as a periodic composite with Y = Y_m ∪ Y_c, where Y_m contains pores (Z_s) and solid matrix (Z_p), and Y_c represents fractures.
• Applies periodicity and uniform ellipticity assumptions to the diffusion coefficient A(y, τ) and reaction term g(y, τ, u), ensuring boundedness and regularity.
• Derives a two-scale expansion for the solution u_ε, decomposing it into u_0(x,t) + εu_1(x,t,y,τ) + ε²u_2(x,t,y,τ) + ..., with y = x/ε and τ = t/ε².
• Solves corrector problems in the cell Y_m × T to determine effective coefficients, including the macroscopic diffusion tensor Â(x,t) and convection-like terms L₁, L₂, L₃.
• Establishes the limit problem as a convection-diffusion-reaction equation in the macroscopic domain Ω × (0,T), with effective coefficients derived from cell problems.

Experimental results

Research questions

• RQ1 How does the presence of large reaction terms affect the homogenization process in fractured porous media with periodic microstructure?
• RQ2 What is the effective macroscopic equation that captures the long-term behavior of reaction-diffusion processes in such multi-scale media?
• RQ3 How do the pore and fracture structures influence the effective diffusion and reaction coefficients in the homogenized model?
• RQ4 What convergence properties hold for the ε-solution u_ε as the microscale parameter ε → 0?
• RQ5 Can the homogenized problem be formulated as a convection-diffusion-reaction equation with explicitly defined effective coefficients?

Key findings

• The solution u_ε of the original problem converges strongly in L²(Ω_T) to the limit solution u₀ as ε → 0.
• The homogenized problem is a convection-diffusion-reaction equation on the fixed domain Ω × (0,T), with a time derivative scaled by |Z_s|∫_{Y_m} ρ(y) dy.
• The effective diffusion tensor Â(x,t) is derived from a cell problem involving the matrix coefficient Â(y,τ) and corrector functions θ and ω₁.
• The convection-like terms L₁, L₂, and L₃ arise from the interaction of the reaction term g and the microstructure, with L₂ and L₃ depending on the gradient of the reaction potential and corrector functions.
• The effective coefficients are uniformly bounded and Lipschitz continuous, ensuring the uniqueness and well-posedness of the homogenized problem.
• The limit problem retains the structure of a reaction-diffusion equation but with modified diffusion, convection, and reaction terms that encapsulate the multi-scale effects.