[Paper Review] Sigma-convergence for thin heterogeneous domains and application to the upscaling of Darcy-Lapwood-Brinkmann flow

Willi Jäger, Jean Louis Woukeng|arXiv (Cornell University)|Sep 16, 2023

Advanced Mathematical Modeling in EngineeringComputer Science3 citations

TL;DR

This paper extends the sigma-convergence method to thin heterogeneous domains with general microstructures (periodic, almost periodic, or deterministic), enabling upscaling of Darcy-Lapwood-Brinkmann flows. It proves compactness results via algebras with mean value and derives effective 2D models: Darcy’s law in the low-permeability regime and Hele-Shaw-type behavior in the high-permeability case.

ABSTRACT

The sigma-convergence concept has been up to now used to derive macroscopic models in full space dimensions. In this work, we generalize it to thin heterogeneous domains given rise to phenomena in lower space dimensions. More precisely, we provide a new approach of the sigma-convergence method that is suitable for the study of phenomena occurring in thin heterogeneous media. This is made through a systematic study of the sigma-convergence method for thin heterogeneous domains. Assuming that the thin heterogeneous layer is made of microstructures that are distributed inside in a deterministic way including as special cases the periodic and the almost periodic distributions, we make use of the concept of algebras with mean value to state and prove the main compactness results. As an illustration, we upscale a Darcy-Lapwood-Brinkmann micro-model for thin flow. We prove that, according to the magnitude of the permeability of the porous domain, we obtain as effective models, the Darcy law in lower dimensions. The effective models are derived through the solvability of either the local Stokes-Brinkmann problems or the local Hele-Shaw problems.

Motivation & Objective

• Develop a rigorous mathematical framework for sigma-convergence in thin heterogeneous domains with non-periodic microstructures.
• Address the challenge of compactness in thin domains where microstructures are distributed deterministically, not just periodically.
• Generalize homogenization techniques to lower-dimensional effective models in thin layers using algebras with mean value.
• Apply the framework to upscale the Darcy-Lapwood-Brinkmann system in thin porous layers.
• Derive effective models depending on the scaling of permeability, distinguishing between Darcy and Hele-Shaw regimes.

Proposed method

• Introduce a new sigma-convergence framework tailored for thin domains $G_\varepsilon = G_1 \times \varepsilon G_2$, where $G_1 \subset \mathbb{R}^{d_1}$, $G_2 \subset \mathbb{R}^{d_2}$, and $\varepsilon \to 0$.
• Use algebras with mean value $\mathcal{A}$ on $\mathbb{R}^{d_1}$ to describe general microstructures, including almost periodic and asymptotic periodic distributions.
• Establish compactness via generalized Besicovitch spaces $\mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d_1}; L^p(G_2))$ and mean value operators $M$.
• Apply the method to the Darcy-Lapwood-Brinkmann system in $\Omega^\varepsilon = \Omega \times (-\varepsilon, \varepsilon)$, with $\Omega \subset \mathbb{R}^{d-1}$.
• Use weak $\Sigma_{\mathcal{A}}$-convergence to pass to the limit in the variational formulation, identifying effective equations.
• Solve local cell problems in the $y$-variable (transverse direction) to derive effective coefficients, distinguishing between Darcy and Hele-Shaw regimes.

Experimental results

Research questions

• RQ1 How can sigma-convergence be generalized to thin heterogeneous domains with non-periodic microstructures?
• RQ2 What compactness results hold for sequences in thin domains when the microstructure is described by an algebra with mean value?
• RQ3 What effective macroscopic model emerges for the Darcy-Lapwood-Brinkmann system in a thin layer when permeability scales as $O(\varepsilon^2)$?
• RQ4 What effective model is obtained when permeability scales faster than $\varepsilon^2$, leading to dominant viscous or pressure-driven flow?
• RQ5 Can the local cell problems be solved in the general algebra-with-mean-value framework, or are additional structural assumptions required?

Key findings

• A new compactness result is proven: for sequences $u_\varepsilon$ in $L^p(G_\varepsilon)$ with uniform $\varepsilon^{-d_2/p}\|u_\varepsilon\|_{L^p(G_\varepsilon)} \leq C$, there exists a subsequence such that $\varepsilon^{-d_2}\int_{G_\varepsilon} u_\varepsilon f(x/\varepsilon) dx \to \int_{G_0} \int_{G_2} M(u_0(\cdot, \cdot, \zeta) f(\cdot, \cdot, \zeta)) d\zeta d\overline{x}$.
• When $K_\varepsilon = O(\varepsilon^2)$ and $K_\varepsilon / \varepsilon^2 \to K \in (0, \infty)$, the rescaled velocity $\mathbf{u}_\varepsilon / \varepsilon^2$ weakly $\Sigma_{\mathcal{A}}$-converges to $\mathbf{u}_0$, and the effective model is Darcy’s law in the limit domain $\Sigma = \Omega \times \{0\}$.
• In the regime $K_\varepsilon >> \varepsilon^2$, the effective pressure $p_0$ is independent of the transverse variable $\zeta$, and the limit system reduces to a Hele-Shaw-type problem in the $\overline{x}$-plane.
• The effective velocity $\mathbf{u}^\prime$ is obtained by averaging $\mathbf{u}_0$ over the transverse direction, with $u_d = 0$ in the limit.
• The local problem $-\mathrm{div}_y(A\nabla_y \mathbf{w}_j) + \overline{\nabla}_y \pi_j = e_j$ in $\mathbb{R}^{d-1} \times I$ is not always solvable in general algebras with mean value, but is solvable for $\mathcal{A} = \mathcal{C}_{\text{per}}(Y')$ or $\mathcal{C}_{\text{per}}(Y') + \mathcal{C}_0(\mathbb{R}^{d-1})$.
• The effective pressure $p_0$ satisfies $\nabla_{\overline{x}} p_0 = \int_{-1}^1 M(\boldsymbol{f}_1 - A\nabla_y \mathbf{u}_1) d\zeta$, linking macroscopic pressure to microscale flow.