Multicontinuum Homogenization for Coupled Flow and Transport Equations

TL;DR

The paper develops a multicontinuum homogenization framework to derive a coupled macroscopic model for flow and transport in highly heterogeneous media. By introducing two-term multicontinuum expansions and oversampled coupled constraint cell problems, it constructs localized multiscale basis functions that yield coarse-grid elliptic and convection-diffusion-reaction equations with homogenized properties. The approach handles nonlinear aspects of transport through linearization by macroscopic fluxes and demonstrates accuracy on layered and circular coefficient fields under varied boundary conditions, with errors remaining small and decreasing as the coarse grid is refined. This method enables efficient offline/online computation and potential incorporation of data-driven predictions for effective properties, while preserving fine-scale information through the multicontinuum expansions.

Abstract

In this paper, we present the derivation of a multicontinuum model for the coupled flow and transport equations by applying multicontinuum homogenization. We perform the multicontinuum expansion for both flow and transport solutions and formulate novel coupled constraint cell problems to capture the multiscale property, where oversampled regions are utilized to avoid boundary effects. Assuming the smoothness of macroscopic variables, we obtain a multicontinuum system composed of macroscopic elliptic equations and convection-diffusion-reaction equations with homogenized effective properties. Finally, we present numerical results for various coefficient fields and boundary conditions to validate our proposed algorithm.

Figures (13)

• Figure 1: Illustration of computational domain $\Omega$, coarse block $\omega$, RVE $R_\omega$ and oversampled RVE $R_\omega^+$
• Figure 2: Layered field for Example 1 ($\Omega_1$: blue regions; $\Omega_2$: yellow regions)
• Figure 3: Initial concentration when $t=0$
• Figure 4: Solutions of concentration when $H=1/40$ for Case 1 in Example 1. First column: multiscale solution $C_1$ at $t=0.02$, $0.1$, $0.5$, $1$, $2$. Second column: reference averaged solution in $\Omega_1$ at the corresponding time instants.
• Figure 5: Solutions of concentration when $H=1/40$ for Case 1 in Example 1. First column: multiscale solution $C_2$ at $t=0.02$, $0.1$, $0.5$, $1$, $2$. Second column: reference averaged solution in $\Omega_2$ at the corresponding time instants.