Upscaling of master equations in heterogeneous environments. Multicontinuum master equations

In this paper, we study master equations in heterogeneous porous media. The spatial heterogeneities are represented via highly varying transition rates or high particle concentrations. The regions of high densities can have very irregular forms in porous media representing high conductivity channels or fractures. Due to very small widths of these regions, this requires very dense particle distributions. In this paper, we present a new multicontinuum master equation model for upscaling. The main idea is to divide the region into coarse agglomerates (called coarse grid) and represent each agglomerate with a macroscopic point with several properties and the transition probabilities among them. We present a rigorous derivation of the macroscopic model and discuss conditions for the convergence. The macroscopic model has a form of master equations with multiple variables at each macroscale point. We discuss the derivation of macroscopic system of PDEs that can be derived from the discrete system. By choosing appropriate continua, we can obtain localized effective transition probabilities. This, in turn, is important for practical modeling, allows using Taylor expansion and deriving continuous-in-space PDE formulations of the resulting multicontinua master equations. Numerical results are presented to demonstrate the accuracy of macroscopic master equations, where we also show localization properties of macroscopic coefficients. © 2025 Elsevier Inc.