Automated Symbolic Upscaling: 1. Model Generation for Extended Applicability Regimes

In porous media theory, upscaling techniques are fundamental to deriving rigorous Darcy-scale models for flow and reactive transport in subsurface systems. Due to limitations in classical techniques, a number of ad hoc approaches have been proposed to address physical regimes in which reactive time scales are similar to, or faster than, diffusive time scales. In Part 1 of this two part series, we present a strategy for expanding the applicability of classical homogenization theory by generalizing the assumed closure form. We detail the implementation of this strategy on two reactive mass transport problems with moderately reactive physics. The strategy produces nontrivial homogenized models with emergent terms and effective parameters that couple reactive, diffusive, and advective transport. The differences in equation forms between the macroscopic and pore-scale descriptions advise caution to further studies where the forms of macroscopic equations are assumed, as opposed to rigorously derived. Numerical validation is provided for each problem to show that the error estimates of homogenization theory are satisfied, and to justify the implemented strategy. In Part 2, the presented strategy is automated using symbolic computing to expedite its implementation.

Automated summary

In porous media theory, upscaling techniques play an important role in obtaining accurate models for flow and reactive transport in subsurface systems. This study presents a strategy to enhance the classical homogenization theory by expanding its applicability and implementing it on reactive mass transport problems. The strategy produces nontrivial homogenized models that couple reactive, diffusive, and advective transport, and caution is advised when assuming the forms of macroscopic equations without rigorous derivation. Numerical validation and automated implementation using symbolic computing are provided in the second part of the series.