Universal transport laws in buoyancy-driven porous mixing

TL;DR

This paper develops a theoretical framework for predicting transient buoyancy-driven mixing in porous media, deriving universal laws validated by high-resolution simulations, advancing understanding of natural and engineered flow systems.

Contribution

It introduces exact time-dependent balances for transient porous buoyancy-driven mixing and derives a minimal closure predicting universal self-similar profiles and transport laws.

Findings

Exact balances accurately describe active mixing layers.

Universal self-similar mean profiles are predicted.

Validated by high-resolution numerical simulations.

Abstract

Buoyancy-driven convection in porous media governs heat and mass transport in a wide range of natural and engineered systems, from groundwater aquifers and geothermal reservoirs to carbon storage in geological formations and flows through planetary interiors. Yet the transient regime, in which fingering flows emerge and transport is strongly enhanced, is still described largely through empirical scaling laws, limiting predictive capability across conditions. Here we show that transient porous buoyancy-driven mixing obeys exact time-dependent balances that couple transport, flow intensity, and scalar dissipation. These balances remain accurate when restricted to the actively mixing layer, revealing that the essential dynamics are localized within a finite region. Leveraging these results, we derive a minimal one-parameter closure for the mean scalar field. The theory we propose predicts self-similar mean profiles, universal second-order statistics, and a linear transport law without case-by-case tuning. Direct numerical simulations up to spatial resolution of $2048\times2048\times16384$ points validate these predictions. Our results place transient porous mixing on a predictive footing, showing how macroscopic transport laws emerge from exact balances and self-similar dynamics, and provide a general framework for buoyancy-driven transport in porous media.