Is Chaotic Advection Inherent to Heterogeneous Darcy Flow?

TL;DR

This paper proves that chaotic advection is an inherent feature of steady 3D Darcy flow in heterogeneous porous media, significantly impacting transport and mixing processes in such systems.

Contribution

It establishes a rigorous link between heterogeneity, anisotropy, and chaotic advection in steady 3D Darcy flow using braid theory and Lyapunov exponents.

Findings

Chaotic advection occurs in all realistic heterogeneous Darcy flows.

Transverse dispersion is linked to chaotic advection through Lyapunov exponents.

The phenomena persist across various levels of heterogeneity and anisotropy.

Abstract

At all scales, porous materials stir interstitial fluids as they are advected, leading to complex distributions of matter and energy. Of particular interest is whether porous media naturally induce chaotic advection at the Darcy scale, as these stirring kinematics profoundly impact basic processes such as solute transport and mixing, colloid transport and deposition and chemical, geochemical and biological reactivity. While many studies report complex transport phenomena characteristic of chaotic advection in heterogeneous Darcy flow, it has also been shown that chaotic dynamics are prohibited in a large class of Darcy flows. In this study we rigorously establish that chaotic advection is inherent to steady 3D Darcy flow in all realistic models of heterogeneous porous media. Anisotropic and heterogenous 3D hydraulic conductivity fields generate non-trivial braiding of stream-lines, leading to both chaotic advection and (purely advective) transverse dispersion. We establish that steady 3D Darcy flow has the same topology as unsteady 2D flow, and so use braid theory to establish a quantitative link between transverse dispersivity and Lyapunov exponent in heterogeneous Darcy flow. We show that chaotic advection and transverse dispersion occur in both anisotropic weakly heterogeneous and in heterogeneous weakly anisotropic conductivity fields, and that the quantitative link between these phenomena persists across a broad range of conductivity anisotropy and heterogeneity. The ubiquity of macroscopic chaotic advection has profound implications for the myriad of processes hosted in heterogeneous porous media and calls for a re-evaluation of transport and reaction methods in these systems.