Is chaotic advection inherent to heterogeneous Darcy flow?

At all scales, porous materials stir interstitial fluids as they are advected, leading to complex (and chaotic) distributions of matter and energy. Of particular interest is whether porous media naturally induce chaotic advection in Darcy flows at the macroscale, 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 the prevalence of pore-scale chaotic advection has been established, and 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 three-dimensional (3-D) Darcy flow with anisotropic and heterogeneous hydraulic conductivity fields. These conductivity fields generate non-trivial braiding of streamlines, leading to both chaotic advection and (purely advective) transverse macro-dispersion. We establish that steady 3-D Darcy flow has the same topology as unsteady 2-D flow and use braid theory to establish a quantitative link between transverse dispersivity and Lyapunov exponent in heterogeneous Darcy flow. Our main results 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 fields. As the ubiquity of macroscopic chaotic advection has profound implications for the myriad processes hosted in porous media, these results call for re-evaluation of transport and reaction methods in these systems.