InterPore2026

Chaotic Advection is Inherent to Heterogeneous Darcy Flow

19 May 2026, 14:35

15m

Oral Presentation (MS08) Mixing, dispersion and reaction processes across scales in heterogeneous and fractured media MS08

Speaker

Daniel Lester

Description

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 in Darcy flows at the macroscale, as these stirring kinematics profoundly impact basic processes such as solute transport and mixing, colloid transport and deposition, chemical reactions, 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 an important class of heterogeneous 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 (as shown in Figure 1(d)), 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 topological braid theory to establish a quantitative link between transverse macro-dispersivity $D_{T,\infty}$ and Lyapunov exponent $\lambda_\infty$ in heterogeneous Darcy flow.

We show that chaotic advection and transverse macro-dispersion occur in both anisotropic weakly heterogeneous and in heterogeneous weakly anisotropic conductivity fields, and that the quantitative link between chaotic advection and transverse dispersion persists across a broad range of conductivity fields.

Conversely, isotropic heterogeneous Darcy flows are not chaotic and exhibit zero transverse macro-dispersion (as shown in Figure 1(c)). As field experiments report non-zero transverse dispersion in the limit of large Peclet number, we conclude that the corresponding hydraulic conductivity fields must be anisotropic and hence the stirring kinematics are chaotic.

We demonstrate that such chaotic advection profoundly augments mixing, transport and reactions in heterogeneous porous media. Specifically, the concentration variance of a solute plume decays exponentially as $\langle c^2\rangle\sim\exp(-\lambda_\infty t/3)$ rather than algebraically, and dilution index of a Gaussian plume grows exponentially as $E(t)\sim\exp(\lambda_\infty t)$ rather than algebraically. Similarly, transverse dispersivity $D_T$ of diffusive solutes is exponentially amplified by chaotic advection. Mixing-limited reactions are impacted in the same manner as solute dilution, whereas more complex reaction systems that involve autocatalysis, oscillatory reactions, bistable and competitive reactions are qualitatively altered by chaotic advection.

The recognition that chaotic dynamics are inherent to porous media flow across all scales opens the door to the development of a broad class of upscaling methods that explicitly honour these kinematics and new class of tuneable engineered porous materials that exploit these phenomena. The ubiquity of macroscopic chaotic advection has profound implications for the myriad processes hosted in heterogeneous porous media and calls for a fundamental re-evaluation of transport and reaction methods in macroscopic porous systems.

Figure 1. (a) Iso-surfaces of typical heterogeneous log-conductivity field used to model isotropic and anisotropic conductivity tensors, (b) iso-surfaces of associated potential field for heterogeneous Darcy flow driven by a uniform mean potential gradient. Associated streamlines of heterogeneous Darcy flow with (c) isotropic conductivity field and (d) anisotropic conductivity field. Adapted from (1).