Is Chaotic Advection Inherent to Heterogeneous Darcy Flow?

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.

Figures (8)

• Figure 1: (a) Isosurfaces of the typical normalised heterogeneous log-conductivity field $f(\mathbf{x})=\ln K(\mathbf{x})/\sigma^2_{\ln K}$ used to model isotropic $k(\mathbf{x})\mathbf{I}$ and anisotropic $\mathbf{K}(\mathbf{x})$ conductivity tensors and (b) associated potential field $\phi(\mathbf{x})$ for anisotropic Darcy flow driven by a uniform mean potential gradient. Associated streamlines for heterogeneous Darcy flow with (c) isotropic conductivity field ($\delta=0$ in (\ref{['eqn:perturb']})) and (d) anisotropic conductivity field ($\delta=1$ in (\ref{['eqn:perturb']})) with log-conductivity variance $\sigma^2_{\ln K}=4$ and parameters $N=4$, $N_i=2$ in (\ref{['eqn:logKfield']}).
• Figure 2: (a) Schematic of streamline (or pathline) braiding in a unidirectional steady 3D flow (or unsteady 2D flow). The set of four thick coloured streamlines (or pathlines) braid with each other as they evolve with the mean flow direction $x_3$ (or time $t$). This topological braiding motion stirs the fluid continuum (grey planes) and the length of the black rectangular material boundary grows exponentially with the number of braiding motions. Adapted from Thiffeault:2006aa. This schematic also depicts streamline braiding in steady 3D anisotropic Darcy flow with intrinsic coordinates $\boldsymbol\xi=(\chi_1,\chi_2,\phi)$, where the grey planes depict isopotential surfaces. (b) Absence of braiding in isotropic Darcy flow in intrinsic coordinates $\boldsymbol\xi=(\chi_1,\chi_2,\phi)$. As the velocity field is everywhere orthogonal to level sets of $\phi$ denoted by grey planes, streamlines in this coordinate system do not move laterally or undergo braiding.
• Figure 3: (a) Top left: schematic of streamlines $n-1$, $n$, $n+1$ in the transverse $x_2-x_3$ plane with clockwise (black) $\sigma_n$, anti-clockwise (blue) $\sigma_n^{-1}$ braid generators in the 1D streamline model. Bottom left: stretching of material elements due to braiding motions (adapted from braidbook) that evolve in the longitudinal $\phi$ direction. Right: braid diagram (black) depicting stretching of material elements (red) as braid evolves in longitudinal $\phi$ direction. (b) Braid diagram depicting evolution of $x_2$ coordinate of $N_p=20$ streamlines over $N_b=20$ random braid actions in the longitudinal $\phi$ direction, leading to non-trivial braiding and dispersion of streamlines. (c) Growth of topological entropy $h$ (black line) and transverse variance $\sigma_{x_2}^2$ (blue line) with braid number $N_b$, which agrees well with theory Lester:2024aa (red line). Inset: Brownian motion of streamlines due to streamline braiding with braid number $N_b$. Adapted from Lester:2024aa.
• Figure 4: (a) Perturbation of $\delta=0.1$ streamlines (red) from $\delta=0$ (zero helicity) streamlines (green) and associated $\psi_1$ streamsurface (blue) for the conductivity field given in (\ref{['eqn:perturb']}). Similar perturbation of $\delta=0.1$ streamlines away from the $\psi_2$ streamsurfaces (not shown) also occurs. (b) Growth of mean absolute helicity $\langle|\mathcal{H}|\rangle$ with $\delta$, inset shows $|\mathcal{H}|$ fields for $\delta =0.9, 1$ (Adapted from Lester:2024aa). (c) Growth of transverse dispersivity $D_T/\langle v_1\rangle \ell$ with $\delta$ from simulations (red points) and fitted exponential (\ref{['eqn:DT_delta']}) (red curve). Inset shows temporal evolution of transverse variance. (d) Growth of Lyapunov exponent $\lambda_\infty$ with perturbation parameter $\delta$ from simulations (black points) and fitted exponential (\ref{['eqn:lambda_delta']}) (blue curve). (red dotted curve) dimensionless Lyapunov exponent $\lambda_\infty$ predicted from fitted exponential in (b) and (\ref{['eqn:model']}). (c), (d) Adapted from Lester:2024aa.
• Figure 5: (a) Variation of normalised mean longitudinal velocity $\langle v_1\rangle/v_h$ (red points) with log-variance $\sigma^2_{\ln K}$ and linear fit (\ref{['eqn:vmean']}), (grey dashed line) in anisotropic Darcy flow. (Inset) Variation of small velocity scaling index $\beta$ with $\sigma^2_{\ln K}$. (b) PDFs of normalised velocity magnitude $p_v(v/\langle v\rangle)$ as a function of log-variance $\sigma^2_{\ln K}$ and fitted log-normal distribution (black lines). (c) Variation of dimensionless Lyapunov exponent $\lambda_\infty$ (red dots) with log-variance $\sigma^2_{\ln K}$ and nonlinear fit (\ref{['eqn:lyapunov_fit']}), (grey dashed line). (d) Variation of dimensionless stretching variance $\sigma^2_\lambda$ with log-variance $\sigma^2_{\ln K}$ and nonlinear fit (\ref{['eqn:lnrhovar']}), (grey dashed line). (Inset) Variation of Protean velocity gradient variance $\sigma^2_\epsilon$ with log-variance $\sigma^2_{\ln K}$ and linear fit (\ref{['eqn:velgradvar']}), (grey dashed line).