How pore-scale disorder controls fluid stretching in porous media

Abstract

Fluid stretching in porous media governs the mixing of reactants, contaminants, and nutrients, yet how the solid microstructure controls the stretching statistics remains poorly understood. We investigate how porous-medium heterogeneity controls stretching using (i) particle-tracking velocimetry experiments in 3D-printed millifluidic cells, (ii) numerical simulations of solute-plume deformation in the measured flow fields, and (iii) analytical calculations of fluid stretching. The cells contain arrays of cylindrical rods with systematically-varying disorder levels, from ordered to random. Velocity and shear-rate measurements reveal that fluid deformation is strongly localized near solid boundaries for all disorder levels, suggesting that near-wall flow is the main driver of stretching. The mean stretching grows linearly in time for ordered media and quadratically for disordered media, while the stretching distributions are approximately log-normal. We analytically describe the stretching produced by flow around an isolated cylinder and embed this description in a random-walk model that reproduces the observed stretching statistics in random media. These results provide the first quantitative connection between porous-medium structure and fluid-stretching statistics, revealing the extent to which disordered media accelerate mixing relative to ordered media and enabling progress beyond the common mean-field description of stretching in two-dimensional media as a simple shear flow.

Figures (7)

• Figure 1: (A) Schematic of the experimental setup for particle-tracking velocimetry. (B) Detail of the progressively-disordered stereolithography-printed cells, with vertical rods of diameter $a$ arranged according to a disorder parameter $\varepsilon$. (C) Zoomed-in view of an experimental image, with fluorescent microspheres visible as white dots. (D) Overlay of successive zoomed-in images, showing particles tracing out streamlines.
• Figure 2: The streamwise velocity and shear rate fields are displayed for three of the observed heterogeneity levels. In (A), the ordered model displays two velocity modes, the channel-like pathways where the velocity is high, and the wake-like pathways behind the grains where the velocity is low. Comparison of the velocity fields in (A), (B), and (C) shows an increase in velocity heterogeneity with disorder that relates to a blending of the high and low-velocity pathways visible in (A). Likewise, the shear rate $\sigma$ becomes more spatially heterogeneous with increasing disorder parameter $\varepsilon$. Shear magnitudes are typically large on the side walls of cylinders and drop to zero in the widest pores where the velocity is high.
• Figure 3: Panel (A) displays the velocity statistics for different model heterogeneities (denoted by colors). Ordered media display two peaks in the velocity probability density function associated with wake and channel modes. Inset (i) shows the exponential tail of $P(u)$ that broadens with $\varepsilon$. Inset (ii) highlights the wake and channel modes that contribute to the $\varepsilon=0$ velocity distributions; these modes increasingly blend together as $\varepsilon$ increases. Panel (B) shows the streamwise shear-rate statistics. The sharp channel mode $\sigma\approx \overline{\sigma}$, visible for $\varepsilon=0$, progressively transfers to low and high $\sigma$ as disorder increases. Inset (i) displays the probability distribution of the dimensionless deformation kernel $k = (|\sigma|\bar{u}^2)/(u^2\bar{\sigma})$ (Eq. \ref{['eq:stretch']}). The distribution of $k$ displays a power-law decay over more than three decades, regardless of the disorder level. Inset (ii) shows locations where $k>50$, indicating that stretching potential is concentrated in thin margins near solid boundaries.
• Figure 4: Plume development for completely ordered ($\varepsilon=0$) and disordered ($\varepsilon=1$) models. Each gray cylinder has diameter $a=1.0$ mm. The number of accumulated wraps of the plume over cylinders is a key control over the stretching statistics. Wraps are denoted by the symbol ($\ast$). In ordered media, the number of wraps tends toward a constant (Panel A), whereas in disordered media, the number of wraps steadily accumulates (Panel B). In all cases, an overall spatial organization of the stretching is visible, with the earliest stagnation points to be wrapped originating the most highly-stretched sections of the plume.
• Figure 5: First two moments of the stretching for all disorder levels. In (A) the asymptotic mean stretching grows quadratically for disordered media, approaching linear for ordered media. (B) displays the relative variation $\sigma_\rho/\langle \rho\rangle$. The scaling $\sigma_\rho/\langle \rho\rangle\sim t^{1/2}$ is superimposed for disordered media, while $\sigma_\rho/\langle \rho\rangle\sim t^1$ is superimposed for ordered media. The stretching model developed in Section \ref{['sec:aggregates']} predicts these scaling relations. The inset of panel (B) shows an increased density of neighboring filaments in disordered media, reflecting an increased propensity for lamella aggregation during mixing.