Anomalous percolation flow transition of yield stress fluids in porous media

Yield stress fluid (YSF) flows through porous materials are fundamental to biological, industrial, and geophysical processes, from blood and mucus transport to enhanced oil recovery. Despite their widely recognized importance across scales, the emergent transport properties of YSFs in porous environments remain poorly understood due to the nonlinear interplay between complex fluid rheology and pore microstructure. Here, we combine microfluidic experiments and nonlinear network theory to uncover an anomalous, hierarchical yielding process in the fluidization transition of a generic YSF flowing through a random medium. Percolation of a single fluidized filament gives way to pathways that branch and merge to form a complex flow network within the saturated porous medium. The evolution of the fluidized network with the flowing fraction of YSF results in a highly nonlinear flow conductivity and reveals a novel dispersion mechanism, resulting from the rerouting of fluid streamlines. The identified flow percolation phenomenon has broad implications for YSF transport in natural and engineered systems, and provides a tractable archetype for a diverse class of breakdown phenomena.

Figure 1

Fluidization of a yield stress fluid in saturated porous media. (a) Images of microfluidic porous media channels. Porosity $\sigma$ is controlled by adjusting pillar radii $r=25$, 32.5, 40, and $47.5\mu \mathrm{m}$, respectively. Pillar locations in experiments and simulations are generated by randomizing a hexagonal lattice (see also Supplemental Material [25]). Porosity color coding is carried throughout all figures. Scale bars, $250\mu \mathrm{m}$. (b) Experimentally measured flow speed fields generated from particle tracking data [25]. Flow direction is from left to right in all figures. Rows correspond to the porosities in panel (a) and columns correspond to four selected yielded fractions $\mathrm{\Phi }$, with pressure drop $\mathrm{\Delta }P$ increasing from left to right. Flow speed $v$ is normalized by the median flow speed $v_m$ for each map. Black zones are arrested microgel and pillars. Scale bar, 1 mm. (c) Flow speed maps from pore network simulations [25] for parameters corresponding to panel (b).

Figure 2

Coexistence of yielded and arrested regions reveals the nature of nonlinear conductivity in porous media. Color coding for the different porosities is defined in Fig. 1. (a) Yielded fraction $\mathrm{\Phi }$ of microgel, measured from flow speed maps [Fig. 1 and 1], transitions smoothly from 0 to 1 with increasing pressure drop $\mathrm{\Delta }P$. Symbols are experimental data measured from individual flow field realizations, solid lines are predictions from pore network simulations, and vertical dashed lines are scaling predictions for percolation onset pressure. Color coding for the different porosities is defined in Fig. 1. Inset: Collapse of the fluidization transition for all porosities with nondimensional pressure $P^{*}$. (b) Total flow rate $Q$, computed from individual measured velocity fields, increases nonlinearly with pressure drop for the four different porosities. The black line indicates expected asymptotic slope $1/n\approx 1.8$ from the shear-thinning rheology of the microgel. Error bars indicate standard deviation of flow rates measured at different cross sections along the microchannel for a single measured flow field per data point, where some error bars are occluded by the data points. (c) Flow speed distributions for the flow maps in Fig. 1. For each porosity, the normalized speed distributions for all yielded fractions $\mathrm{\Phi }\ge 0.1$ collapse (mean flow speed $\langle v\rangle$). Black lines are normalized speed distributions for a Newtonian fluid (50% wt. aqueous glycerol) measured in the same microchannels. Inset: Standard deviations of YSF flow speed probability density functions (PDFs) with error bars indicating the variation for different yielded fractions within a porosity.

Figure 3

Yielded fraction of flowing microgel exhibits hysteresis upon pressure cycling. Experiments are performed by ramping pressure up from 0 mbar to a maximum applied pressure, then back to 0 mbar. Upon cycling the applied pressure, a hysteresis of the flowing fraction $\mathrm{\Phi }$ is observed for all porosities: (a) $\sigma =0.84$, (b) $\sigma =0.73$, (c) $\sigma =0.60$, and (d) $\sigma =0.46$.

Figure 4

Flowing microgel exhibits no slip along unyielded microgel, but shows significant partial slip on solid PDMS surfaces. (a) Flow field through a porous network ($\sigma =0.84$, $\mathrm{\Phi }\approx 0.5$) reconstructed from high resolution particle tracking. Gray circles represent the pillars, and black zones indicate unyielded microgel. The color map shows the measured flow speed, and light blue lines are streamlines. The magenta line corresponds to the velocity profile transect in (b) with the white arrow indicating the positive $\lambda$ direction. Scale bar, $65\mu \mathrm{m}$. (b) Velocity profile corresponding to the magenta transect in (a), where blue and green lines are linear fits to five points of the velocity profile near unyielded microgel and PDMS pillars, respectively. Intersections of the fits with the lateral axis show minimal slip length for the microgel, but significant slip length (${\ell }_{\mathrm{slip}}\sim 50\mu \mathrm{m}$) along the PDMS surfaces.

Figure 5

Flow network topology and transport evolve with yielding. (a) Representative example of an experimentally measured pore-network exhibiting filamentous flow path topology ($\sigma =0.62$, $\mathrm{\Phi }=0.60$, $\mathrm{\Delta }P=200$ mbar). Pores are vertices of the Voronoi tessellation and connected by active (blue) or inactive (white) throats (edges). Throat activity is determined from the underlying measured flow speed map. Grey dashed edges are permanently inactive due to pillar merging (grey filled circles). Filaments are defined as contiguous chains of active edges connecting degree-2 vertices (green circles) and capped by degree-3 vertices (yellow squares). Cyan box indicates region shown in (b). Scale bar, $250\mu \mathrm{m}$. (b) Comparison of fluid streamlines at $\mathrm{\Phi }=0.60$ ($\mathrm{\Delta }P=200$ mbar; green) and $\mathrm{\Phi }=1.0$ ($\mathrm{\Delta }P=600$ mbar; blue) illustrating the rerouting of flow during the yielding transition. Streamlines are one mean lattice spacing ($a=120\mu \mathrm{m}$). Rerouting is quantified by $\alpha =A/a^2$, where $A$ is the magenta shaded area bounded by streamlines at $\mathrm{\Phi }=0.60$ and $\mathrm{\Phi }=1.0$. Blue arrows show the velocity field for $\mathrm{\Phi }=1.0$. Scale bar, $125\mu \mathrm{m}$. (c) Probability distribution of the measured rerouting parameter $\alpha$ for various yielded fractions $\mathrm{\Phi }$ at fixed porosity $\sigma =0.62$. (d) Mean rerouting parameter $\langle \alpha \rangle$ from (c) as a function of the unyielded fraction $1-\mathrm{\Phi }$. Error bars are standard deviations of five random subsamplings of the data, where some error bars are occluded by the data points. Dashed lines have slopes 1/2 and 1/4, respectively. (e) Cumulative distribution functions (CDFs) of fluidized filament length from the experimentally measured pore network (dashed lines and symbols) and from simulations (solid lines) for $\sigma =0.73$. (f) Evolution of the average excess filament length $\langle N-1\rangle$ with pressure drop in experiments (symbols) and pore-network simulations (lines).