Flow of yield stress fluid in a percolating network

Abstract

We study the flow of a Bingham yield-stress fluid in a pore network model where the throats have radii drawn from a uniform distribution. We consider the case in which a fraction of the largest radii is blocked. The fluid can flow only through the percolating cluster that exists when the fraction is above the percolation threshold. Two distinct flow regimes are identified: above the percolation threshold the flow curve can be characterized by deterministic values of the critical pressure drop, permeability, and other observables, with subleading fluctuations that we quantify. At the percolation threshold these quantities become non-self-averaging, and their scaling is governed exclusively by the critical percolation backbone, independent of the specific realization of the radii.

Figures (11)

• Figure 1: Sketch of a pore network of size $L$ and width $W$. The bold links indicate a larger radius, and cut links are represented by dashed lines. The pressure $P = P_{in}>0$ is imposed at the inlet (left boundary), while $P = 0$ is imposed at the outlet (right boundary). Here, the flow goes from left to right.
• Figure 2: Snapshots of simulations on a $128 \times 128$ network at the percolation threshold $p=p_c$. The cut links in the network are represented in black. The blue nodes belong to the percolating cluster, while the white ones denote the non-spanning clusters : fluid can not flow through these. With increasing pressure drops $\Delta P$, more paths start flowing, increasing the medium's effective permeability.
• Figure 3: Left: Average pressure threshold $\Delta P_0$ as a function of the system size $L$, at fixed distance from the percolation threshold $\delta p = p - p_c$. Right: Standard deviation of $\Delta P_0$ versus $L$. Averages are performed over 10 000 disorder realizations.
• Figure 4: Left: Average length $\ell$ of the first open path as a function of the system size $L$ on a log-log scale. Middle: Standard deviation of $\ell$ versus $L$. Right:$\Delta P_0$ as a function of the path length $\ell$. To reduce fluctuations, we average $\Delta P_0$ over different realizations with the same path length. The averages are computed from 10 000 samples. Inset: Average local threshold along the selected path (blue) and average local threshold over the system (yellow). Owing to optimization, the former is smaller than the latter. Only at $\delta p = 0$ do the two become comparable, as the optimization is less effective.
• Figure 5: Left: Critical pressure threshold $\Delta P_0$ as a function of $\delta p = p- p_c$ for multiple system sizes $L$. Averages were performed for each value of $\delta p$ and $L$ over $300$ realizations of the disorder. Right: Collapse using Eq. \ref{['eq:DeltaP0_scaling']}. The power-law decay $x^{-\theta}$, with $\theta = 0.17$ is indicated by a solid line.