Darcy’s Law for Yield Stress Fluids

Predicting the flow of non-Newtonian fluids in a porous structure is still a challenging issue due to the interplay between the microscopic disorder and the nonlinear rheology. In this Letter, we study the case of a yield stress fluid in a two-dimensional structure. Thanks to an efficient optimization algorithm, we show that the system undergoes a continuous phase transition in the behavior of the flow, controlled by the applied pressure difference. In analogy with studies of plastic depinning of vortex lattices in $\mathrm{high}\text{−}{T}_c$ superconductors, we characterize the nonlinearity of the flow curve and relate it to the change in the geometry of the open channels. In particular, close to the transition, a universal scale-free distribution of the channel length is observed and explained theoretically via a mapping to the Kardar-Parisi-Zhang equation.

Figure 1

Sketch of porous media. Left: realistic porous medium in which the solid phase consists of an assembly of grains (in black). Right: model of a pore network in which large open pores are connected by straight tubes (throats) with random radii and unit length.

Figure 2

The flowing path network at different applied pressures for a system of size $L=100$.

Figure 3

(a) The mean flow curve $\overline{Q}$ for a given $\mathrm{\Delta }P=P-P_0$ averaged over more than 200 realizations. The thresholds ${\tau }_{ij}$ are uniformly distributed in $[2-\sqrt{3}/5,2+\sqrt{3}/5]$. Circles and triangles correspond to $L=64$ and $L=128$, respectively. Inset: the flow curve of a single realization ($L=50$). (b) The averaged pressure increments $\overline{\mathrm{\Delta }{P}_k}$ as function of number of open paths $k$. Inset: the averaged permeability $\overline{{\kappa }_k}$ as function of the number of open paths $k$. Circles, squares, triangles, and crosses correspond to different system sizes $L=64$, 100, 128, 256; blue, red, green correspond to different threshold distributions respectively: uniform, Gaussian, exponential. (c) The PDF of lengths of the second open paths $P_{\ell }$. Inset: the PDFs of gaps for different system sizes renormalized by their mean values which follow a clear exponential distribution. $○$, $\square$, and $△$ correspond to different threshold distributions: uniform, Gaussian, and exponential, respectively.