Network modelling of yield-stress fluid flow in randomly disordered porous media

Abstract

Yield-stress fluid flow through porous media is governed by a strong coupling between rheology and pore-scale geometry, leading to nonlinear, non-Darcy transport and pronounced channelisation near yielding. We develop a pore-network model for Herschel-Bulkley flow in two-dimensional disordered porous media, including optional wall slip. The network is closed by a physics-based pressure-flow relation for a converging-diverging throat, so that yielding and post-yield transport emerge directly from the pore-scale fluid mechanics without fitted resistance parameters. Benchmarking against direct numerical simulations shows that the model captures both the bulk pressure drop and the evolution of the flow topology from spatially distributed transport to strongly channelised flow. The framework also captures the leading effect of wall slip, which lowers the pressure gradient required for transport and reactivates pathways that remain blocked in the no-slip case. Using the model across different porous geometries, we show that near-yield pressure losses are governed by constriction statistics rather than by an obstacle-scale length. In particular, rescaling with the domain-averaged minimum throat width collapses the plastic-dominated response across porosities, identifying the dissipation-relevant geometric scale for viscoplastic transport in this regime.

Figures (4)

• Figure 1: (a) Two-dimensional porous medium formed by randomly placed circular obstacles. (b) Voronoi-based pore network: vertices are pores and edges are throats, with resistance set by the converging--diverging gap between neighbouring obstacles.
• Figure 2: (a) No-slip response in geometry CT: dimensionless bulk pressure gradient $\mathcal{G}$ versus bulk Bingham number $\mathcal{B}$. Solid line, present network model; markers, direct numerical simulations of Chaparian-Tammisola:2021. The horizontal dashed line denotes the Newtonian limit ($\mathcal{B}\to0$) from the fully resolved flow simulations. (b-d) Velocity magnitude maps in geometry CT without slip, comparing direct numerical simulations (left) and the network model (right) for (b) $\mathcal{B}=0$, (c) $\mathcal{B}=1.5$, and (d) $\mathcal{B}=150$. The left-hand panels in (c) and (d) are reproduced from Chaparian-Tammisola:2021 under CC BY 4.0.
• Figure 3: Effect of wall slip on the bulk response and flow topology in geometry G1. (a) Dimensionless bulk pressure gradient $\mathcal{G}$ as a function of bulk Bingham number $\mathcal{B}$ for the no-slip case and for a slipping wall with $\alpha=0.1\,R/K$ and $\tau_s=0.2\,\tau_0$. The dashed lines indicate the large-$\mathcal{B}$ asymptotic scalings, $\mathcal{G}\sim\mathcal{B}$ without slip and $\mathcal{G}\sim(\tau_s/\tau_0)\mathcal{B}$ with slip. (b,c) Velocity maps, normalised by the mean inlet velocity $\bar{U}$, at the same imposed pressure gradient, corresponding to $\mathcal{B}=500$ in the no-slip case: (b) no slip and (c) wall slip. Wall slip lowers the bulk resistance and activates a substantially larger number of flow pathways.
• Figure 4: Comparison of scalings for the bulk response of porous-media geometries G1 (blue), G2 (red), and G3 (green). Panels (a-c) show the dimensionless bulk pressure gradient versus bulk Bingham number using (a) the scaling of Chaparian-Tammisola:2021, (b) the revised scaling of Chaparian:2024, and (c) the present scaling based on $\bar{h}_{\min}$. (d) Dimensionless geometric length scale $h_c/R$ versus porosity $\phi$. Symbols denote random realisations of porous media (green) and the three geometries of Chaparian-Tammisola:2021 (red). The solid black curve corresponds to the scaling of Chaparian:2024, with prefactor $1/\pi$.