[Paper Review] Network modelling of yield-stress fluid flow in randomly disordered porous media

Cláudio P. Fonte, Elliott Sutton+4 more|arXiv (Cornell University)|Mar 10, 2026

Rheology and Fluid Dynamics Studies0 citations

TL;DR

A 2D pore-network model for Herschel-Bulkley flow in disordered porous media is developed with wall slip, closed by a physics-based converging-diverging throat law, reproducing yielding, channelisation, and non-Darcy behavior without fitted parameters.

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.

Motivation & Objective

• Capture yield-stress rheology and pore-scale geometry coupling in 2D disordered porous media.
• Develop a pore-network closure that yields directly from pore-scale fluid mechanics without fitted resistances.
• Incorporate optional wall slip and assess its influence on transport thresholds and topology.
• Benchmark network predictions against direct numerical simulations to validate bulk and flow-structure predictions.
• Identify geometric scales controlling near-yield dissipation and transport in viscoplastic regimes.

Proposed method

• Represent the void space via a Voronoi tessellation of obstacle centers to form a network of pores (vertices) and throats (edges).
• Model throat flow with a physics-based converging-diverging pressure–flow relation derived from Herschel–Bulkley rheology and one-dimensional flow in the gap.
• Incorporate wall slip through a slip velocity boundary condition that depends on local shear stress and a slip yield stress.
• Solve the resulting nonlinear network system using a Newton-type method with trust-region globalization and continuation in inlet pressure.
• Non-dimensionalise using obstacle radius and define bulk Bingham number and dimensionless pressure gradient to compare with direct numerical simulations.
• Provide an openly available Julia implementation for replication.

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.

Experimental results

Research questions

• RQ1 How does yield-stress rheology couple with random pore-scale geometry to produce non-Darcy transport in 2D porous media?
• RQ2 Can a fully predictive pore-network model (without fitted resistance parameters) capture yielding, channelisation, and post-yield transport including wall slip?
• RQ3 What is the role of wall slip in altering the pressure threshold and flow topology in viscoplastic flow through porous media?
• RQ4 What geometric quantity governs near-yield pressure losses across porosities, and can a universal scaling be established?
• RQ5 How well does the network model reproduce bulk pressure drop and flow-path evolution compared with direct numerical simulations?

Key findings

• The network model reproduces the Newtonian limit and the linear viscoplastic scaling G ~ B at large B, matching DNS trends.
• The model captures the transition from spatially distributed flow to strongly channelised transport as B increases.
• Wall slip lowers the required pressure gradient and reactivates pathways blocked in the no-slip case, altering both bulk resistance and topology.
• Near yield, pressure losses are governed by constriction statistics, not an obstacle-scale length, with a master scaling achieved when using the domain-averaged minimum throat width h_min.
• Rescaling with h_min collapses the plastic-dominated response across porosities, identifying the dissipation-relevant geometric scale for viscoplastic transport.
• The network predictions show good quantitative agreement with DNS, with deviations of about 17% at B = 100 and 31% at B = 10^3, highlighting remaining near-yield modeling challenges.

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 and Tammisola ( 2021 ) . The horizontal dashed line denotes the Newtonian