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

This paper presents a physics-based pore-network model for yield-stress fluid flow in disordered porous media that accurately captures nonlinear transport and channelization by deriving pressure-flow relations from pore-scale mechanics, revealing that near-yield pressure losses are governed by constriction statistics rather than obstacle-scale length.

The Gist

Imagine trying to push a giant, thick blob of toothpaste through a sponge made of randomly scattered marbles. That's essentially what this paper is about, but with a scientific twist: they are studying how yield-stress fluids (like toothpaste, ketchup, or certain drilling muds) move through porous media (like soil, rocks, or filters).

Here is the story of their discovery, broken down into simple concepts and everyday analogies.

1. The Problem: The "Stuck" Toothpaste

Most fluids, like water, flow easily as soon as you push them. But yield-stress fluids are stubborn. They act like a solid until you push hard enough to break their internal structure. Once you push hard enough, they suddenly turn into a liquid and flow.

When these fluids try to squeeze through a messy, random maze of rocks (porous media), things get complicated:

• The Bottleneck: The fluid doesn't flow everywhere at once. It only flows through the "easy" paths where the rocks are far apart.
• The Traffic Jam: As the fluid gets stickier (higher yield stress), it stops flowing in most places and gets forced into just a few narrow, high-speed highways. This is called channelisation.
• The Slippery Factor: Sometimes, the fluid doesn't stick to the walls of the rocks; it slides along them (like a wet bar of soap). This "wall slip" changes everything, making it easier to push the fluid through.

2. The Old Way vs. The New Way

Scientists have tried to simulate this in two ways:

• The "Supercomputer" Method: You can try to calculate the movement of every single drop of fluid around every single rock. This is incredibly accurate but takes so much computing power that it's like trying to count every grain of sand on a beach to predict how the tide moves. It's too slow for real-world use.
• The "Map" Method (Network Modeling): Instead of tracking every drop, you draw a simplified map. You treat the gaps between rocks as "throats" (narrow tunnels) and the open spaces as "pores" (rooms). You then write rules for how fluid moves from one room to the next.

The Problem with Old Maps: Previous maps were too simple. They used "fitted parameters," which is like guessing the speed limit on a road based on a guess rather than the actual physics of the car. They often failed right at the moment the fluid started to move (the "yielding" point).

3. The New Solution: A Physics-Based Map

The authors of this paper built a smarter map.

• No Guessing: Instead of guessing the rules, they derived them directly from the physics of how fluid squeezes through a narrowing gap between two rocks.
• The "Converging-Diverging" Tunnel: They realized that the space between two rocks isn't a straight pipe; it's a tunnel that gets narrow and then wide again. They built a mathematical model specifically for this shape.
• The Slippery Slide: They also added a rule for wall slip. Imagine the fluid sliding along the rock walls like a skater on ice. This lowers the friction and allows the fluid to find new paths that were previously blocked.

4. The Big Discovery: It's About the "Neck," Not the "Body"

The researchers tested their new map against the "Supercomputer" method and found it was surprisingly accurate, even though it was much simpler.

But the biggest surprise came when they tried to understand what controls the pressure needed to push the fluid through.

• Old Theory: Scientists thought the size of the rocks (the obstacles) determined how hard it was to push the fluid.
• New Reality: They found that the rock size didn't matter as much as the size of the narrowest gap (the "neck") between the rocks.

The Analogy: Imagine a crowd of people trying to leave a stadium.

• The Old Theory says the difficulty depends on how big the stadium is.
• The New Theory says the difficulty depends entirely on the width of the exit door. Even if the stadium is huge, if the door is tiny, the crowd is stuck. If the door is wide, the crowd flows easily.

They discovered that if you measure the average width of the narrowest gaps in the entire maze, you can predict exactly how the fluid will behave, no matter how messy the maze looks. This "gap width" is the true ruler for these fluids.

5. Why This Matters

This new model is a game-changer because:

1. It's Fast: It runs on a laptop in seconds, whereas the supercomputer method takes days.
2. It's Accurate: It predicts exactly when the fluid will start moving and how it will channelize.
3. It's Practical: This helps engineers design better oil recovery systems (getting oil out of rock), improve soil cleaning (removing pollutants), and create better filters.

In a nutshell: The authors built a smart, physics-based map that treats the fluid like a stubborn crowd trying to escape a maze. They discovered that the crowd's movement isn't controlled by the size of the maze, but by the size of the narrowest doorways. And if the walls are slippery, the crowd escapes much faster!

Technical Summary

Here is a detailed technical summary of the paper "Network modelling of yield-stress fluid flow in randomly disordered porous media."

1. Problem Statement

The flow of yield-stress fluids (e.g., Herschel-Bulkley fluids) through porous media is critical for applications like enhanced oil recovery, filtration, and soil remediation. Unlike Newtonian fluids, these systems exhibit non-Darcy behavior:

• Flow only initiates above a finite pressure threshold.
• The relationship between flow rate and pressure drop is nonlinear.
• Near the yield point, transport becomes highly heterogeneous, characterized by channelization where only a subset of the pore space is mobilized.
• Wall slip (common in soft glassy and particulate materials) further complicates the physics by reducing friction and altering the yield threshold.

Current Limitations:

• Direct Numerical Simulations (DNS): While accurate, DNS is computationally expensive and difficult to apply to large or strongly disordered domains.
• Existing Pore-Network Models (PNM): Current models often rely on fitted resistance parameters or fail to accurately capture the transition near yielding. They frequently lack a physics-based closure that incorporates both yield-stress rheology and wall slip without empirical calibration.

2. Methodology

The authors developed a physics-based pore-network model for Herschel-Bulkley flow in 2D disordered porous media.

• Geometry Construction:

  • The porous medium is modeled as a 2D domain ($L \times L$) containing randomly placed, non-overlapping circular obstacles of radius $R$.
  • A Voronoi tessellation of the obstacle centroids defines the network: vertices represent pores, and edges represent throats.
  • The throat geometry is modeled as a converging-diverging gap between neighboring obstacles, defined by a minimum half-width $h_{min}$.

• Throat-Scale Physics (The Closure):

  • Instead of using fitted parameters, the model derives the pressure-flow relation ($P(Q)$) directly from the fluid mechanics of a converging-diverging throat.
  • Governing Equations:
    • Rheology: Herschel-Bulkley law ($\eta(\dot{\gamma}) = \tau_0/\dot{\gamma} + K\dot{\gamma}^{n-1}$).
    • Slip Condition: A slip velocity $u_s$ is imposed at the walls, dependent on the wall shear stress exceeding a slip yield stress $\tau_s$.
    • Momentum Balance: Solved for steady, unidirectional flow to relate local pressure gradient $G(x)$ to flow rate $Q$.
  • Numerical Implementation: The local relation is inverted numerically to find $G(x)$ for a given $Q$, and the total pressure drop across the throat is obtained by integrating $G(x)$ along the throat length. A regularization parameter is used to ensure numerical robustness near zero flow.

• Network Solution:

  • The system solves mass conservation at each pore and the nonlinear pressure-flow relations across all throats.
  • A Newton-type method with trust-region globalization (implemented in Julia) solves the coupled system.
  • Continuation: Solutions are computed by sweeping the inlet pressure, using previous solutions as initial guesses to handle stiffness near the yield limit.

• Validation:

  • Benchmarked against Direct Numerical Simulations (DNS) from Chaparian and Tammisola (2021) and the authors' own Newtonian limit simulations.
  • Tested across multiple geometries (Cases CT, G1, G2, G3) with varying porosities ($\phi$).

3. Key Contributions

1. Parameter-Free Pore-Network Model: The first PNM for viscoplastic flow that incorporates wall slip and yield-stress rheology without empirical fitting. The throat resistance is derived entirely from first principles (geometry and rheology).
2. Inclusion of Wall Slip: Explicitly models the effect of wall slip on both the bulk pressure drop and the flow topology, a feature previously absent in predictive PNM for this regime.
3. Identification of the Governing Length Scale: The study challenges the use of obstacle radius ($R$) or porosity-based metrics as the primary scaling length for near-yield flow. It identifies the domain-averaged minimum throat width ($\bar{h}_{min}$) as the correct geometric scale governing dissipation in the plastic-dominated regime.

4. Key Results

• Benchmarking & Accuracy:

  • The model accurately reproduces the macroscopic departure from Darcy behavior, capturing the monotonic increase of the dimensionless pressure gradient ($\mathcal{G}$) with the Bingham number ($\mathcal{B}$).
  • It correctly predicts the transition from spatially distributed flow (low $\mathcal{B}$) to strongly channelized flow (high $\mathcal{B}$).
  • Discrepancy: The model slightly underpredicts resistance in the highly channelized regime (up to ~31% at high $\mathcal{B}$). This is attributed to the 1D flow assumption in wider constrictions and the neglect of dissipation within the pore bodies (nodes). However, it remains a highly efficient reduced-order tool.

• Effect of Wall Slip:

  • Slip significantly lowers the pressure gradient required to sustain flow.
  • Topology Change: In the no-slip case, flow is confined to very few pathways. With slip, secondary pathways are reactivated, and the flow network becomes more connected.
  • Scaling: The asymptotic scaling of pressure gradient changes from $\mathcal{G} \sim \mathcal{B}$ (no slip) to $\mathcal{G} \sim (\tau_s/\tau_0)\mathcal{B}$ (with slip), indicating that the slip yield stress becomes a dominant threshold.

• Scaling Laws (The "Collapse"):

  • Traditional scaling using obstacle radius $R$ fails to collapse data across different porosities.
  • Scaling using porosity-based metrics ($\phi/(1-\phi)$) improves collapse but leaves residual offsets at high porosity.
  • New Scaling: Rescaling the Bingham number and pressure gradient using the average minimum throat width ($\bar{h}_{min}$) (defined as $\mathcal{B}^*$ and $\mathcal{G}^*$) results in a universal collapse of the data.
  • In the plastic-dominated regime, the relationship becomes a master curve: $\mathcal{G}^* = \mathcal{B}^*$. This confirms that near-yield pressure losses are governed by constriction statistics rather than the obstacle scale.

5. Significance

• Computational Efficiency: The model provides a predictive framework for complex porous media at a fraction of the cost of DNS, making it suitable for large-scale or real-time applications.
• Physical Insight: It clarifies that the "bottleneck" for yield-stress fluids is not the average pore size or obstacle spacing, but specifically the minimum constriction width available in the network.
• Practical Application: By quantifying the impact of wall slip, the model offers a better tool for designing processes involving soft materials (e.g., foams, pastes, gels) where slip is prevalent, potentially leading to more efficient filtration or injection strategies.
• Theoretical Advancement: The work bridges the gap between microscopic fluid mechanics and macroscopic transport laws, demonstrating that reduced-order models can capture complex non-Newtonian phenomena if the local closures are physically rigorous.