Local Thermal Non-Equilibrium Models in Porous Media: A Comparative Study of Conduction Effects

This study compares Local Thermal Non-Equilibrium (LTNE) models against a pore-resolved reference for purely conductive porous media, demonstrating that REV-scale models utilizing homogenization-based effective parameters accurately capture interfacial resistance effects, whereas dual-network models with fixed spatial resolution show greater deviation.

The Gist

Imagine a porous material, like a sponge or a rock, as a busy city. This city has two types of residents: the solid grains (the buildings) and the fluid (the water or air flowing through the streets).

When you heat up one side of this city, you want to know how the temperature spreads. The big question this paper asks is: Do the buildings and the water inside them heat up at the exact same speed, or do they lag behind each other?

Here is a breakdown of what the researchers did, using simple analogies.

1. The Two Ways to Think About Heat

In the past, scientists usually assumed Local Thermal Equilibrium (LTE).

• The Analogy: Imagine a room full of people holding hands. If one person gets hot, everyone else instantly feels it. In this model, the "buildings" and the "water" are so perfectly connected that they always have the exact same temperature at any given spot. It's like they are sharing a single brain.

However, the researchers knew this isn't always true. Sometimes, the connection between the building and the water is "sticky" or slow. This is Local Thermal Non-Equilibrium (LTNE).

• The Analogy: Imagine the people are in separate rooms with thick, insulated doors between them. If you heat the water in the hallway, the buildings might stay cool for a while because the heat has to struggle to get through the door. The water gets hot, but the building stays cold for a bit. They have different temperatures at the same location.

2. The Three "Maps" Used to Predict Heat

To figure out when this "lag" happens and how to predict it, the team compared three different ways of drawing a map of this city:

• Map A: The "Street-Level" View (Pore-Resolved Model)

  • What it is: This is the most detailed map. It draws every single building and every single street. It sees the exact shape of the rock and the water.
  • The Catch: It's incredibly slow and computationally expensive, like trying to simulate every single grain of sand in a beach. The researchers used this as their "Gold Standard" or reference to see if the other maps were right.

• Map B: The "Neighborhood" View (Dual-Network Model)

  • What it is: Instead of drawing every street, this map simplifies the city into a network of dots (representing the buildings and water pockets) connected by lines (representing the connections between them).
  • The Catch: It's faster, but it has a fixed resolution. It's like looking at a city through a grid of windows; you can't zoom in closer than the size of the window. The paper found that because this grid is fixed, it sometimes misses the sharp temperature changes happening right at the edges.

• Map C: The "Aerial" View (REV-Scale Model)

  • What it is: This is a high-level, averaged map. It doesn't see individual buildings; it sees "blocks" of the city. It uses math to guess the average behavior of the whole block.
  • The Catch: To make this work, you have to guess the "average properties" of the block. If you guess wrong, the whole map is wrong.

3. The Big Experiment

The researchers ran simulations on a computer to see how heat moved through this "city" under two different conditions:

Scenario 1: The Open Door (Low Resistance)

• The Setup: The connection between the water and the rock was perfect (like a wide-open door). Heat flowed freely.
• The Result: The "Open Door" meant the water and rock heated up instantly together. The LTE assumption (the single brain) worked perfectly. All three maps gave almost the same answer. The "lag" didn't exist.

Scenario 2: The Insulated Door (High Resistance)

• The Setup: The connection was blocked or "sticky" (like a thick, insulated door). Heat had a hard time jumping from the water to the rock.
• The Result: Now, the water got hot, but the rock stayed cool for a while. The LTE assumption failed completely.
  • The Street-Level Map showed the exact lag.
  • The Aerial Map (if calculated correctly using a specific math method called homogenization) matched the Street-Level Map very well.
  • The Neighborhood Map was okay, but because its "windows" were fixed in size, it smoothed out the sharp differences a bit too much.

4. The Key Takeaway

The most important finding is about how you calculate the "Aerial" map.

• Some old ways of calculating the average properties for the Aerial map ignored the "sticky door." They assumed the heat transfer was always perfect. When the researchers used these old formulas, the Aerial map failed to show the lag between the water and the rock.
• However, when they used a specific, more advanced math method (homogenization) that did account for the "sticky door" (the interfacial resistance), the Aerial map became incredibly accurate. It matched the detailed Street-Level view almost perfectly, even though it was much simpler.

Summary

• If the connection is perfect: You can use simple models; everything heats up together.
• If the connection is slow/sticky: You must use models that allow the water and rock to have different temperatures.
• The Best Shortcut: If you need to model a huge system (like a whole aquifer or a fuel cell) and can't simulate every grain, use the "Aerial" model, but make sure you use the specific math that accounts for the resistance between the materials. If you do that, your simple model will be just as accurate as the super-detailed one.

Note: The paper explicitly states that this study only looked at heat moving through still materials (conduction). They did not look at heat moving with flowing water (convection), which they say they will study in a future paper.

Technical Summary

1. Problem Statement

Modeling heat transfer in porous media often relies on the Local Thermal Equilibrium (LTE) assumption, which posits that fluid and solid phases share the same temperature instantaneously within a control volume. While valid for many scenarios, this assumption fails in systems characterized by:

• Large temperature gradients.
• Significant differences in thermal properties between phases.
• High interfacial thermal resistances (low heat transfer coefficients).
• Fast dynamics.

When LTE fails, Local Thermal Non-Equilibrium (LTNE) models are required to resolve distinct temperatures for the fluid ($T_f$) and solid ($T_s$) phases and the heat exchange between them. However, there is a lack of comprehensive comparative studies validating continuum-scale LTNE models (Dual-Network and REV-scale) against high-fidelity pore-resolved references, particularly regarding the accuracy of effective parameter formulations (e.g., homogenization vs. empirical correlations).

2. Methodology

The study compares three continuum-scale modeling approaches on a fully saturated system (one fluid, one solid) considering pure heat conduction (neglecting convection and mass transport).

A. The Three Model Classes

1. Pore-Resolved Model (Reference):

   • Approach: Solves energy balance equations separately for fluid ($\Omega_f$) and solid ($\Omega_s$) domains on a grid resolving the exact pore-scale geometry.
   • Interface: Explicitly resolves the sharp fluid-solid interface ($\Gamma_{fs}$).
   • LTNE Implementation: Uses a finite heat transfer coefficient ($h$) to allow a temperature jump across the interface ($\lambda_f \nabla T_f \cdot n = h(T_s - T_f)$).
   • Solver: Finite Element Method (FEM) using Netgen/NGSolve.

2. Dual-Network Model:

   • Approach: Represents the porous medium as two interconnected networks (fluid and solid) composed of idealized bodies (spheres) and throats (cylinders).
   • Interface: Does not resolve the sharp interface; instead, it uses interfacial throats with thermal transmissibility based on shape factors and effective areas.
   • Solver: Control Volume Finite Element (CVFE) method using DuMux.

3. REV-Scale Model (Representative Elementary Volume):

   • Approach: Uses averaged balance equations over a control volume where fluid and solid temperatures overlap.
   • LTNE Implementation: Solves two coupled energy equations with an interfacial heat exchange term ($q_{cond} = \lambda_{eff,I} a_{fs} (T_s - T_f)$).
   • Effective Parameters: The study evaluates three different formulations for effective thermal conductivities and interfacial terms:
     • Nuske et al.: Simple volume-fraction scaling.
     • Nakayama et al.: Includes tortuosity and conductivity ratios.
     • Homogenization Theory: Derives effective parameters mathematically from pore-scale equations, explicitly incorporating the interfacial heat transfer coefficient ($h$).
   • Solver: Finite Volume Method (FV) using PorePy.

B. Simulation Setup

• Geometry: A periodic 3D domain consisting of cubic reference cells containing a sphere connected to six tubes.
• Materials: Granite (solid) and Water (fluid).
• Boundary Conditions: Quasi-1D setup with a temperature step on the left boundary (fluid heated by 10K, solid insulated).
• Test Cases:
  • Case 1 (Low Resistance): High heat transfer coefficient ($h \approx 8.3 \times 10^7$ W/m²K), mimicking standard water-granite interfaces.
  • Case 2 (High Resistance): Low heat transfer coefficient ($h = 100$ W/m²K), mimicking high-resistance interfaces (e.g., copper-helium) to induce strong LTNE effects.

3. Key Contributions

• Systematic Comparison: Provides a rigorous benchmark of Dual-Network and REV-scale models against a pore-resolved reference for LTNE conditions.
• Parameter Formulation Analysis: Evaluates the efficacy of different effective parameter formulations. It demonstrates that homogenization-based approaches are superior for capturing LTNE behavior because they mathematically incorporate the interfacial heat transfer coefficient, whereas empirical formulations (Nuske, Nakayama) often fail to predict LTNE in purely conductive systems without explicit $h$-dependency.
• Resolution Impact: Highlights the critical role of spatial resolution. The Dual-Network model showed deviations from the reference due to its fixed spatial resolution, which cannot be refined to capture steep temperature gradients near boundaries, unlike the REV-scale model where grid refinement is possible.

4. Key Results

• Low Interfacial Resistance (High $h$):

  • The system behaves as LTE; fluid and solid temperatures are nearly identical across all models.
  • The REV-scale model (using homogenized parameters) closely matches the pore-resolved reference.
  • The Dual-Network model showed slightly lower temperatures and less steep gradients compared to the reference, attributed to its fixed spatial resolution and numerical diffusion.

• High Interfacial Resistance (Low $h$):

  • LTNE Effects: Significant temperature differences between fluid and solid phases were observed, particularly near the heated boundary.
  • Model Performance:
    • The REV-scale model with homogenized parameters accurately captured the LTNE behavior and the temperature jump, closely matching the pore-resolved reference (though slightly faster due to numerical diffusion).
    • The Dual-Network model again showed a less steep gradient and lower temperatures due to its inability to refine the mesh at the interface.
    • LTE vs. LTNE: Models ignoring interfacial resistance (LTE) failed to capture the temperature jump, confirming the necessity of LTNE models for high-resistance scenarios.

• Effective Parameter Formulations:

  • In the high-resistance case, formulations by Nuske and Nakayama (which do not inherently include $h$ in the conductive limit) failed to produce LTNE effects, predicting LTE instead.
  • Only the homogenization-based approach successfully captured the LTNE behavior because the upscaled interfacial term ($H = h \cdot a_{fs}$) explicitly depends on the heat transfer coefficient.

5. Significance and Conclusion

• Validation of Homogenization: The study confirms that homogenization theory provides the most reliable effective parameters for REV-scale LTNE models, as it rigorously bridges the pore-scale physics (including interfacial resistance) to the continuum scale.
• Limitations of Dual-Network Models: While useful for pore-scale insights, Dual-Network models are limited by their fixed spatial resolution. They struggle to resolve sharp thermal gradients near boundaries compared to continuum models that allow grid refinement.
• Practical Implications: For applications involving high interfacial resistance (e.g., specific geothermal or cryogenic systems), relying on empirical effective parameters without considering the heat transfer coefficient can lead to significant errors (predicting LTE when LTNE exists).
• Future Work: The authors note that convection is a dominant factor in most practical applications and plan to extend this comparative study to include mass transport and convective heat transfer in a follow-up article.

In summary, the paper establishes that REV-scale models utilizing homogenized effective parameters are the most robust continuum approach for simulating LTNE in porous media, provided that the interfacial heat transfer coefficient is explicitly accounted for in the model formulation.