Upscaling the Navier-Stokes-Cahn-Hilliard model for incompressible multiphase flow in inhomogeneous porous media

This paper rigorously derives a macroscopic two-phase flow model for inhomogeneous porous media by upscaling the Navier-Stokes-Cahn-Hilliard equations via volume averaging, incorporating wetting behavior into the averaged chemical potential and validating the framework through numerical simulations.

The Gist

The Big Picture: From Microscopic Chaos to Macroscopic Rules

Imagine you are trying to predict how oil and water move through a sponge.

The Problem:
If you look at the sponge under a microscope, it's a chaotic mess. The oil and water are fighting for space in tiny, winding tunnels (pores). They stick to the walls, they swirl around obstacles, and they form complex shapes. To simulate this perfectly, you would need a supercomputer to track every single drop of liquid in every tiny hole. This is called Direct Numerical Simulation (DNS). While accurate, it's impossible to do for a whole oil field or a large groundwater aquifer because the computer would need to run for centuries.

The Goal:
Engineers need a "big picture" view. They want to know how the fluids move through the entire sponge without worrying about the tiny twists and turns inside every single pore. This is called the Darcy scale (or macroscopic scale).

The Challenge:
For decades, scientists used "rules of thumb" (empirical formulas) to describe this big-picture flow. They guessed how the fluids behaved based on experiments. But these rules often failed when things got complicated, like when the fluids had different stickiness (viscosity) or when the sponge's material made the fluids stick to the walls differently (wettability).

The Solution: A "Smart Zoom" Technique

This paper introduces a new, mathematically rigorous way to "zoom out" from the microscopic chaos to the macroscopic rules. Think of it as a smart averaging technique.

Instead of just guessing the rules, the authors took the fundamental laws of physics (the Navier-Stokes equations for flow and the Cahn-Hilliard equation for how fluids mix and separate) and mathematically "averaged" them over a representative chunk of the sponge.

Here is how they did it, using a few metaphors:

1. The "Crowded Room" Analogy (Volume Averaging)

Imagine a crowded room where people (fluid molecules) are moving around.

• Micro-scale: You track every single person's exact path, speed, and who they bump into.
• Macro-scale: You want to know the "average flow" of the crowd. Are they moving left or right? How fast is the crowd moving as a whole?

The authors used a method called Volume Averaging. They took a "snapshot" of a small section of the sponge (a Representative Elementary Volume, or REV) and calculated the average behavior. But here's the catch: when you average, you lose information. You don't know exactly how the people bumped into each other or the walls.

2. The "Missing Puzzle Pieces" (Closure)

When you average the equations, you get some "missing pieces" (unclosed terms). It's like trying to solve a math problem where you know the answer is "5 plus something," but you don't know what the "something" is.

• In the past, scientists just guessed the "something" based on experiments.
• In this paper, the authors derived what that "something" is. They solved a mini-problem inside that small chunk of the sponge to figure out exactly how the tiny bumps and wall-sticking affect the big-picture flow.

3. The "Sticky Wall" Secret (Wetting Behavior)

This is the paper's biggest breakthrough.
Imagine the sponge is made of a material that loves water (hydrophilic). The water will try to hug the walls, while the oil tries to stay in the middle.

• Old Models: Treated the wall-sticking as a separate, messy force added later.
• New Model: The authors figured out how to bake the "stickiness" directly into the Chemical Potential.
  • Analogy: Think of "Chemical Potential" as the "desire" of the fluid to move. Usually, fluids move from high pressure to low pressure. This paper shows that if the walls are sticky, the fluid's "desire" to move changes. They mathematically combined the wall-sticking effect into the fluid's internal "desire" equation. This means the model naturally knows that water wants to hug the walls without needing a separate, clunky rule.

What Did They Prove? (The Experiments)

To make sure their new "smart zoom" model actually works, they ran computer simulations (using a method called Lattice Boltzmann, which is like simulating fluid flow with a grid of tiny digital particles) on three scenarios:

1. Flow through a uniform sponge: They checked if their model could predict simple flow through a sponge. It matched the known math perfectly.
2. Flow between a sponge and open water: They simulated water flowing next to a sponge. The model correctly predicted how the speed changes as the water moves from the open space into the tight sponge pores.
3. The "Fingering" Effect: This is the cool part. Imagine pushing a thick fluid (like honey) into a thin fluid (like water) inside a sponge. Usually, the honey tries to push straight through, but it gets unstable and forms "fingers" that wiggle around.
   • They tested different "stickiness" levels (contact angles).
   • Result: They found that if the sponge is very "water-loving" (hydrophilic), it actually stops the fingers from forming as much. The water hugs the walls so tightly that it blocks the oil from making those wiggly fingers. This matches what we see in real life but is hard to predict with old models.

Why Does This Matter?

This paper provides a unified bridge between the tiny world of pores and the big world of engineering.

• For Oil Companies: It helps predict how to get more oil out of the ground by understanding how fluids interact with rock at a fundamental level, not just by guessing.
• For Environmentalists: It improves predictions of how pollutants move through soil or how to store carbon underground.
• For Science: It moves us away from "guessing rules" based on experiments and toward "deriving rules" from the laws of physics.

In a nutshell: The authors built a mathematical microscope that can zoom out from the chaotic dance of fluids in tiny pores to give us a clear, accurate, and physics-based rulebook for how those fluids move through the whole system, including how they stick to the walls.

Technical Summary

1. Problem Statement

Multiphase flow in porous media is critical for applications like groundwater remediation, oil recovery, and CO2 sequestration. While Direct Numerical Simulation (DNS) at the pore scale captures detailed physics, it is computationally prohibitive for large-scale engineering problems due to the vast disparity in time and space scales.

Current macroscopic models, such as the two-phase Darcy law, rely heavily on empirical correlations for relative permeability and capillary pressure. These models often fail to capture complex phenomena like wetting hysteresis, history-dependent displacement, and non-equilibrium effects because they treat capillary pressure as a simple function of saturation, neglecting the underlying pore-scale thermodynamics and interfacial forces. There is a need for a rigorous, thermodynamically consistent macroscopic model that bridges the pore scale (governed by Navier-Stokes and Cahn-Hilliard equations) and the Darcy scale without relying solely on empirical fitting.

2. Methodology

The authors employ the Volume Averaging Method (VAM) to upscale the pore-scale governing equations to the macroscopic (Darcy) scale.

• Pore-Scale Governing Equations:

  • Fluid Dynamics: Incompressible Navier-Stokes equations.
  • Phase Evolution: Cahn-Hilliard equation, utilizing a diffuse-interface theory where an order parameter $\phi$ (ranging from 0 to 1) represents the fluid phases.
  • Thermodynamics: The system is driven by a free-energy functional that includes bulk free energy, interfacial energy, and wall free energy (to account for wetting).
  • Boundary Conditions: No-slip velocity at solid walls and a specific wetting boundary condition (cubic form) derived from the variation of the wall free energy.

• Upscaling Process:

  1. Averaging: The governing equations are integrated over a Representative Elementary Volume (REV).
  2. Decomposition: Variables are decomposed into intrinsic phase averages (slow-varying) and spatial deviations (fast-varying) using Gray's decomposition.
  3. Theorems: Spatial and temporal averaging theorems are applied to handle derivatives and integrals, introducing unclosed terms related to spatial deviations and surface integrals at the fluid-solid interface.
  4. Closure:
     • Momentum Equation: The unclosed drag force term is modeled using a generalized Darcy-Forchheimer formulation, incorporating relative permeability ($K_r$) and a nonlinear inertial term.
     • Chemical Potential: A key innovation is the rigorous derivation of the averaged chemical potential. The authors explicitly incorporate the wetting boundary condition into the averaged chemical potential, linking pore-scale surface tension and contact angle directly to the macroscopic capillary effects.
     • Assumptions: Local equilibrium is assumed within the REV (chemical potential is constant locally), and length-scale constraints ($l \ll r_0 \ll L$) are applied to simplify the deviation equations.

3. Key Contributions

• Unified Single-Fluid Model: Unlike previous upscaling efforts that resulted in two-fluid models with separate velocities and pressures, this work derives a single-fluid macroscopic model. This simplifies the numerical implementation while retaining the physics of phase interaction.
• Formal Incorporation of Wetting: The study formally derives the averaged chemical potential, $\langle \mu \rangle_\alpha$, which explicitly includes terms for the contact angle ($\theta$) and specific surface area ($a_v$). This allows wetting behavior to be intrinsically coupled to the macroscopic momentum equation, rather than being an empirical add-on.
• Thermodynamic Consistency: The model is derived from a free-energy functional, ensuring that the macroscopic equations are thermodynamically consistent with the pore-scale physics.
• Generalized Capillary Pressure: The interfacial force term in the momentum equation acts as a generalized capillary pressure that naturally recovers classical Darcy formulations under specific limits (e.g., homogeneous media, low velocity) but extends to complex, heterogeneous scenarios.
• Lattice Boltzmann Implementation: A specialized Lattice Boltzmann Method (LBM) was developed to solve the derived volume-averaged equations, capable of handling the complex source terms and variable porosity.

4. Numerical Results and Validation

The authors validated the model through several numerical tests using the developed LBM solver:

1. Poiseuille Flow in Homogeneous Porous Media:

   • Simulated flow driven by gravity in a channel filled with porous media.
   • Result: Excellent agreement with analytical solutions for velocity profiles across different Darcy numbers, validating the hydrodynamic terms (Brinkman/Darcy-Forchheimer).

2. Poiseuille Flow in Fluid-Porous Channels:

   • Simulated flow in a channel partially filled with porous media (free fluid on top, porous medium below).
   • Result: The model successfully captured the velocity continuity and stress jump at the interface, matching analytical solutions for various porosities.

3. Buckley-Leverett Problem (1D Displacement):

   • Simulated the displacement of a non-wetting phase by a wetting phase.
   • Result: The model accurately predicted the shock-front saturation and propagation speed, closely matching the classical analytical solution. It demonstrated the ability to capture the mixing region between phases.

4. Viscous Fingering in Porous Media:

   • Investigated the instability of the interface between fluids of different viscosities.
   • Result: The simulations qualitatively reproduced viscous fingering patterns. Crucially, the model showed that wettability significantly influences instability: hydrophilic conditions (low contact angle) dispersed the fingers and hindered their formation, while hydrophobic conditions promoted distinct finger splitting. This highlights the model's capability to capture wetting-dependent interfacial dynamics.

5. Significance and Conclusion

This work provides a rigorous theoretical framework that bridges the gap between pore-scale physics and macroscopic engineering models. By deriving the macroscopic equations directly from the Navier-Stokes-Cahn-Hilliard system, the authors eliminate the need for arbitrary empirical closures for capillary pressure and wetting effects.

• Theoretical Impact: It establishes a single-fluid description where capillary forces and wetting are naturally embedded in the chemical potential and momentum equations.
• Practical Impact: The model offers a more predictive tool for simulating complex multiphase flows in heterogeneous porous media, particularly where wetting history, contact angle variations, and interfacial instabilities play a dominant role.
• Future Work: The authors note that while closure coefficients were estimated, future research should focus on determining these coefficients for complex, real-world geometries and establishing explicit links to conventional capillary pressure models (e.g., Leverett J-function).

In summary, the paper successfully demonstrates that volume averaging of the NSCH system yields a powerful, thermodynamically consistent macroscopic model capable of capturing essential two-phase flow characteristics, including wetting effects, without relying on ad-hoc empirical correlations.