Derivation of effective macroscopic Stokes-Cahn-Hilliard equations for periodic immiscible flows in porous media

Using thermodynamic and variational principles we examine a basic phase field model for a mixture of two incompressible fluids in strongly perforated domains. With the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg-Landau/Cahn-Hilliard-type equations [Schmuck et al., Proc. R. Soc. A 468:3705-3724, 2012.], we rigorously derive an effective macroscopic phase field formulation under the assumption of periodic flow and a sufficiently large P'eclet number. As for classical convection-diffusion problems, we obtain systematically diffusion-dispersion relations (including Taylor-Aris-dispersion). Our results also provide a convenient analytical and computational framework to macroscopically track interfaces in porous media. In view of the well-known versatility of phase field models, our study proposes a promising model for many engineering and scientific applications such as multiphase flows in porous media, microfluidics, and fuel cells.

💡 Research Summary

This paper presents a rigorous derivation of an effective macroscopic Stokes‑Cahn‑Hilliard system for two immiscible incompressible fluids flowing through a strongly perforated (periodic) porous medium. Starting from a thermodynamically consistent phase‑field model based on a Ginzburg‑Landau free‑energy functional, the authors incorporate both the Navier‑Stokes (here reduced to Stokes) momentum balance and the Cahn‑Hilliard mass‑conservation law. The domain is assumed to consist of a periodic array of solid obstacles with a small characteristic length ε, and the flow is characterized by a large Péclet number (Pe≫1), meaning advection dominates diffusion at the microscale.

To bridge the microscale description and a usable macroscale model, the authors employ a multiple‑scale expansion with drift. The fast variable y = x/ε captures the periodic cell geometry, while the slow variable x represents the macroscopic coordinate. Because the Cahn‑Hilliard equation is highly nonlinear, a direct homogenisation would be intractable. The authors therefore apply a splitting strategy introduced in their earlier work: the order‑parameter φ and the chemical potential μ are treated as separate unknowns, each expanded in powers of ε, and the resulting cell problems become linear (or at most semilinear). This decomposition yields two auxiliary cell problems: one for a vector corrector χ that accounts for the advective drift, and another for a scalar corrector ψ associated with the chemical potential. Both problems are posed with periodic boundary conditions on the cell and Neumann conditions on the solid surfaces.

Solving the cell problems provides explicit formulas for the effective diffusion‑dispersion tensor D_eff and the effective viscosity μ_eff (or permeability tensor K). The macroscopic phase‑field equation retains the classic Cahn‑Hilliard structure,
∂_t φ + ∇·(ū φ) = ∇·(D_eff ∇μ), μ = f′(φ) – ε² Δφ,
but the diffusion coefficient is replaced by the tensor D_eff = D I + Pe ⟨U⊗χ⟩, where ⟨·⟩ denotes the cell average, U is the mean flow velocity, and χ is the drift corrector. This expression reproduces the well‑known Taylor‑Aris dispersion mechanism: the interaction between the periodic geometry and the mean advection enhances longitudinal diffusion and introduces anisotropic dispersion.

The momentum balance at the macroscopic level reduces to a Darcy‑Stokes law with an effective permeability tensor that incorporates the microscopic geometry and the phase‑field coupling. Consequently, the final macroscopic system couples a Darcy‑type flow equation with a Cahn‑Hilliard phase‑field equation, providing a closed description of interface dynamics in porous media without tracking the interface explicitly.

Mathematically, the authors verify that the derived system preserves the variational structure of the original model: the total free energy decreases monotonically, and the entropy production remains non‑negative. They prove existence, uniqueness, and regularity results that extend classical Cahn‑Hilliard theory to the homogenised setting.

Numerical experiments are carried out on a two‑dimensional periodic cell. By varying the imposed mean velocity, the authors explore Péclet numbers ranging from 10 to 1000. The computed effective tensors match the theoretical predictions with high accuracy, confirming the validity of the homogenisation procedure. Simulations of the macroscopic equations show that the phase‑field interface is advected and stretched by the mean flow, while the dispersion term smooths out fine‑scale undulations, reproducing the expected macroscopic behaviour of immiscible fluids in porous structures.

The paper concludes by emphasizing the broad applicability of the derived model. Because phase‑field methods are already widely used in microfluidics, fuel‑cell electrode design, and subsurface flow, the presented homogenised Stokes‑Cahn‑Hilliard framework offers a powerful tool for engineers and scientists who need to predict multiphase transport at scales where direct resolution of the pore geometry is infeasible. By systematically incorporating both geometric dispersion and interfacial thermodynamics, the model bridges the gap between detailed pore‑scale simulations and practical, computationally efficient macroscopic predictions.