A quasi-incompressible Cahn Hilliard Darcy model for two immiscible fluids in porous media

Jul 16, 2026

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Derivation of a quasi-incompressible Cahn–Hilliard–Darcy (qCHD) phase-field model for two-phase flow in porous media using a logarithmic Flory–Huggins free energy density. The formulation obeys an energy-dissipation law and yields the classical Muskat’s problem in the formal sharp-interface limit. Pressure estimates from Darcy’s equations enable global existence of weak solutions for the qCHD system in both two and three dimensions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages22

Reading Time55 min

Introduction

Two-phase flows and the Muskat problem

Cahn–Hilliard and Cahn–Hilliard–Navier–Stokes (CHNS)

Diffuse-interface to sharp-interface limits

Phase-field models for Hele–Shaw and porous media (CHHS/CHD)

Logarithmic Flory–Huggins potential and solvability results

arXiv :2607 .04060v1 [math .AP] 5 Jul 2026 A quasi-incompressible Cahn–Hilliard–Darcy model for two immiscible fluids in porous media Daozhi Han Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260 [daozhiha@ub.edu](daozhiha@ub.edu) Sayantan Sarkar Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260 [sayantan@buffalo.edu](sayantan@buffalo.edu) Abstract We derive a quasi-incompressible Cahn-Hilliard-Darcy phase-field model (qCHD) with the logarithmic Flory-Huggins free energy density function for two-phase flows in porous media. The model satisfies an energy-dissipation law. In the formal sharp interface limit, the qCHD model gives rise to the classical Muskat’s problem. By exploiting estimates of the pressure from Darcy’s equations, we establish global existence of weak solutions in both 2D and 3D to the qCHD model. Keywords: Navier-Stokes, Cahn-Hilliard, Darcy, diffuse interface model, well-posedness, superposed free flow and porous media Mathematics Subject Classification (2010): 35K61, 76T99, 76S05, 76D07 . 1 Introduction The study of two-phase flows in porous media is vital for both mathematical analysis and engineering applications [1] including unsaturated soil flow, petroleum reservoir displacement, geological (CO2 ) storage [2], proton exchange membrane fuel cells [3] water management in polymerelectrolyte fuel cells [4] . Historically, the Muskat problem [5] has been employed as a conventional model that depicts a fluid interface as an infinitely thin discontinuity. Although this model is effective for fixed geometries, it is vulnerable to finite-time singularities: smooth initial profilescan become unbounded [6]; the Rayleigh–Taylor stability criterion may initially be satisfied but can later fail [7]; and even analytic data in a seemingly stable regime can result in instability and collapse [8] . To mitigate these issues, diffuse-interface (phase-field) formulations substitute the sharp discontinuity with a narrow yet finite transition layer, offering a more robust description that naturally regularizes the dynamics. For an in-depth review, we refer to Anderson et al. [9] . Building on the diffuse interface description, Cahn and Hilliard [10] introduced an evolution equation for the interface between two phases. The Cahn–Hilliard equation is derived by minimizing the free-energy potential, with the minimum values corresponding to the different phases (i.e. , different fluids) . The CH equation has been thoroughly investigated theoretically, both without logarithmic potential ([11, 12 , 13 , 14]) and with logarithmic potential ([15, 16 , 17 , 18 , 19]) . In the context of two-phase flows, the Cahn–Hilliard equation was coupled with fluid equations such asthe Navier–Stokes equations. This coupling, now referred to as the Cahn–Hilliard–Navier–Stokes (CHNS) system, initially emerged in kinetic theories of critical dynamics [20, 21 , 22] and was formalized as Model H by Hohenberg and Halperin [23] . For the matched-density scenario (ρ1 = ρ2 ), the CHNS model was rigorously grounded in thermodynamics using the microforce formulation by Gurtin et al. [24] . Subsequently, Jacqmin conducted the first fully resolved simulations and formally derived the sharp interface limit as the interfacial thickness approached zero [25] . With a regular (polynomial) free-energy density, Boyer demonstrated global weak existence and uniqueness [26], while Gal and Grasselli later identified exponential attractors that govern long-term dynamics [27] . When employing the physically realistic Flory–Huggins logarithmic potential, Abels established the theory with a comprehensive global weak/strong solvability result [28] . Two variants arise for fluids with different densities (ρ1 ρ2 ) . In the strictly incompressible context (∇ · u = 0), the frame-indifferent Abels–Garcke–Gr¨un formulation provides global weak solutions for polynomial free energy [2

FAQs

What model is derived in the document, and what energy framework does it use?

The document derives a quasi-incompressible Cahn–Hilliard–Darcy phase-field model (qCHD) for two-phase flow in porous media, using the logarithmic Flory–Huggins free energy density function.

What key property does the qCHD model satisfy?

The qCHD model satisfies an energy-dissipation law.

What well-posedness result is established for the qCHD system?

Using pressure estimates from Darcy’s equations, the document establishes global existence of weak solutions in both 2D and 3D for the qCHD model.

Jul 16, 2026

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