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

Chunhua Zhang, Peiyao Liu, Cheng Peng, Lian-Ping Wang, Zhaoli Guo

TL;DR

The paper develops a macroscopic model for two immiscible incompressible fluids in inhomogeneous porous media by upscaling the pore-scale Navier–Stokes–Cahn–Hilliard equations via volume averaging. It derives closed Darcy-scale equations that embed wetting through the averaged chemical potential, and it introduces a lattice Boltzmann method to solve the resulting equations at the representative elementary volume scale. Closure is achieved for both the phase-field and momentum equations through structured ansatzes, linking microstructure to macro-scale coefficients and recovering Darcy- and Brinkman-type limits. Numerical tests on viscous fingering and bubble rise demonstrate the influence of wetting and capillary forces on interface dynamics, while highlighting the need to calibrate certain closure parameters for realistic flows.

Abstract

In this work, we present a macroscopic model for the flow of two immiscible and incompressible fluids in inhomogeneous porous medium. At the pore scale, the flow is governed by the fully Navier-Stokes equations while the evolution of the phase interface is captured by the Cahn-Hilliard equation. Using the volume averaging method, the upscaled equations describing the averaged behavior of two fluids at the Darcy scale are obtained, with unclosed terms related to spatial deviations. Then, spatial derivations are carefully modeled up to some undetermined coefficients, which could be evaluated by solving simplified closure problems in each representative volume element. In particular, the wetting behavior is incorporated into the averaged chemical potential. The differences between the proposed equations and the empirical two-phase Darcy-type models are discussed. Finally, a phase-field-based lattice Boltzmann model for the averaged equations is presented, and numerical results demonstrate the abilities of the proposed model.

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

TL;DR

The paper develops a macroscopic model for two immiscible incompressible fluids in inhomogeneous porous media by upscaling the pore-scale Navier–Stokes–Cahn–Hilliard equations via volume averaging. It derives closed Darcy-scale equations that embed wetting through the averaged chemical potential, and it introduces a lattice Boltzmann method to solve the resulting equations at the representative elementary volume scale. Closure is achieved for both the phase-field and momentum equations through structured ansatzes, linking microstructure to macro-scale coefficients and recovering Darcy- and Brinkman-type limits. Numerical tests on viscous fingering and bubble rise demonstrate the influence of wetting and capillary forces on interface dynamics, while highlighting the need to calibrate certain closure parameters for realistic flows.

Abstract

In this work, we present a macroscopic model for the flow of two immiscible and incompressible fluids in inhomogeneous porous medium. At the pore scale, the flow is governed by the fully Navier-Stokes equations while the evolution of the phase interface is captured by the Cahn-Hilliard equation. Using the volume averaging method, the upscaled equations describing the averaged behavior of two fluids at the Darcy scale are obtained, with unclosed terms related to spatial deviations. Then, spatial derivations are carefully modeled up to some undetermined coefficients, which could be evaluated by solving simplified closure problems in each representative volume element. In particular, the wetting behavior is incorporated into the averaged chemical potential. The differences between the proposed equations and the empirical two-phase Darcy-type models are discussed. Finally, a phase-field-based lattice Boltzmann model for the averaged equations is presented, and numerical results demonstrate the abilities of the proposed model.

Paper Structure

This paper contains 18 sections, 93 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Pore-scale equations
  3. Derivation of the upscaled equations
  4. Overview of the volume average method
  5. Volume averaging of Cahn-Hilliard equation
  6. Volume averaging of the continuity equation
  7. Volume averaging of the momentum equation
  8. Closure of the two-phase Darcy scale equations
  9. Closure of the averaged Cahn-Hilliard equation
  10. Closure of the averaged momentum equation
  11. Equilibrium profile of the averaged order parameter for a single capillary tube
  12. Lattice Boltzmann method
  13. Numerical results
  14. Viscous fingering in porous media
  15. bubble rising in a porous media
  16. ...and 3 more sections

Figures (7)

  • Figure 1: Schematic representation of the hierarchy of length scales of a model porous medium and of a typical representative volume
  • Figure 2: Schematic of a circular capillary tube.
  • Figure 3: Viscous fingering with different contact angles and $\eta_2/\eta_1=16.7$. From left to right, $t\sqrt{g}/\sqrt{L_x}=0, 100, 200,300, 400.$
  • Figure 4: Viscous fingering with different contact angles and $\eta_2/\eta_1=41.7$. From left to right, $t\sqrt{g/L_x}=0, 100, 200,300, 400$.
  • Figure 5: The bubble rising velocity with time under different contact angles. From left to right, $t\sqrt{g/L_x}=0,358, 716, 1073$.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6