Homogenization of Two-Phase Flow in Porous Media From Pore to Darcy Scale: A Phase-Field Approach


It is well known that the generalized Darcy law describing multiphase flow in porous media has some shortcomings. In particular, it cannot explain hysteresis effects in the capillary pressure--saturation curve which have been observed in measurements. In this work, we derive a numerically tractable micro-macro model including coupled generalized Darcy's laws that still includes the microscale dynamics which are responsible, e.g., for hysteresis effects. For this purpose, we extend the two-scale expansion approach of periodic homogenization to include different time scales which allows us to start from a fully instationary Navier--Stokes--Cahn--Hilliard model at the pore scale as microscale. Identifying and separating the time scales allows us to derive local fast scale equations describing the microscale dynamics and global slow-scale equations giving rise to the macroscopic Darcy law.


  1. homogenization
  2. time scales
  3. two-phase flow
  4. porous media
  5. Navier--Stokes
  6. phase-field

MSC codes

  1. 35B27
  2. 76S05
  3. 76D05
  4. 65L60
  5. 35G20
  6. 35Q35

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Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 320 - 343
ISSN (online): 1540-3467


Submitted: 18 September 2019
Accepted: 16 October 2020
Published online: 16 February 2021


  1. homogenization
  2. time scales
  3. two-phase flow
  4. porous media
  5. Navier--Stokes
  6. phase-field

MSC codes

  1. 35B27
  2. 76S05
  3. 76D05
  4. 65L60
  5. 35G20
  6. 35Q35



Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS 1759536

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