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Updated: May 24, 2026

The Diffusion of Passive Tracers in Laminar Shear Flow
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Generalized nonequilibrium capillary relations for two-phase flow through heterogeneous media.

Brahim Amaziane1, Josipa Pina Milišić, Mikhail Panfilov

  • 1Laboratoire de Mathématiques et leurs Applications, CNRS-UMR 5142, Université de Pau, France. brahim.amaziane@univ-pau.fr

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 10, 2012
PubMed
Summary

Natural medium heterogeneity causes capillary nonequilibrium effects in two-phase flow. This study introduces macroscopic flow models with long-term memory, offering exact relationships for capillary pressure and bounds for model analysis.

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Area of Science:

  • Multiphase flow
  • Porous media physics
  • Mathematical modeling

Background:

  • Heterogeneity in porous media leads to capillary nonequilibrium effects.
  • Relaxation to equilibrium is slow and requires inclusion in macroscopic flow models.
  • Existing models often use phenomenological approaches, but rigorous mathematical methods exist.

Purpose of the Study:

  • To develop macroscopic flow models accounting for long-term memory effects in porous media.
  • To derive exact relationships for nonequilibrium capillary pressure under conditions of slow relaxation.
  • To establish bounds for the macroscopic model, providing a more general framework than previous work.

Main Methods:

  • Utilizing a rigorous mathematical formalism based on homogenization of microscale flow equations.
  • Analyzing the case of long relaxation times, leading to nonlocality in time.
  • Deriving exact relationships for capillary pressure and proving a comparison theorem for model bounds.

Main Results:

  • An exact relationship for nonequilibrium capillary pressure in two-phase flow with long-term memory was derived in two independent forms.
  • The derived relationship is more general than those previously obtained.
  • A comparison theorem was proven, establishing upper and lower bounds for the macroscopic model, which are completely homogenized linear flow models.

Conclusions:

  • The study provides a more general and rigorous macroscopic model for two-phase flow in heterogeneous porous media with long relaxation times.
  • The derived capillary pressure relationship and model bounds are valuable for applications requiring accurate simulation of nonequilibrium effects.
  • Numerical simulations validate the theoretical findings and demonstrate the utility of the developed model.