Upscaling of an immiscible non-equilibrium two-phase flow in double porosity media

Abstract

We study an immiscible incompressible two-phase, such as water–oil flow through a -periodic double porosity media. The mesoscopic model consists of equations derived from the mass conservation laws of both fluids, along with a generalized Darcy law in the framework of Kondaurov’s non-equilibrium flow model. The mobility functions as well as the capillary pressure function for each component of the porous medium are the functions of the saturation and an additional non-equilibrium parameter, which satisfies a kinetic equation coming from the definition of the Helmholtz free energy. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the permeability ratio of matrix blocks to fracture planes is of order . Using the method of two-scale asymptotic expansions, we derive the macroscopic model of the flow which is written in terms of the homogenized phase pressures, saturation, and the non-equilibrium parameter. For small relaxation times, we compare our model with the global models obtained earlier by H. Salimi and J. Bruining. We show the novelty of our macroscopic double porosity flow model.

Keywords:

• homogenization
• double porosity media
• immiscible
• two-phase flow
• non-equilibrium model

AMS Subject Classifications:

• 35B27
• 35K65
• 35Q35
• 74Q10
• 74Q15

Acknowledgements

This work was done in the Laboratory of Fluid Dynamics and Seismic of Moscow Institute of Physics and Technology.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Russian Scientific Fund [grant number 15-11-00015]. The work of L. Pankratov was partially supported by the Russian Academic Excellence Project [5top100].