A fully homogenized model for incompressible two-phase flow in double porosity media

Abstract

In this paper, we discuss a model describing the global behavior of the two-phase incompressible flow in fractured porous media. The fractured medium is regarded as a porous medium consisting of two superimposed continua, a connected fracture system, which is assumed to be thin of order , where being the relative fracture thickness, and an –periodic system of disjoint matrix blocks. We derive the global behavior of the fractured medium by passing to the limit as , taking into account that the permeability of the blocks is proportional to , while the permeability of the fractures is of order one and obtain the corresponding global –model, i.e. the homogenized model with the coefficients depending on the small parameter . In the –model, we linearize the cell problem in the matrix block and then by letting , we obtain the macroscopic model which does not depend on and , and is fully homogenized in the sense that all the coefficients are calculated in terms of given data and do not depend on the additional coupling or cell problems.

Keywords:

• homogenization
• incompressible
• two-phase flow
• double porosity media
• thin fractures

AMS Subject Classifications:

• 35B27
• 35K65
• 35Q35
• 74Q15
• 76M50
• 76S05

Acknowledgements

Most of the work on this paper was done when Leonid Pankratov was visiting Faculty of Science, University of Zagreb. We thank Faculty of Science for hospitality.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by University of Zagreb [grant number 202704]. The work ofL. Pankratov was also partially supported by the Russian Academic Excellence Project “5top100”.