Homogenization of Kondaurov’s non-equilibrium two-phase flow in double porosity media

ABSTRACT

We consider a two-phase incompressible non-equilibrium flow in fractured porous media in the framework of Kondaurov’s model, wherein the mobilities and capillary pressure depend both on the real saturation and a non-equilibrium parameter satisfying a kinetic equation. The medium is made of two superimposed continua, a connected fracture system, which is assumed to be thin of order $\epsilon \delta$, where $\delta$ is the relative fracture thickness and an $\epsilon$-periodic system of disjoint cubic matrix blocks. We derive the global behavior of the model by passing to the limit as $\epsilon \to 0$, assuming that the block permeability is proportional to ${(\epsilon \delta )}^2$, while the fracture permeability is of order one, and obtain the global $\delta$–model. In the $\delta$-model we linearize the cell problem in the matrix block and letting $\delta \to 0$, obtain a macroscopic non-equilibrium fully homogenized model, i.e. the model which does not depend on the additional coupling. The numerical tests show that for $\delta$ sufficiently small, the exact global $\delta$-model can be replaced by the fully homogenized one without significant loss of accuracy.

KEYWORDS:

• Double porosity media
• thin fissures
• homogenization
• two-phase flow
• non-equilibrium model

AMS SUBJECT CLASSIFICATIONS:

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

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was funded by the RSCF research [project number 15-11-00015]. The work was done in the Laboratory of Fluid Dynamics and Seismic of Moscow Institute of Physics and Technology (RAEP 5top100).