A two-pressure model for slightly compressible single phase flow in bi-structured porous media

• Upscaling of slightly compressible single phase flow in bi-structured porous media.
• The resulting macroscopic system is a two-pressure equations.
• All the effective coefficients are entirely determined by three closure problems.
• Comparison with pore-scale direct numerical simulations for a particle filter.

Problems involving flow in porous media are ubiquitous in many natural and engineered systems. Mathematical modeling of such systems often relies on homogenization of pore-scale equations and macroscale continuum descriptions. For single phase flow, Stokes equations at the pore-scale are generally approximated by Darcy's law at a larger scale. In this work, we develop an alternative model to Darcy's law that can be used to describe slightly compressible single phase flow within bi-structured porous media. We use the method of volume averaging to upscale mass and momentum balance equations with the fluid phase split into two fictitious domains. The resulting macroscale model combines two coupled equations for average pressures with regional Darcy's laws for velocities. Contrary to classical dual-media models, the averaging process is applied directly to Stokes problem and not to Darcy's laws. In these equations, effective parameters are expressed via integrals of mapping variables that solve boundary value problems over a representative unit cell. Finally, we illustrate the behavior of these equations for model porous media and validate our approach by comparing solutions of the homogenized equations with computations of the exact microscale problem.