A Linearized Porous Media Model Revisited—On Its Derivation via Variationally Consistent Homogenization and Upscaling

ABSTRACT

The classical theory of porous media is based on assumed micro‐homogeneity of the solid phase (skeleton). Based on the drained macroscale elastic stiffness, it is possible to analytically compute auxiliary properties such as the Biot‐Willis and Skempton parameters. The more general case of a micro‐heterogeneous solid phase, which is addressed in this paper, requires computational homogenization of the fluid‐solid interaction. Linearized kinematics is adopted, which is a necessary (but not sufficient) assumption for direct upscaling. Linear elasticity with arbitrary anisotropy is then assigned to the solid skeleton, while the fluid motion is modeled as compressible Stokes' flow. The RVE problem is established based on assumed periodicity for the fluctuation parts of solid displacement, pore pressure and fluid velocity. Although the macroscale problem is transient (quasi‐static), we approximate the RVE‐problem as (micro‐)stationary.

The macroscale problem reveals the classical structure with displacements and pore pressure as the unknown fields.

It is then possible to rigorously show the particular symmetry that the same Biot‐Willis coefficient appears in the “effective stress principle” as well as in the fluid storage function, thereby generalizing the classical model feature.

In the numerical examples, we first verify the classical results for a micro‐isotropic and micro‐homogeneous solid phase. We thereby consider a simplified periodic micro‐architecture, which lends itself to an expected orthotropic macroscale response. Next, we consider a micro‐heterogeneous solid skeleton with inclusions, in which case the classical analysis does not apply.