Poroelasticity equations derived from microstructure

Equations are derived which govern the linear macroscopic mechanical behavior of a porous elastic solid saturated with a compressible viscous fluid. The derivation is based on the equations of linear elasticity in the solid, the linearized Navier–Stokes equations in the fluid, and appropriate conditions at the solid–fluid boundary. The scale of the pores is assumed to be small compared to the macroscopic scale, so that the two-space method of homogenization can be used to deduce the macroscopic equations. When the dimensionless viscosity of the fluid is small, the resulting equations are those of Biot, who obtained them by hypothesizing the form of the macroscopic constitutive relations. The present derivation verifies those relations, and shows how the coefficients in them can be calculated, in principle, from the microstructure. When the dimensionless viscosity is of order one, a different equation is obtained, which is that of a viscoelastic solid.