Flow and deformation in poroelastic media with moderate load and weak inertia

Abstract

The governing equations for the deformation and the fluid flow in a poroelastic medium are derived under (i) a moderately small ratio of external load to elastic stiffness which is of O(ε), where ε = l/l′ is the ratio of the microscale to the macroscale lengths, and (ii) a moderately small pore- (or grain-) size-based Reynolds number which is of O(ε1/2), by employing the theory of homogenization. The macroscale governing equations are nonlinear in general and several canonical boundary–value problems are defined in a unit cell with a periodic structure on the microscale. However, to determine the effects of inertia and deformation on the macroscale seepage flow, only a few cell problems need to be solved. The correction to Darcy's law in deformable media is shown to be cubic in the seepage velocity as in the case of a rigid medium but with modified permeability and inertial effect due to deformation. It is also shown that Darcy's law is stated with permeability which is dependent on the state of elastic strain of the medium throughout the consolidation process. While the changes due to fluid inertia and medium deformation are small but as important as the inertial effects in rigid media, the porosity and permeability changes for a medium isotropic on the macroscale are shown to be linearly proportional to the volume strain of the medium, and also modify the inertial effect in Darcy's law accordingly, as compared with the case for a rigid medium.