Analysis of the consolidation problem of compressible porous media by a macroscopic variational continuum approach

Abstract

This study presents an analysis of the stress-partitioning mechanism for fluid saturated poroelastic media in the transition from drained (e.g. slow deformations) to undrained (e.g. fast deformation) flow conditions. The goal of this analysis is to derive fundamental solutions for the general consolidation problem and to show how Terzaghi’s law is recovered as the limit undrained flow condition is approached. The approach is based on a variational macroscopic theory of porous media (VMTPM). First, the linearized form of VMTPM is expressed in a u–p dimensionless form. Subsequently, the behavior of the poroelastic system is investigated as a function of governing dimensionless numbers for the case of a displacement controlled compression test. The analysis carried out in this study produced two crucial results. First, in the limit of undrained flow, it confirmed that the solutions of the consolidation problem recover Terzaghi’s law. Second, it was found that a dimensionless parameter ( $P_I$ ), which solely depends on mixture porosity and Poisson ratio of the solid phase, discriminates two diametrically opposed mechanic responses of the poroelastic system. More specifically, when $P_I$ is positive, the solid stress in the mixture first increases and then relaxes to an equilibrium value (stress relaxation). In contrast, when $P_I<0$ , the solid stress monotonically tenses up to reach the equilibrium value (stress tension).