Abstract
We study unsaturated poroelasticity,i.e., coupled hydro-mechanical processes in variably saturated porous media, here modeled by a non-linear extension of Biot’s well-known quasi-static consolidation model. The coupled elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media, obtained under similar assumptions as usually considered for Richards’ equation. In this work, existence of weak solutions is established in several steps involving a numerical approximation of the problem using a physically-motivated regularization and a finite element/finite volume discretization. Eventually, solvability of the original problem is proved by a combination of the Rothe and Galerkin methods, and further compactness arguments. This approach in particular provides the convergence of the numerical discretization to a regularized model for unsaturated poroelasticity. The final existence result holds under non-degeneracy conditions and natural continuity properties for the constitutive relations. The assumptions are demonstrated to be reasonable in view of geotechnical applications.
Highlights
Coupled hydro-mechanical processes in porous media are occurring in various applications of societal relevance within, e.g., geotechnical, structural, and biomechanical engineering
In the field of porous media, such microscopically complex processes are typically modeled by a continuum mechanics approach [17]
The main result of this work is the existence of a weak solution for the unsaturated poroelasticity model under the Kirchhoff transformation, cf., Section 2.2
Summary
Coupled hydro-mechanical processes in porous media are occurring in various applications of societal relevance within, e.g., geotechnical, structural, and biomechanical engineering. We consider a non-linear coupled system of partial differential equations, modeling the quasistatic consolidation of variably saturated porous media – called unsaturated poroelasticity. The model is in particular relevant in soil mechanics It can be obtained by simplifying the more general model for two-phase flow in deformable porous media, founded on macroscopic momentum and mass balances combined with constitutive relations [23] – it is assumed that one fluid phase can be neglected. The existence of weak solutions is established under two strict model assumptions: (i) the coupling term in the fluid flow equation is linear; and (ii) after introducing a new pressure variable by applying the Kirchhoff transformation, the coupling and the diffusion terms in the mass balance equation simultaneously become linear. In Appendix B, technical results from the literature used in the proof of the main result are recalled for a comprehensive presentation
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