Phase-field modeling of fracture in partially saturated porous media

Porous media such as soil, rocks and concrete are of great importance in the context of civil engineering and environmental geomechanics. They consist of a solid skeleton and pores \ufb01lled with \ufb02uids, e.g. air and water. Complex mechanisms of \ufb02ow and transport take place within the pore network and can lead to deformation of the solid skeleton and eventually to fracture phenomena [1]. Phase-\ufb01eld modeling of fracture has recently emerged as an alternative to conventional approaches such as remeshing, extended \ufb01nite element methods or cohesive zone modeling. The phase-\ufb01eld framework can be considered a special type of gradient damage modeling approach, where a di\ufb00usive approximation of the crack is taken into account and the continuous phase-\ufb01eld parameter is used to describe the material integrity. The essential advantages are the possibility to describe arbitrarily complicated fracture patterns such as nucleation, branching and merging, without ad-hoc criteria on a \ufb01xed mesh, through the solution of partial di\ufb00erential equations derived from variational principles [2, 3, 4, 5]. Phase-\ufb01eld modeling of fracture in porous media has been addressed in some recent publications [6, 7], which however have only focused on the fully saturated case. Objective of this contribution is to describe fracture in partially saturated porous media using a phase-\ufb01eld approach. In this study, the material is described by its linear-elastic properties. The overall balance of linear momentum, the continuity equation and the phase-\ufb01eld evolution equation constitute a nonlinear coupled and time-dependent system of equations, which needs to be discretized and linearized. We formulate the coupled non-linear system of partial di\ufb00erential equations governing the problem with displacements, capillary pressure and crack phase-\ufb01eld as unknowns. The spatial discretization is carried out with \ufb01nite elements of appropriate order for the di\ufb00erent unknowns. We discuss its solution and present some relevant examples