1. Introduction#

The formulation of the coupled model HM- XFEM is based on the equations of the joint model [R7.02.15] as well as on those of the behavior model THHM [R7.01.11] in the saturated case. Conventionally, the joint model is used to model in 2D the behavior of a hydraulic seal or a discontinuity in the presence of a fluid flow within it, thus generating fluid pressure. The joint model makes it possible to take into account:

  • the preferential flow of the fluid in the discontinuity conditioned by the opening of the latter,

  • the exchange of fluid between the porous medium and the discontinuity crossing it,

  • the spread of discontinuity,

  • the deformation of the porous medium induced by fluid pressure.

In the context of the classical finite element method, the use of such a model has a major drawback. In fact, it is necessary to explicitly represent the discontinuity in the mesh and to make the lips of the latter agree with the edges of the elements constituting the mesh. This implies that during its evolution, it is necessary to use elaborate projection algorithms to update its geometry. This step can be very costly in terms of calculation time for complex geometries.

In order to overcome this constraint linked to the mesh, we are considering the introduction of a new hydromechanical element (HM) coupled with the extended finite element method (XFEM) [1,2,3]. This method, based on the principle of partitioning the unit [4], ensures greater flexibility for models involving more or less complex geometries. Indeed, with this method, the discontinuity is no longer represented physically in the mesh but symbolically by enriching the approximated fields with additional degrees of freedom. For more details on the extended finite element method in Code_Aster, the user can refer to the documentation [R7.02.12] (mechanical case only).

In the literature, some authors have already considered the coupling of method XFEM with the coupled poromechanical model HM. This is the case of [5] in the case of the dynamic analysis of porous media under unsaturated conditions. The extension to case THM for a saturated medium crossed by an impermeable interface is envisaged in [6]. Other authors such as [7] have proposed a model that can take into account the singularity of the pressure field at the point of discontinuity, by adapting the expression of singular functions for the background of discontinuity. However, the models developed by these authors do not take into account either the exchange phenomena that may exist between the surrounding environment (referred to as massive in the rest of this document) and discontinuity, or the propagation of this last one within the porous medium. This point will be taken into account in the model developed in this documentation by the introduction of cohesive laws regularized into the weak formulation of the mechanical equilibrium equation (see§ 3.1.2).

From a practical point of view, the main difficulty in formulating the element HM- XFEM is the construction of the various approximation spaces for mechanical and hydraulic quantities. Compliance with stability condition LBB is essential in order to obtain a unique and convergent solution [8,9]. Violation of this condition (i.e. not choosing the correct degree of interpolation of the various approximated fields see§ 5.2.1) results in oscillations in the solution. The cohabitation of the HM-XFEM elements and the classical HM elements (those referred to in the documentation [R7.01.11] and [R7.01.10]) is also a delicate point, especially with regard to the distribution of the degrees of freedom at the different nodes (vertices or midpoints) of each type of element (see § 5.2.1).

In the rest of this documentation we will recall the framework for studying the problem, then in a second time the fundamental equations of the poromechanical model involved in the formulation of the HM- XFEM model. Finally we will proceed to discretize the variational forms of the equilibrium equations, both in time (thanks to a \(\theta\) -diagram) and in space (thanks to the XFEM method).

The presentation of the HM- XFEM model and its validation were the subject of a scientific publication [15].