Problem overview ======================== Definition of the field of study ------------------------------ Let :math:`\mathrm{\Omega }\subset {\mathrm{ℝ}}^{d}` with :math:`d\in \{\mathrm{2,}3\}` be a domain that is entirely crossed by a crack or a permeable interface. Let :math:`n` be the external normal at the :math:`\partial \Omega` border of the domain, and :math:`{n}_{c}` that of the interface :math:`{\Gamma }_{c}`. It is possible to break down: * the border of domain :math:`\Omega` in :math:`\partial \Omega ={\Gamma }_{u}\cup {\Gamma }_{t}\cup {\Gamma }_{p}\cup {\Gamma }_{F}` where limit conditions (of the Dirichlet and Neumann type) for hydrodynamics (on :math:`{\Gamma }_{p}` and :math:`{\Gamma }_{F}`) and for mechanics (on :math:`{\Gamma }_{u}` and :math:`{\Gamma }_{t}`) are imposed, * the interface in :math:`{\Gamma }_{c}={\Gamma }_{f}\cup {\Gamma }_{1}\cup {\Gamma }_{2}` where :math:`{\Gamma }_{1}` and :math:`{\Gamma }_{2}` represent the lips of discontinuity. Flow conditions are imposed on :math:`{\Gamma }_{1}`, :math:`{\Gamma }_{2}` and :math:`{\Gamma }_{f}` and cohesive surface forces are imposed on :math:`{\Gamma }_{1}` and :math:`{\Gamma }_{2}`. The data is a symbolic representation of the conditions imposed on the border of the domain. .. image:: images/10000000000002AB000001C74E024DFF31F024C3.jpg :width: 2.872in :height: 1.9028in .. _RefImage_10000000000002AB000001C74E024DFF31F024C3.jpg: Figure 2.1-1: Imposition of boundary conditions Assumptions and notations ------------------------ A porous medium saturated with liquid (generally water) is considered. The coupling law associated in *Code_Aster* is therefore LIQU_SATU (for more details refer to the model instructions THM [:external:ref:`U2.04.05 `]). On the other hand, there is a longitudinal flow of fluid at the interface. Assuming that it is permeable, fluid exchanges take place between the massif (part of :math:`\Omega` that is not the interface) and the interface. They are represented on the. .. image:: images/100000000000019A000000E823063B816F477D5E.jpg :width: 2.6937in :height: 1.5118in .. _RefImage_100000000000019A000000E823063B816F477D5E.jpg: Figure 2.2-1: Orientation of exchanges between the massif and the discontinuity :math:`{q}_{1}` and :math:`{q}_{2}` are the flows due to exchanges between the massif and the interface, and are expressed in :math:`{\mathit{kg.m}}^{\text{-2}}\mathrm{.}{s}^{\text{-1}}`. These flows come from the interface and are directed respectively to the upper and lower parts of the massif at the level of lips :math:`{\Gamma }_{1}` and :math:`{\Gamma }_{2}` of the interface. They are directed from the discontinuity to the massif. The pressure field at the level of the massif is noted :math:`p` and that at the interface level is noted :math:`{p}_{f}` (field induced by the fluid circulating at the interface level). The movement field is noted :math:`u` and the movement jump at the interface level is noted :math:`⟦u⟧`. Let :math:`{P}_{1}` be a point of :math:`{\Gamma }_{1}` and :math:`{P}_{2}` a point of :math:`{\Gamma }_{2}` and :math:`{n}_{c}={n}_{c}^{1}` be the external normal to :math:`{\Gamma }_{1}` and :math:`{n}_{c}^{2}` the external normal to :math:`{\Gamma }_{2}`. The normal displacement jump (taken to be negative or zero when opened and positive when the lips interpenetrate) is therefore defined as follows: :math:`⟦u⟧\mathrm{.}{n}_{c}=(u({P}_{1})-u({P}_{2}))\mathrm{.}{n}_{c}⩽0` On the, we indicate the conventions adopted to take account of the displacement jump at the interface level. .. image:: images/100000000000019E000000CFE02A182774D6F506.jpg :width: 2.8335in :height: 1.4165in .. _RefImage_100000000000019E000000CFE02A182774D6F506.jpg: Figure 2.2-2: Definition of the displacement jump at the level of discontinuity For the massif, as for the interface, the hypothesis of effective constraints is taken into consideration. So: * the total stress in the massif is noted :math:`\sigma`, * the total constraint (linked to cohesive efforts) at the interface level is noted :math:`{t}_{c}`. The hypothesis of small disturbances is accepted. On the other hand, mechanical and hydraulic quantities are considered to be isotropic.