Problem overview
========================




Assumptions
----------

We consider a porous medium, noted :math:`\Omega` at the current moment and whose border is noted :math:`\partial \Omega`. This volume is separated into two parts :math:`{\Omega }^{\text{+}}` and :math:`{\Omega }^{\text{-}}` by a :math:`\Gamma` interface. All the quantities in :math:`{\Omega }^{\text{+}}` (resp. :math:`{\Omega }^{\text{-}}`) are indicated by a + (resp. of a -).

We are limited to the study of a saturated porous medium in two dimensions.




Ratings
---------




Mechanical quantities
~~~~~~~~~~~~~~~~~~~~~~

The field of movement in the massif and the discontinuity of movement through the crack are noted respectively :math:`u=(\begin{array}{c}{u}_{x}\\ {u}_{y}\end{array})` and :math:`〚u〛=(\begin{array}{c}〚{u}_{x}〛\\ 〚{u}_{y}〛\end{array})`.

The total constraint and the effective constraint are respectively noted :math:`\sigma` and :math:`\sigma \text{'}` in the massive and :math:`\tau` and :math:`\tau \text{'}` on the interface.




Hydraulic quantities
~~~~~~~~~~~~~~~~~~~~~~~~

The pore pressure of the intersticial fluid is noted :math:`{p}^{\text{+}}` and :math:`{p}^{\text{-}}` in the masses surrounding the crack. The fluid pressure in the crack is noted :math:`p`. The pressure gradient in the mountains is noted :math:`\nabla {p}^{\text{+}}` or :math:`\nabla {p}^{\text{-}}`. The longitudinal crack pressure gradient is noted :math:`{\nabla }_{l}p` and is defined by


.. math::
 :label: eq-2

    {\nabla} _ {l} p=\nabla p-\ frac {\ partial p} {\ partial n} n

where :math:`n` is the normal interface.

The mass contributions of fluid are noted :math:`m` in the massif (unit: :math:`{\mathit{M.L}}^{\mathrm{-}3}`). In the crack, they are noted :math:`w` (unit: :math:`{\mathit{M.L}}^{\mathrm{-}2}`) and are integrated over the :math:`\varepsilon` thickness of the crack:

The mass hydraulic flow in the massif is noted :math:`M` (unit: :math:`{\mathit{M.T}}^{\mathrm{-}1}\mathrm{.}{L}^{\mathrm{-}2}`). In the crack, the hydraulic mass flow is noted :math:`W` (unit: :math:`{\mathit{M.T}}^{\mathrm{-}1}\mathrm{.}{L}^{\mathrm{-}1}`).

:math:`\varepsilon` is the crack opening and is therefore connected to the normal displacement jump by


.. math::
 :label: eq-2

    \ varepsilon = {\ varepsilon} _ {0} + u\ mathrm {.} n

Where :math:`{\varepsilon }_{0}` is the initial thickness.