2. Problem overview#
2.1. Assumptions#
We consider a porous medium, noted \(\Omega\) at the current moment and whose border is noted \(\partial \Omega\). This volume is separated into two parts \({\Omega }^{\text{+}}\) and \({\Omega }^{\text{-}}\) by a \(\Gamma\) interface. All the quantities in \({\Omega }^{\text{+}}\) (resp. \({\Omega }^{\text{-}}\)) are indicated by a + (resp. of a -).
We are limited to the study of a saturated porous medium in two dimensions.
2.2. Ratings#
2.2.1. Mechanical quantities#
The field of movement in the massif and the discontinuity of movement through the crack are noted respectively \(u=(\begin{array}{c}{u}_{x}\\ {u}_{y}\end{array})\) and \(〚u〛=(\begin{array}{c}〚{u}_{x}〛\\ 〚{u}_{y}〛\end{array})\).
The total constraint and the effective constraint are respectively noted \(\sigma\) and \(\sigma \text{'}\) in the massive and \(\tau\) and \(\tau \text{'}\) on the interface.
2.2.2. Hydraulic quantities#
The pore pressure of the intersticial fluid is noted \({p}^{\text{+}}\) and \({p}^{\text{-}}\) in the masses surrounding the crack. The fluid pressure in the crack is noted \(p\). The pressure gradient in the mountains is noted \(\nabla {p}^{\text{+}}\) or \(\nabla {p}^{\text{-}}\). The longitudinal crack pressure gradient is noted \({\nabla }_{l}p\) and is defined by
where \(n\) is the normal interface.
The mass contributions of fluid are noted \(m\) in the massif (unit: \({\mathit{M.L}}^{\mathrm{-}3}\)). In the crack, they are noted \(w\) (unit: \({\mathit{M.L}}^{\mathrm{-}2}\)) and are integrated over the \(\varepsilon\) thickness of the crack:
The mass hydraulic flow in the massif is noted \(M\) (unit: \({\mathit{M.T}}^{\mathrm{-}1}\mathrm{.}{L}^{\mathrm{-}2}\)). In the crack, the hydraulic mass flow is noted \(W\) (unit: \({\mathit{M.T}}^{\mathrm{-}1}\mathrm{.}{L}^{\mathrm{-}1}\)).
\(\varepsilon\) is the crack opening and is therefore connected to the normal displacement jump by
Where \({\varepsilon }_{0}\) is the initial thickness.