Reference solution
=====================

The reference solution is one-dimensional because it only depends on the vertical coordinate (1D loading). The water-saturated system comes down to solving the problem of mass conservation:

     :math:`\frac{\partial (\phi {\rho }_{l})}{\partial t}-\text{div}({K}_{\text{int}}\frac{{\rho }_{l}}{{\mu }_{l}}\nabla {P}_{l})=q`

where q is the flow imposed on the edge.


* The liquid is incompressible: :math:`{\rho }_{l}=c`


* The matrix is incompressible (infinite stiffness): the porosity therefore remains constant :math:`\phi =\mathit{cst}`.

In this case, we therefore obtain a stationary flow which can be summed up in writing the flow:

:math:`-\text{div}({K}_{\text{int}}\frac{{\rho }_{l}}{{\mu }_{l}}\nabla {P}_{l})=q`

If we write :math:`{P}_{L}` the pressure in :math:`Y=L`, we therefore obtain after discretization:

     :math:`{K}_{\text{int}}\frac{{\rho }_{l}}{{\mu }_{l}}\frac{{P}_{L}-{P}_{0}}{L}=q`

Which works with the data entered previously: :math:`{P}_{L}=1.1{10}^{6}\mathit{Pa}`