Benchmark solution
=====================


Calculation method
-----------------

It is an analytical solution. The fluid pressure :math:`{p}_{f}` imposed in the cohesive interface is transmitted to the pore pressure at the level of the lips of the cohesive interface (cf Documentation [:ref:`R7.02.18 <R7.02.18>`]). Taking into account the boundary conditions, everything happens as if a pore pressure step were imposed at the end of the two sub-columns formed by the cohesive interface. So the problem is unidirectional. These two sub-columns are frozen because the movements are blocked. By neglecting gravity, the equation for the conservation of mass in the porous matrix is written as follows:

:math:`\frac{\partial {m}_{w}}{\partial t}+\nabla \cdot M=0`

with :math:`{m}_{w}=\mathrm{\rho }\mathrm{\varphi }(1+{\mathrm{ϵ}}_{v})` the mass inputs into the porous matrix and :math:`M=\frac{-\mathrm{\rho }{K}^{\text{int}}}{{\mathrm{\mu }}_{w}}\nabla p` the flow of Darceen fluid into the porous matrix. Since the movements are blocked, the volume deformation :math:`{\mathrm{ϵ}}_{v}` is zero. Moreover, :math:`d\mathrm{\varphi }=(b-\mathrm{\varphi })(d{\mathrm{ϵ}}_{v}+\frac{\mathit{dp}}{{K}_{s}})` with :math:`{K}_{s}` the compressibility modulus of the solid matrix. However, the solid matrix is supposed to be incompressible because we have taken :math:`b=1` (and :math:`b=1-\frac{{K}_{m}}{{K}_{s}}` with :math:`{K}_{S}` the compressibility module of the porous medium). So finally :math:`d\mathrm{\varphi }=0`. Finally :math:`\frac{d\mathrm{\rho }}{\mathrm{\rho }}=\frac{\mathit{dp}}{{K}_{w}}`. The mass conservation equation can therefore be rewritten:

:math:`\frac{\mathrm{\rho }\mathrm{\varphi }}{{K}_{w}}\frac{\partial p}{\partial t}-\frac{{K}^{\text{int}}}{{\mathrm{\mu }}_{w}}[\nabla p\cdot \nabla \mathrm{\rho }+\mathrm{\rho }\mathrm{\Delta }p]=0`

Overlooking the second-order term :math:`\nabla p\cdot \nabla \mathrm{\rho }`, it comes from:

:math:`\frac{\mathrm{\varphi }{\mathrm{\mu }}_{w}}{{K}_{w}{K}^{\text{int}}}\frac{\partial p}{\partial t}=\mathrm{\Delta }p`

It is a diffusion equation. The coefficient :math:`\frac{\mathrm{\varphi }{\mathrm{\mu }}_{w}}{{K}_{w}{K}^{\text{int}}}` has the dimension of time over a length squared. In our problem, the characteristic dimension for broadcasting is :math:`\frac{\mathit{LZ}}{2}`. A characteristic diffusion time :math:`\mathrm{\tau }=\frac{\mathrm{\varphi }{\mathrm{\mu }}_{w}}{{K}_{w}{K}^{\text{int}}}\ast {(\frac{\mathit{LZ}}{2})}^{2}` is deduced from this. This characteristic time is worth :math:`\mathrm{\tau }=\mathrm{0,459842}s`. At the end of :math:`t=10s`, the diffusion is therefore complete, the pore pressure in the porous matrix then verifies :math:`\mathrm{\Delta }p=0`.

According to the boundary conditions, below the interface:


* the pore pressure is :math:`p(y)=\frac{2y}{\mathit{LZ}}{p}_{f}` and the fluid flow in the matrix is :math:`M=\frac{-2\mathrm{\rho }{K}^{\text{int}}{p}_{f}}{\mathit{LZ}{\mathrm{\mu }}_{w}}`.

And at the top of the interface:


* the pore pressure is :math:`p(y)=(1-\frac{y}{\mathit{LZ}})2{p}_{f}` and the fluid flow in the matrix is :math:`M=\frac{2\mathrm{\rho }{K}^{\text{int}}{p}_{f}}{\mathit{LZ}{\mathrm{\mu }}_{w}}`.


Reference quantities and results
-----------------------------------

The value of the outflows from the cohesive interface to the lower and upper parts of the porous column as well as the value of the pore pressure in :math:`z=4m` and :math:`z=6m` are tested.


.. csv-table::

    "Tested sizes", "Value"
    "LAG_FLI (Flow leaving the interface to the bottom of the column)", "4.07748 kg·m²/s"
    "LAG_FLS (Flow leaving the interface to the top of the column)", "4.07748 kg·m²/s"
    "PRE1 (at Z=4m)", "8Mpa"
    "PRE1 (at Z=6m)", "8Mpa"



Uncertainties about the solution
----------------------------

The reference solution is analytical.


Bibliographical references
---------------------------


1. Reference documentation R7.02.18 (Hydromechanical elements coupled with the Extended Finite Element Method).