Reference problem
=====================


Geometry
---------

The field studied represents a section of land around a storage cell.

D

C

02

O

01

Y

X

B

A

Coordinates of the points (:math:`m`):


.. csv-table::

    ":math:`A` ", "0", "-500", "", "-500", "", ":math:`C` ", "10", "-400"
    ":math:`B` ", "10", "-500", "", "-500", "", ":math:`D` ", "0", "-400"
    ":math:`O` ", "0", "-450", "", "", "", "", ""


Cell radius: :math:`5.6m`

**Note:**

*The alveolus is not scaled on the diagram.*


Material properties
----------------------

Only the properties on which the solution depends are given here. The command file contains other material data (temperatures,...) that play no role in solving the problem at hand.


+----------------------+---------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Liquid water          |Density (                                                                                                      |1000                                                                                                                                                                                                                                                                                                                      |
+                      + :math:`{\mathit{kg.m}}^{\mathrm{-}3}`                                                                         +                                                                                                                                                                                                                                                                                                                          +
|                      |)                                                                                                              | 1                                                                                                                                                                                                                                                                                                                        |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      | Viscosity                                                                                                     |                                                                                                                                                                                                                                                                                                                          |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               |                                                                                                                                                                                                                                                                                                                          |
+----------------------+---------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|Homogenized parameters|Permeability                                                                                                   |:math:`{10}^{\mathrm{-}18}{m}^{\mathrm{-}2}`                                                                                                                                                                                                                                                                              |
+                      + :math:`K`                                                                                                     +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               |                                                                                                                                                                                                                                                                                                                          |
+                      + Sorption isothermal                                                                                           + .. image:: images/Object_95.svg                                                                                                                                                                                                                                                                                          +
|                      | Relative permeability                                                                                         |    :width: 229                                                                                                                                                                                                                                                                                                           |
+                      + Porosity Storage                                                                                              +    :height: 47                                                                                                                                                                                                                                                                                                           +
|                      |                                                                                                               |                                                                                                                                                                                                                                                                                                                          |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               |                                                                                                                                                                                                                                                                                                                          |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               |                                                                                                                                                                                                                                                                                                                          |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               | .. image:: images/Object_96.svg                                                                                                                                                                                                                                                                                          |
+                      +                                                                                                               +    :width: 229                                                                                                                                                                                                                                                                                                           +
|                      |                                                                                                               |    :height: 47                                                                                                                                                                                                                                                                                                           |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               |                                                                                                                                                                                                                                                                                                                          |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               |                                                                                                                                                                                                                                                                                                                          |
+                      +                                                                                                               +                                                                                                                                                                                                                                                                                                                          +
|                      |                                                                                                               | 0.14 :math:`4.{10}^{\mathrm{-}10}{m}^{\mathrm{-}1}`                                                                                                                                                                                                                                                                      |
+----------------------+---------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



Initial conditions
---------------------

The problem has two phases:


* A first phase of 15 years of desaturation corresponding to the operation of the underground structure.


* A second phase of restoration after backfilling the cell corresponding to exploitation (the saturation of the cell is initialized to 0.7).

The initial conditions are as follows:

For phase 1


* Cell :math:`{P}_{c}\mathrm{=}\mathrm{9,4}{.10}^{7}\mathit{Pa}` (:math:`S\mathrm{=}\mathrm{0,49}`)


* Geological barrier :math:`{P}_{c}\mathrm{=}{1.10}^{5}\mathit{Pa}` (:math:`S\mathrm{=}\mathrm{0,999}`)

For phase 2 (:math:`t>15\mathit{ans}`)


* Cell :math:`{P}_{c}\mathrm{=}\mathrm{3,015}{.10}^{7}\mathit{Pa}` (:math:`S\mathrm{=}\mathrm{0,7}`)


Boundary conditions
-----------------------

They are expressed in capillary pressure.

Stage 1:

On :math:`\mathrm{[}\mathit{AB}\mathrm{]}` :math:`{P}_{c}\mathrm{=}{1.10}^{5}\mathit{Pa}`

On :math:`\mathrm{[}\mathit{CB}\mathrm{]}` Zero hydraulic flow

On :math:`\mathrm{[}\mathit{CD}\mathrm{]}` :math:`{P}_{c}\mathrm{=}{1.10}^{5}\mathit{Pa}`

On :math:`\mathrm{[}\mathit{A01}\mathrm{]}\mathrm{\cup }\mathrm{[}\mathrm{02D}\mathrm{]}` Zero hydraulic flow

On the whole cell :math:`{P}_{c}\mathrm{=}\mathrm{9,4}{.10}^{7}\mathit{Pa}` (:math:`S\mathrm{=}\mathrm{0,49}`).

Stage 2:

On :math:`\mathrm{[}\mathit{AB}\mathrm{]}` :math:`{P}_{c}\mathrm{=}{1.10}^{5}\mathit{Pa}`

On :math:`\mathrm{[}\mathit{CB}\mathrm{]}` Zero hydraulic flow

On :math:`\mathrm{[}\mathit{CD}\mathrm{]}` :math:`{P}_{c}\mathrm{=}{1.10}^{5}\mathit{Pa}`

On :math:`\mathrm{[}\mathit{AD}\mathrm{]}` Zero hydraulic flow