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


Geometry
---------

The domain is a :math:`[\mathrm{0m}\mathrm{,10}m]\times [\mathrm{0m};\mathrm{10m}]` square with a :math:`[\mathrm{1m}\mathrm{,1}m]` hole at the bottom left.


.. image:: images/100000000000028C000001CF9A50FBB65EF1CE54.jpg
    :width: 2.1728in
    :height: 1.5429in


.. _RefImage_100000000000028C000001CF9A50FBB65EF1CE54.jpg:

Figure 1.1-a: Domain representation

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

Only the properties on which the solution depends are given here, knowing that the command file contains other material data that plays no role in solving the problem at hand.


.. csv-table::

    "", "", ""
    "Liquid water", "Density :math:`(\mathit{kg}\mathrm{.}{m}^{\mathrm{-}3})`
    Molar mass :math:`(\mathit{kg}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1})`
    Viscosity :math:`(\mathit{kg}\mathrm{.}{m}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1})` "," :math:`1000`
    :math:`{10}^{-2}`
    :math:`{10}^{-3}`"
    "Gas", "Density :math:`(\mathit{kg}\mathrm{.}{m}^{\mathrm{-}3})`
    Molar mass :math:`(\mathit{kg}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1})`
    Viscosity :math:`(\mathit{kg}\mathrm{.}{m}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1})` "," :math:`8{10}^{-2}`
    :math:`2{10}^{-3}`
    :math:`9{10}^{-5}`"
    "Dissolved gas", "Henry coefficient :math:`(\mathit{Pa}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1}\mathrm{.}{m}^{3})` "," :math:`130719`"
    "Steam", "Density :math:`(\mathit{kg}\mathrm{.}{m}^{\mathrm{-}3})` "," :math:`{10}^{-4}`"
    "Homogenized parameters", "Permeability :math:`k` :math:`({m}^{2})`
    Porosity
    Fick gas :math:`({m}^{2}\mathrm{.}{s}^{\mathrm{-}1})`
    Liquid Fick :math:`({m}^{2}\mathrm{.}{s}^{\mathrm{-}1})` "," :math:`5{10}^{-20}`
    :math:`0.15`
    :math:`0`
    :math:`\mathrm{0,45}{10}^{-9}`"
    "Van-Genuchten Settings", ":math:`N`
    :math:`{P}_{r}` :math:`\mathit{MPa}`
    :math:`{S}_{rl}`
    :math:`{S}_{gr}`
    :math:`\mathrm{Smax}` "," :math:`\mathrm{1,49}`
    :math:`2`
    :math:`0`
    :math:`0`
    :math:`\mathrm{0,999}`"
    "Initial state", "Capillary pressure :math:`(\mathit{Pa})`
    Gas pressure :math:`(\mathit{Pa})` "," :math:`\{\begin{array}{}{P}_{l}^{0}={10}^{-6}\\ {S}_{l}=1\end{array}` :math:`\iff \{\begin{array}{}{P}_{c}^{0}=-{10}^{-6}\\ {P}_{g}^{0}=0\end{array}`"


Table 1.2-1: material properties

The saturation and permeability curves follow the Mualem-Van-Genuchten model (HYDR_VGM). It is therefore necessary to define the :math:`n,\mathrm{Pr},\mathrm{Sr},\mathrm{Smax}` parameters in the materials.

Remember that these models are: :math:`{S}_{le}=\frac{{S}_{l}-{S}_{\mathrm{lr}}}{1-{S}_{\mathrm{lr}}}` and :math:`m\mathrm{=}1\mathrm{-}\frac{1}{n}`

:math:`{S}_{\mathrm{we}}=\frac{1}{{\left[1+{(\frac{{P}_{c}}{{P}_{r}})}^{n}\right]}^{m}}`

Relative permeability to water is expressed by integrating the prediction model proposed by Mualem (1976) into Van Genuchten's capillarity model.

:math:`{k}_{r}^{l}\mathrm{=}\sqrt{{S}_{le}}{(1\mathrm{-}{(1\mathrm{-}{S}_{le}^{\frac{1}{m}})}^{m})}^{2}`

Permeability to gas is formulated in a similar way by a Parker law:

:math:`{k}_{r}^{g}=\sqrt{(1-{S}_{le})}{(1-{S}_{le}^{\frac{1}{m}})}^{2m}`

We recall that for :math:`S>\mathrm{Smax}`, these curves are interpolated by a polynomial of degree 2 :math:`\mathrm{C1}` in :math:`\mathrm{Smax}`.

For modeling E, Parker's law for relative gas permeability is replaced by a cubic law (me 'HYDR_VGC'):

                         :math:`{k}_{r}^{g}={(1-{S}_{l})}^{3}`


Boundary and initial conditions
-----------------------------------

The boundary conditions are as follows:


* Neumann conditions on the right and left of the domain:

:math:`({F}_{l}^{w}+{F}_{g}^{w})\mathrm{.}n\mathrm{=}0`

:math:`({F}_{l}^{c}+{F}_{g}^{c})\mathrm{.}n\mathrm{=}0`


* Neumann conditions in hole :math:`\tau`:


*
    *
        *
            * If :math:`0<t<T` then :math:`({F}_{l}^{w}+{F}_{g}^{w})\mathrm{.}n=0`

If :math:`0<t<T` then :math:`({F}_{l}^{c}+{F}_{g}^{c})\mathrm{.}n=Q`

If :math:`T<t<T` then :math:`({F}_{l}^{c}+{F}_{g}^{c})\mathrm{.}n=0`


* Dirichlet conditions on the upper right part of the domain:


*
    *
        *
            * :math:`{P}_{l}(9\mathrm{\le }x\mathrm{\le }\mathrm{10,}y\mathrm{=}\mathrm{10,}t)\mathrm{=}{10}^{6}\mathit{Pa}`

:math:`{P}_{g}(9\mathrm{\le }x\mathrm{\le }\mathrm{10,}y\mathrm{=}\mathrm{10,}t)\mathrm{=}0\mathit{Pa}`

The initial conditions are as follows:

:math:`{P}_{l}(x,y,t\mathrm{=}0)\mathrm{=}{10}^{6}\mathit{Pa}`

:math:`{P}_{g}(x,y,t\mathrm{=}0)\mathrm{=}0\mathit{Pa}`

The hydrogen flow imposed on the left hand side, :math:`Q`, is equal to:

:math:`Q=\mathrm{0,44}{.10}^{\text{-11}}{\mathrm{kg.m}}^{\text{-2}}\mathrm{.}{s}^{\text{-1}}`

The injection time, :math:`\mathrm{TINJ}` is :math:`{5.10}^{5}` years and the simulation time is :math:`{10}^{6}` years.