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


Geometry
---------

We place ourselves in the framework of 2D modeling with the hypothesis of plane deformations.

The geometry considered corresponds to a rectangular sample of 20x5 cm (the Y dimension does not matter because of the one-dimensional nature of the problem).

A

B


.. image:: images/Shape2.gif


.. _RefSchema_Shape2.gif:

D

C

Point coordinates (:math:`m`):

A (0; 0.05)

B (0.2; 0.05)

C (0.2; 0)

D (0; 0)


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


Only the properties on which the solution depends are given here, knowing that the command file contains other material data that ultimately does not play a role in the solution of the problem. The modeling is purely thermo-hydraulic (there is no role for mechanics).
---------------------------------------------


The sorption isotherms are explained after this table as is the relative permeability to the liquid.
---------------------


.. csv-table::

    "Initial reference state", "Porosity
    Temperature T0 (:math:`K`)
    Vapor pressure (:math:`\mathit{Pa}`)", "0.15
    293.
    1000"
    "Liquid water", "Density (:math:`{\mathrm{kg.m}}^{-3}`)
    Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`)
    Compressibility (:math:`{\mathrm{Pa}}^{-1}`)
    Specific heat (:math:`J\mathrm{.}{K}^{-1}`)", "1000
    10-3
    0.5 10-9
    4180"
    "Gas", "Density (:math:`{\mathrm{kg.m}}^{-3}`)
    Viscosity (:math:`{\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}`)
    Specific heat (:math:`J\mathrm{.}{K}^{-1}`)", "2 10-3
    910-6
    1017"
    "Dissolved gas", "Henry's coefficient (:math:`\mathrm{Pa.}{\mathrm{mol}}^{-1}{m}^{3}`)", "130719"
    "Steam", "Density (:math:`{\mathrm{kg.m}}^{-3}`)
    Specific heat (:math:`J\mathrm{.}{K}^{-1}`)", "18 10-3
    1900"
    "Homogenized coefficients", "Homogenized density (:math:`{\mathit{kg.m}}^{\mathrm{-}3}`)
    Intrinsic permeability :math:`({m}^{2})`
    Relative gas permeability
    Thermal conductivity (:math:`W\mathrm{.}{K}^{-1.}{m}^{-1}`)
    Specific heat (:math:`J\mathrm{.}{K}^{-1}`)
    Diffusion in liquid mixture :math:`({m}^{2}\mathrm{.}{s}^{-1})`
    Diffusion in the gas mixture :math:`({m}^{2}\mathrm{.}{s}^{-1})` ", "1737
    :math:`{K}_{\text{int}}=1.2{10}^{-20}`
    1
    :math:`{\mathrm{\lambda }}_{\text{T}}=0.09S+1.34`
    1500
    :math:`{D}_{l}(S)=S\ast {10}^{-10}`
    :math:`{D}_{g}(S,T)`
    (see below)"


The sorption curve relating capillary pressure to saturation is given by the following relationship:

:math:`\mathit{Pc}(S,T)=\frac{-{\mathrm{\rho }}_{l}({T}_{0})\mathrm{.}R\mathrm{.}{T}_{0}}{\mathrm{\alpha }{M}_{\mathit{vap}}}{({S}^{-1/\mathrm{\beta }}-1)}^{1-\mathrm{\beta }}\frac{{\mathrm{\gamma }}_{\mathit{lv}}({T}_{0})}{{\mathrm{\gamma }}_{\mathit{lv}}(T)}\mathrm{.}\sqrt{\frac{m({T}_{0})}{m(T)}}`

where :math:`{T}_{0}` is the reference temperature (20° C.) where the parameters :math:`\mathrm{\alpha }` and :math:`\mathrm{\beta }` are evaluated, R is the ideal gas constant, :math:`{M}_{\mathit{vap}}` is the molar mass of steam, and:

:math:`\frac{m(T)}{m({T}_{0})}={10}^{{A}_{d}[{2.10}^{-3}(T-{T}_{0})-{10}^{-6}{(T-{T}_{0})}^{2}]}` and :math:`{\mathrm{\gamma }}_{\mathit{lv}}(T)=0.1558{(1-\frac{T}{647.1})}^{1.26}`

Moreover, the relative permeabilities are given by a Van-Genuchten relationship, such as:

:math:`{k}_{\mathit{rl}}(S)={S}^{p}{[1-{(1-{S}^{1/\mathrm{\beta }})}^{\mathrm{\beta }}]}^{2}`

and

:math:`{k}_{\mathit{rg}}(S)={(1-S)}^{p}{[1-{S}^{1/\mathrm{\beta }}]}^{2.\mathrm{\beta }}`

The constants used are [:ref:`3 <RefNumPara__1561_2056665313>`]: :math:`p=2.91` :math:`\mathrm{\beta }=0.389`, :math:`\mathrm{\alpha }=9.334` and :math:`{A}_{d}=10.378`


.. image:: images/10000000000001DA000001201944C7EC775ACEB8.png
    :width: 4.4681in
    :height: 2.7118in


.. _RefImage_10000000000001DA000001201944C7EC775ACEB8.png:

Figure 1: S (Pc) at 20°C and 60°C

Finally, the diffusion in the gas mixture [:ref:`3 <RefNumPara__1561_2056665313>`] is such that :math:`{D}_{g}(S,T)={D}_{0}(T){\mathrm{\varphi }}^{a}{(1-S)}^{b}` with a and b the Milington coefficients such as a = 2.607 and b = 7 and :math:`{D}_{0}(T)=0.217\mathrm{.}{10}^{-4}{(\frac{T}{273})}^{1.88}`.


Boundary conditions and loads
-------------------------------------

On the left edge of the sample (edge AD), a capillary pressure is applied which corresponds to a fixed relative humidity (using Kelvin's law) as well as a temperature:


* HR = 70% (Pc = 48.2 MPa)


* T = 60°C


* Pg = 0.5 MPa (or 0.4 MPa compared to the initial state)


* Zero flow is applied everywhere else


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

Initially, we uniformly consider a saturation S = 0.98, which corresponds to :math:`{P}_{c}(x)={P}_{c0}=2.4{10}^{6}\mathit{Pa}`, a gas pressure :math:`{P}_{g}(x)={P}_{g0}={10}^{5}\mathit{Pa}`

and a temperature :math:`T(x)={T}_{0}=20°C`.