1. Reference problem

1.1. Geometry

C

D

B

A

Point coordinates (\(m\)):

\(A\)	\(0\)	\(0\)		\(C\)	\(1\)	\(\mathrm{0,5}\)
\(B\)	\(1\)	\(0\)		\(D\)	\(0\)	\(\mathrm{0,5}\)

1.2. Material properties

We only give here the properties on which the solution depends, knowing that the command file contains other material data (elasticity modules, thermal conductivity, etc.) which ultimately play no role in the solution of the problem being treated.

Liquid water	Density ( \({\mathrm{kg.m}}^{-3}\) ) Specific heat at constant pressure (\({\mathrm{J.K}}^{-1}\)) Dynamic viscosity of liquid water (\(\mathrm{Pa.s}\)) coefficient of thermal expansion of liquid (\({K}^{-1}\)) Relative permeability to water	\({10}^{3}\) \(0\) \(0.001\) \(0\) \({\mathrm{kr}}_{w}(S)=0.5\)
Vapeur	Specific heat ( \({\mathrm{J.K}}^{-1}\) ) Molar mass (\({\mathrm{kg.mol}}^{-1}\))	\(0\) \(0.01\)
Gaz	Specific heat ( \({\mathrm{J.K}}^{-1}\) ) Molar mass (\({\mathrm{kg.mol}}^{-1}\)) Relative gas permeability Gas viscosity (\({\mathrm{kg.m}}^{-1}\mathrm{.}{s}^{-1}\))	\(0\) \(0.01\) \(0.001\)
Dissolved air	Specific heat ( \({\mathit{J.K}}^{\mathrm{-}1}\) ) Henry’s constant (\({\text{Pa.m}}^{3}{\text{.mol}}^{-1}\))	\(0\) \(50000\)
Initial state	Porosity Temperature (\(K\)) Gas Pressure (\(\mathrm{Pa}\)) Gas Pressure () Vapor Pressure (\(\mathrm{Pa}\)) Capillary Pressure (\(\mathrm{Pa}\)) Initial Liquid Saturation	\(1\) \(300\) \(1.01{10}^{5}\) \(1000\) \({10}^{6}\) \(0.4\)
Constants	Ideal gas constant	\(8.32\)
Homogenized coefficients	Homogenized density ( \({\mathrm{kg.m}}^{-3}\) ) Sorption isothermal sorption coefficient Biot Fick Vapor (\({m}^{2.}{s}^{-1}\)) Fick dissolved air (\({m}^{2.}{s}^{-1}\)) Intrinsic permeability (\({m}^{2}\))	\(2200\) \(S({p}_{c})=0.4\) \(0\) \(0\) \(\mathrm{FA}={6}^{-10}\) \({1}^{-19}\)

1.3. Boundary conditions and loads

Across the entire domain, we want to:

\({p}_{w}=\mathrm{cte}={p}_{w}^{0}\)

\(\frac{1}{{K}_{w}}\mathrm{=}0\mathrm{\Rightarrow }{\rho }_{w}\mathrm{=}\mathit{cte}\mathrm{=}{\rho }_{w}^{0}\)

\({p}_{\mathrm{vp}}=\mathrm{cte}={p}_{\mathrm{vp}}^{0}\)

\({F}_{\mathrm{vp}}=0\)

\(S({p}_{c})=\mathrm{cte}={S}_{0}\)

\(T=\mathrm{cte}={T}_{0}\)

\(\phi =1\)

\({M}_{\mathrm{as}}^{\mathrm{ol}}={M}_{\mathrm{ad}}^{\mathrm{ol}}={M}_{\mathrm{vp}}^{\mathrm{ol}}\)

On all sides: Zero hydraulic and thermal flows.

We are now going to linearize \({p}_{\mathrm{vp}}\) according to \({p}_{w}\).

Writing of \({p}_{\mathrm{vp}}\) linear function of \({p}_{w}\) :

Section 4.2.3 of the Code_Aster reference document [R7.01.11] gives us the relationship: \(\frac{{\mathrm{dp}}_{\mathrm{vp}}}{{p}_{\mathrm{vp}}}=\frac{{M}_{\mathrm{vp}}^{\mathrm{ol}}}{\mathrm{RT}}\frac{{\mathrm{dp}}_{w}}{{\rho }_{w}}\). If we linearize this expression we get: \({p}_{\mathrm{vp}}=\frac{{p}_{\mathrm{vp}}^{0}}{\mathrm{RT}}\frac{{M}_{\mathrm{vp}}^{\mathrm{ol}}}{{\rho }_{w}^{0}}{p}_{w}+({p}_{\mathrm{vp}}^{0}-\frac{{p}_{\mathrm{vp}}^{0}}{\mathrm{RT}}\frac{{M}_{\mathrm{vp}}^{\mathrm{ol}}}{{\rho }_{w}^{0}}{p}_{w}^{0})\) which can be written in the form:

\({p}_{\mathrm{vp}}=A{p}_{w}+B\) eq 1.3-1

with \(A=\frac{{p}_{\mathrm{vp}}^{0}}{\mathrm{RT}}\frac{{M}_{\mathrm{vp}}^{\mathrm{ol}}}{{\rho }_{w}^{0}}\) and

On edge \(\mathrm{AB}\): \({p}_{\mathrm{vp}}=A{p}_{w}+B\)

\({p}_{\mathrm{gz}}=115000\) and \({p}_{c}={10}^{6}\)