1. Reference problem

1.1. Geometry

We consider a bar with a length \(\mathrm{1m}\) and a cross section \(\mathrm{0.1m}\mathrm{\times }\mathrm{0.1m}\).

H

G

B

C

D

F

E

Z

A

B

Coordinates of the points (\(m\)):

	\(X\)	\(Y\)	\(Z\)			\(X\)	\(Y\)	\(Z\)
\(A\)	0	0	0	0	\(C\)	1	0.1	0
\(B\)	1	0	0	0	\(D\)	0	0.1	0
\(E\)	0	0	-0,1	-0,1	\(G\)	1	0,1	-0,1
F	1	0	-0,1	-0,1	\(H\)	0	0,1	-0,1

1.2. Material properties

Only the properties on which the solution depends are given here. The command file contains other material data (elasticity modules, thermal conductivity…) that play no role in the solution of the problem being treated.

Liquid water	Density ( \({\mathit{kg.m}}^{\mathrm{-}3}\) ) Specific heat at constant pressure (\({\mathit{J.K}}^{\mathrm{-}1}\)) Dynamic viscosity of liquid water (\(\mathit{Pa.s}\)) coefficient of thermal expansion of liquid (\({K}^{\mathrm{-}1}\)) Relative permeability to water	103 0.001 0.
Vapeur	Specific heat ( \({\mathit{J.K}}^{\mathrm{-}1}\) ) Molar mass (\({\mathit{kg.mol}}^{\mathrm{-}1}\))	0.01
Gaz	Specific heat ( \({\mathit{J.K}}^{\mathrm{-}1}\) ) Molar mass (\({\mathit{kg.mol}}^{\mathrm{-}1}\)) Relative gas permeability Gas viscosity (\({\mathit{kg.m}}^{\mathrm{-}{\mathrm{1.s}}^{\mathrm{-}1}}\))	#. 0.01 #. .. image:: images/Object_2.svg width: 74 height: 21 0.001
Dissolved air	Specific heat ( \({\mathit{J.K}}^{\mathrm{-}1}\) ) Henry’s constant (\(\mathit{Pa.}{m}^{3.}{\mathit{mol}}^{\mathrm{-}1}\))	50000
Initial state	Porosity Temperature (\(K\)) Gas Pressure (\(\mathit{Pa}\)) Gas Pressure () Vapor Pressure (\(\mathit{Pa}\)) Capillary Pressure (Pa) Initial Liquid Saturation	1 300 1.01E5 1000 1.E6 0.4
Constants	Ideal gas constant	8.32
Homogenized coefficients	Homogenized density ( \({\mathit{kg.m}}^{\mathrm{-}3}\) ) Sorption isothermal sorption coefficient Biot Fick Vapor (\({m}^{2.}{s}^{\mathrm{-}1}\)) Fick dissolved air (\({m}^{2.}{s}^{\mathrm{-}1}\)) Intrinsic permeability (\({m}^{2}\))	2200 0 \(\mathit{FV}\mathrm{=}0\) \(\mathit{FA}\mathrm{=}\mathrm{6 }\mathrm{.}E\mathrm{-}10\) \(\mathit{Kint}\mathrm{=}1.E\mathrm{-}19\)

1.3. Boundary conditions and loads

Across the entire domain, we want to:

On all sides: Zero hydraulic and thermal flows.

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

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

Section 4.2.3 of the Aster [R7.01.11] reference document gives us the relationship:

.

If we linearize this expression we get:

that can be written in the form:

Eq 1.3-1

with

and

On the left edge of the bar (\(\mathit{AEHD}\)) we impose