1. Reference problem#

The objective of this test case is to compare the solution obtained with the various schemes to an analytical solution.

1.1. Geometry#

We consider a \(\mathrm{5m}\) long \(\mathrm{1D}\) bar. Concretely, the mesh domain will be \([\mathrm{0m}\mathrm{,5}m]\times [\mathrm{0m};\mathrm{0,05}m]\) (in the case of triangle modeling, it is important not to have triangles that are too « flattened », so the choice of the height of the domain is not trivial).

_images/10000000000002A80000010B865D03F1FF68356E.png

The simulation duration is \(\mathrm{100s}\) and the number of time steps is 100.

1.2. Material properties#

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

Gas

Molar mass \((\mathit{kg}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1})\) Viscosity \((\mathit{kg}\mathrm{.}{m}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1})\) Relative permeability \(({m}^{2})\)

\(\mathrm{0,0001}\)

\(1\) \(1\)

Dissolved gas

Henry coefficient \((\mathit{Pa}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1}\mathrm{.}{m}^{3})\)

\(10000000000\)

Liquid

Relative Permeability \(({m}^{2})\)

\(1\)

Homogenized parameters

Permeability \({K}_{\mathit{int}}({m}^{2})\) Porosity Fick gas \(({m}^{2}\mathrm{.}{s}^{\mathrm{-}1})\) Liquid Fick \(({m}^{2}\mathrm{.}{s}^{\mathrm{-}1})\)

\({10}^{-7}\)

\(1\) \(0\) \(0\)

Table 1.2-1 : Material Properties

1.3. Boundary conditions and loads#

The boundary conditions are as follows:

  • Neumann conditions to the right of the domain:

\(\frac{\partial (\delta {P}_{g})}{\partial x}(t,x=\mathrm{5,}y)=0P\)

  • Dirichlet conditions on the left side of the domain:

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

1.4. Initial conditions#

The initial gas pressure variation with respect to the reference pressure is \(\delta {P}_{g}(t\mathrm{=}\mathrm{0,}x,y)\mathrm{=}{10}^{4}\mathit{Pa}\).

We also have \({P}_{g}^{\mathit{ref}}(t\mathrm{=}\mathrm{0,}x,y)\mathrm{=}{10}^{4}\mathit{Pa}\) which is equivalent to studying a weakly non-linear problem (to be linear we should have chosen \({P}_{g}^{r}(t=\mathrm{0,}x,y)={10}^{10}P\)) (because we have: \({P}_{g}(t=\mathrm{0,}x,y)={P}_{g}^{r}+\delta {P}_{g}(t=\mathrm{0,}x,y)\)).