1. Reference problem#
This test case represents the desaturation of a column initially saturated with water by gravity effect (Liakopoulos experiment). Here we consider that air can dissolve in water (version proposed by VAUNAT in 1997).
1.1. Geometry#
The domain is a \(\mathrm{[}\mathrm{0m};\mathrm{0,1}m\mathrm{]}\mathrm{\times }\mathrm{[}\mathrm{0m};\mathrm{1m}\mathrm{]}\) size bar.
1.2. 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.
Liquid water |
Density \((k\mathrm{.}{m}^{-3})\) Molar mass \((k\mathrm{.}{m}^{-1})\) Viscosity \((\mathit{kg}\mathrm{.}{m}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1})\) compressibility |
\(1000\) \({10}^{-2}\) \({10}^{-3}\) \(\mathrm{0,5}{10}^{-9}\) |
Gas |
Density \((\mathit{kg}\mathrm{.}{m}^{\mathrm{-}3})\) Molar mass \((\mathit{kg}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1})\) Viscosity \((\mathit{kg}\mathrm{.}{m}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1})\) |
\(\mathrm{28,96}{10}^{-3}\) \(\mathrm{1,8}{10}^{-5}\) |
Dissolved gas |
Henry coefficient \((\mathit{Pa}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1}\mathrm{.}{m}^{3})\) |
|
Steam |
Density \((\mathit{kg}\mathrm{.}{m}^{\mathrm{-}3})\) |
|
Homogenized parameters |
Permeability \(k\) \(({m}^{2})\) Porosity Fick gas \(({m}^{2}\mathrm{.}{s}^{\mathrm{-}1})\) Liquid Fick \(({m}^{2}\mathrm{.}{s}^{\mathrm{-}1})\) |
\(\mathrm{0,2975}\) \(0\) \(0\) |
Van-Genuchten Settings |
\(N\) \({P}_{r}\) \(\mathit{MPa}\) \({S}_{rl}\) \({S}_{gr}\) \(\mathit{Smax}\) |
\({10}^{4}\) \(0\) \(0\) \(\mathrm{0,999}\) |
Initial state |
Liquid pressure Gas pressure |
\({P}_{l}^{0}\mathrm{=}\rho gY+1\mathit{atm}\mathrm{-}\rho g\) \({P}_{g}^{0}\mathrm{=}1\mathit{atm}\) |
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 in the materials the parameters \(n\), \(\mathit{Pr}\),, \(\mathit{Sr}\), \(\mathit{Smax}\).
Remember that these models are: \({S}_{le}\mathrm{=}\frac{{S}_{l}\mathrm{-}{S}_{\mathit{lr}}}{1\mathrm{-}{S}_{\mathit{lr}}}\) and \(m\mathrm{=}1\mathrm{-}\frac{1}{n}\)
\({S}_{\mathit{we}}\mathrm{=}\frac{1}{{\left[1+{(\frac{{P}_{c}}{{P}_{r}})}^{n}\right]}^{m}}\)
Relative permeability to water is expressed by integrating the prediction model proposed by Mualem (1976) into Van Genuchten’s capillarity model: \({k}_{r}^{l}\mathrm{=}\sqrt{{S}_{le}}{(1\mathrm{-}{(1\mathrm{-}{S}_{le}^{\frac{1}{m}})}^{m})}^{2}\)
Permeability to gas is formulated in a similar way: \({k}_{r}^{g}\mathrm{=}\sqrt{(1\mathrm{-}{S}_{le})}{(1\mathrm{-}{S}_{le}^{\frac{1}{m}})}^{2m}\)
We recall that for \(S>\mathit{Smax}\), these curves are interpolated by a polynomial of degree 2 \(\mathit{C1}\) in \(\mathrm{Smax}\).
1.3. Boundary and initial conditions#
The boundary conditions are as follows:
Neumann conditions on the right and left edges of the domain:
\(({F}_{l}^{w}+{F}_{g}^{w})\mathrm{.}n\mathrm{=}0\)
\(({F}_{l}^{c}+{F}_{g}^{c})\mathrm{.}n\mathrm{=}0\)
Dirichlet conditions on the upper part of the domain (open air surface):
\({P}_{g}(x,y\mathrm{=}\mathrm{1,}t)\mathrm{=}{10}^{5}\mathit{Pa}\)
Dirichlet conditions on the lower part of the domain (saturated environment; water flows):
\({P}_{g}(x,y\mathrm{=}\mathrm{0,}t)\mathrm{=}{10}^{5}\mathit{Pa}\)
\({P}_{l}(x,y\mathrm{=}\mathrm{0,}t)\mathrm{=}{10}^{5}\mathit{Pa}\)
The initial state corresponds to a water-saturated state at hydrostatic equilibrium. The initial conditions are as follows:
\({P}_{l}(x,y,t\mathrm{=}0)\mathrm{=}\rho g(Y\mathrm{-}1)+{10}^{5}\)
\({P}_{g}(x,y,t)\mathrm{=}0\mathrm{=}{10}^{5}\mathit{Pa}\)
1.4. Simulation time#
The calculation is carried out over a year \(({\mathrm{3.1536.10}}^{7}s)\).