1. Reference problem#

1.1. Geometry#

The domain is a \(\mathrm{[}\mathrm{0m}\mathrm{,200}m\mathrm{]}\mathrm{\times }\mathrm{[}\mathrm{0m},\mathrm{1m}\mathrm{]}\) 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 \((\mathit{kg}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1})\) Viscosity \((\mathit{kg}\mathrm{.}{m}^{\mathrm{-}1}\mathrm{.}{s}^{\mathrm{-}1})\)

\(1000\)

\({10}^{-2}\) \({10}^{-3}\)

Gaz
Density

\((k\mathrm{.}{m}^{-3})\)

Molar mass \((k\mathrm{.}{m}^{-1})\) Viscosity \((k\mathrm{.}{m}^{-1}\mathrm{.}{s}^{-1})\)

\(8{10}^{-2}\)

\(2{10}^{-3}\) \(9{10}^{-5}\)

Dissolved gas

Henry’s coefficient \((P\mathrm{.}{\mathrm{mol}}^{-1}\mathrm{.}{m}^{3})\)

\(130719\)

Vapeur

Density \((k\mathrm{.}{m}^{-3})\)

\({10}^{-4}\)

Homogenized parameters
Permeability k

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

Porosity Fick gas \(({m}^{2}\mathrm{.}{s}^{-1})\) Fick liquid \(({m}^{2}\mathrm{.}{s}^{-1})\)

\(5{10}^{-20}\)

\(0.15\) \(0\) \(\mathrm{0,45}{10}^{-9}\)

Van-Genuchten parameters

\(N\)

\({P}_{r}\) \((\mathit{MPa})\) \({S}_{rl}\) \({S}_{gr}\) \(\mathrm{Smax}\) \({P}_{e}\) \((\mathrm{MPa})\)

\(\mathrm{1,49}\)

\(2\) \(\mathrm{0,4}\) \(0\) \(\mathrm{0,999}\) 0

Initial state
Capillary pressure

\((\mathit{Pa})\)

Gas pressure \((\mathit{Pa})\)

 _images/Object_1.svg

The saturation and permeability curves follow the Mualem-Van-Genuchten model (HYDR_VGM). It is therefore necessary to define in the materials the parameters \(n\), \(\mathit{Pr}\),, \(\mathit{Sr}\), \(\mathit{Smax}\). It should be noted that these models are:

(1.1)#\[ {S} _ {the} =\ frac {{S} _ {l} _ {l} - {S} _ {\ mathit {lr}}}} {1- {S} _ {\ mathit {lr}}}\]

Relative permeability to water is expressed by integrating the prediction model proposed by Mualem (1976) into Van Genuchten’s capillarity model:

(1.2)#\[ {k} _ {r} ^ {g}\ mathrm {=}\ mathrm {=}\ sqrt {(1\ mathrm {-} {the})} {(1\ mathrm {-} {S} {S} {S} _ {the} _ {the} _ {the} _ {the} _ {the})} {k}} {k}\]

The permeability to gas is formulated in a similar way:

(1.3)#\[ {k} _ {r} ^ {w}\ mathrm {=}\ mathrm {=}\ sqrt {(1\ mathrm {-} {we})} {(1\ mathrm {-} {s} {S} {S}} _ {S}} _ {S}} {S}} {s}} {\ mathrm {-} {S}} {S}} {s}} {\]

We recall that for \(S>\mathit{Smax}\), these curves are interpolated by a polynomial of degree two \({C}^{1}\) in \(\mathit{Smax}\).

1.3. Boundary and initial conditions#

The boundary conditions are as follows:

  • Neumann conditions at the top and bottom of the domain:

\(\begin{array}{c}({F}_{l}^{w}+{F}_{g}^{w})\mathrm{\cdot }n\mathrm{=}0\\ ({F}_{l}^{c}+{F}_{g}^{c})\mathrm{\cdot }n\mathrm{=}0\end{array}\)

  • Neumann conditions on the left side of the domain:

If \(0<t<\mathit{TSIM}\) then \(({F}_{l}^{w}+{F}_{g}^{w})\mathrm{\cdot }n\mathrm{=}0\)

If \(0<t<\mathit{TINJ}\) then \(({F}_{l}^{c}+{F}_{g}^{c})\mathrm{\cdot }n\mathrm{=}Q\)

If \(\mathit{TINJ}<t<\mathit{TSIM}\) then \(({F}_{l}^{c}+{F}_{g}^{c})\mathrm{\cdot }n\mathrm{=}0\)

  • Dirichlet conditions on the right-hand side of the domain:

\(\begin{array}{c}{P}_{l}(x\mathrm{=}\mathrm{200,}y,t)\mathrm{=}{10}^{\mathrm{-}6}\mathit{Pa}\\ {P}_{g}(x\mathrm{=}\mathrm{200,}y,t)\mathrm{=}0\mathit{Pa}\end{array}\)

The initial conditions are as follows:

\(\begin{array}{c}{P}_{l}(x,y,t\mathrm{=}0)\mathrm{=}{10}^{\mathrm{-}6}\mathit{Pa}\\ {P}_{g}(x,y,t\mathrm{=}0)\mathrm{=}0\mathit{Pa}\end{array}\)

The hydrogen flow imposed on the left hand side, \(Q\), is equal to:

\(Q\mathrm{=}\mathrm{1,76}{.10}^{\mathrm{-}13}\mathit{kg}\mathrm{/}{m}^{2}s\)

The injection time, \(\mathit{TINJ}\) is \(5.{10}^{5}\mathit{ans}\) and the simulation time is \({10}^{6}\mathit{ans}\).