2. Problem Overview: Assumptions, Notations#

The entire hydraulic physical model is presented in detail in R7.01.11. We are content here with recalling the main variables and hypotheses as well as the equations treated. The formalism is essentially the result of the work of Coussy [5]. The routines managing the laws of behavior are not impacted by the use of the finite volume schema.

2.1. Modeling framework#

These developments only concern models with 2 phases (liquid gas) and 2 miscible components (for example water and hydrogen). We therefore place ourselves within the framework of *_ HH2 models (cf. U2.04.05). More precisely, we will talk here about modeling HH2SUDA (cf. section 4).

We recall that the 2 laws of hydraulic behavior that can be used are then:

  • LIQU_AD_GAZ_VAPE: 2 components per phase

  • LIQU_AD_GAZ: 2 components in the liquid phase, only one in the gas phase (neglected vapor)

In the following, we will only talk about the complete model with 2 components, 2 phases. The case without steam is just like cancelling the terms concerning it.

2.2. Notations#

We assume that the pores of the solid are occupied by two components noted \(w\) (for water) and \(h\) (for hydrogen), each coexisting in two phases at most, one liquid marked \(l\) and the other gaseous one noted and the other gaseous noted \(g\).

The quantities \(A\) associated with the phase \(p\) (\(p=l,g\)) of the component \(c\) will be noted: \({X}_{p}^{c}\). Concretely, this gives:

  • \({A}_{l}^{w}\): size \(A\) for liquid water,

  • \({A}_{g}^{w}\): size \(A\) for water vapor,

  • \({A}_{l}^{h}\): quantity \(A\) for the component h dissolved in the liquid,

  • \({A}_{g}^{h}\): quantity \(A\) for the component h in gaseous form (e.g. dry hydrogen).

The general assumptions made are as follows:

  • anisotropic behavior,

  • gases are ideal gases,

  • ideal mixture of ideal gases (total pressure = sum of partial pressures),

  • thermodynamic balance between the phases of the same constituent.

The various notations are explained below.

2.2.1. State variables#

The variables are:

  • the pressures of each component \({P}_{p}^{c}\)

  • the temperature of medium \(T\).

These different variables are not completely independent. In fact, if we consider a single component, the thermodynamic balance between its phases imposes a relationship between the pressure of the vapor and the pressure of the liquid of this component. Finally, there is only one independent pressure per component, just as there is only one equation for the conservation of mass. The number of independent pressures is therefore equal to the number of independent components. The choice of these pressures is free (combinations of the pressures of the components) provided that the pressures selected, associated with the temperature, form a system of independent variables.

We chose - and in order to be homogeneous with the finite element formulation - as independent and descriptive variables of the environment:

  • the total pressure of gas \({P}_{\text{g}}\text{=}{P}_{g}^{w}+{P}_{g}^{h}\), (Dalton’s law)

  • The total liquid pressure \({P}_{\text{l}}\text{=}{P}_{l}^{w}+{P}_{l}^{h}\)

  • capillary pressure \({P}_{c}\text{=}{P}_{\text{g}}\text{-}{P}_{\text{l}}\text{=}{P}_{g}^{w}\text{+}{P}_{g}^{h}\text{-}({P}_{l}^{w}\text{+}{P}_{l}^{h})\).

2.2.2. Characteristic quantities of the solid phase#

We note:

The porosity: \(\phi\),

The intrinsic permeability tensor: \(k\)

2.2.3. Characteristic quantities of fluids#

We note:

  • The density of phase \(p\): \({\rho }_{\text{p}}\), \({\rho }_{\text{p}}\text{=}{\rho }_{p}^{w}+{\rho }_{p}^{h}\)

  • The viscosity of phase \(p\): \({\mu }_{\text{p}}\)

  • The saturation of phase \(p\):, \({S}_{\text{l}}\text{+}{S}_{\text{g}}=1\) which is a decreasing function of capillary pressure. So we have \({S}_{\text{l}}=f({P}_{c})\).

  • The relative permeability of phase \(p\): \({k}_{\text{rp}}\) is a function of saturation

  • The hydraulic conductivity of phase \(p\): \({\lambda }_{\text{p}}^{H}\) such as: \({\lambda }_{\text{p}}^{H}=\frac{k{k}_{\mathrm{rp}}}{{\mu }_{p}}\)

  • The mobility of component \(c,c=(h,w)\) associated with phase \(p\) \({k}_{\text{p}}^{c}\mathrm{=}\frac{{\rho }_{\text{p}}^{c}{k}_{\mathit{rp}}}{{\mu }_{p}}\)

  • The molar mass of component \(c\): \({M}^{c}\)

  • Molar concentration \({c}_{p}^{c}\text{=}\frac{{\rho }_{p}^{c}}{{M}^{c}}\)

  • The mass fraction of phase \(p\) and component \(c\): \({\zeta }_{p}^{c}\text{=}\frac{{\rho }_{p}^{c}}{{\rho }_{p}}\)

  • The molar fraction: \({X}_{p}^{c}\text{=}\frac{{c}_{p}^{c}}{{c}_{p}}\) or \({c}_{p}\text{=}{c}_{p}^{h}+{c}_{p}^{w}\) (in the literature, these concentrations are sometimes also noted \({C}_{p}^{c}\))

  • The \({K}_{w}\) water compressibility module

2.3. Constitutive equations of the model#

The details needed to arrive at the final equations of the model will not be given here. For the intermediate steps, refer to [R7.01.11]. We are therefore content here with a brief summary of the main equations.

The equilibrium equations are given here by the conservation of the mass of each component, i.e.:

\(\{\begin{array}{}\dot{{m}_{l}^{w}}+{\dot{m}}_{g}^{w}+\text{Div}({F}_{l}^{w}+{F}_{g}^{w})\text{=}0\\ \dot{{m}_{l}^{h}}+{\dot{m}}_{g}^{h}+\text{Div}({F}_{l}^{h}+{F}_{g}^{h})\text{=}0\end{array}\)

with \({m}_{p}^{c}\) the mass supply of the component \(c\) in phase \(p\), such as \({m}_{p}^{c}=\phi \mathrm{.}{S}_{p}\mathrm{.}{\rho }_{p}^{c}\) and \({F}_{p}^{c}\) the mass flow of the phase \(p\) for the constituent \(c\).

\({F}_{p}^{c}\) consists of the Fickian mass flow \({J}_{p}^{c}\) and the Darcaean mass flow \({F}_{p}\) such that:

\({F}_{p}^{c}={J}_{p}^{c}+{\rho }_{p}^{c}\frac{{F}_{p}}{{\rho }_{l}},c=(h,w);p=(l,g)\)

2.3.1. Laws of fluid behavior#

Evolution of porosity:

In the absence of mechanics, it is nevertheless possible to change the porosity via the storage coefficient \({E}_{m}\), such as:

\(d\phi \mathrm{=}{E}_{m}d{P}_{l}\)

Liquid behavior:

It is considered that water can be compressible: \(\frac{d{\rho }_{l}^{w}}{{\rho }_{l}^{w}}\text{=}\frac{{\text{dP}}_{l}^{w}}{{K}_{w}}\)

Gas behavior:

It is considered that gas is subject to the ideal gas law:

\(\frac{{P}_{g}^{c}}{{\rho }_{g}^{c}}\text{=}\frac{RT}{{M}^{c}};c=(w,h)\) where \(R\) is the ideal gas constant.

Water-steam balance law:

The water-steam balance is written by equality of the free enthalpies, which, for an isothermal problem, gives (confer R7.01.11):

\(\frac{d{P}_{g}^{w}}{{\rho }_{g}^{w}}\text{=}\frac{{\text{dP}}_{l}^{w}}{{\rho }_{l}^{w}}\)

Equilibrium law dry/dissolved gas:

Henry’s law relates component \(c\) in its gaseous form to its liquid form such as:

\({P}_{g}^{h}\text{=}{K}_{h}\frac{{\rho }_{l}^{h}}{{M}^{h}}\) where \({K}_{h}\) Henry’s coefficient.

Note: this coefficient is often found in the literature in the form \(H=\frac{1}{{K}_{h}}\)

2.3.2. Diffusion equations#

Darcy’s law:

Darcy’s law relates flow \({F}_{p}\) from phase \(p\) to its pressure gradient, such as:

\(\frac{{F}_{\text{p}}}{{\rho }_{p}}=\frac{-k{k}_{\mathrm{rp}}}{{\mu }_{p}}(\nabla {P}_{p}-{\rho }_{p}g)\)

with \(g\) gravity

Fick’s law:

Fickian mass flows are written as:

\({J}_{p}^{c}=-\phi {M}^{c}{S}_{p}{D}_{p}{c}_{p}\nabla {X}_{p}^{c}\) where \({D}_{p}\) is the diffusion coefficient (or Fick coefficient) for phase \(p\).

We neglect the Fickian flow of water in the liquid (concentration of water in the liquid assimilable to 1). In the end, we therefore get:

\({F}_{l}^{w}=-{\rho }_{l}^{w}{\lambda }_{\text{l}}^{H}(\nabla {P}_{l}-{\rho }_{l}g)\)

\({F}_{l}^{h}=-{\rho }_{l}^{h}{\lambda }_{\text{l}}^{H}(\nabla {P}_{l}-{\rho }_{l}g)-\phi {M}^{h}{S}_{l}{c}_{l}{D}_{l}\nabla {X}_{l}^{h}\)

\({F}_{g}^{w}=-{\rho }_{g}^{w}{\lambda }_{\text{g}}^{H}(\nabla {P}_{g}-{\rho }_{g}g)-\phi {M}^{w}{S}_{g}{c}_{g}{D}_{g}\nabla {X}_{g}^{w}\)

\({F}_{g}^{h}=-{\rho }_{g}^{h}{\lambda }_{\text{g}}^{H}(\nabla {P}_{g}-{\rho }_{g}g)-\phi {M}^{h}{S}_{g}{c}_{g}{D}_{g}\nabla {X}_{g}^{h}\)