2. Problem Overview: Assumptions, Notations#

In this chapter, the main focus is on presenting the porous medium and its characteristics.

2.1. Description of the porous medium#

The porous medium in question is a volume consisting of a solid matrix that is more or less homogeneous, more or less coherent (very coherent in the case of concrete, not very coherent in the case of sand). Between the solid elements, there are pores. A distinction is made between closed pores that exchange nothing with their neighbors and connected pores in which exchanges are numerous. When we talk about porosity, we are talking about these connected pores.

Inside these pores are a certain number of fluids (the solidification of these fluids is excluded), possibly present in several phases (liquid or gaseous exclusively), and presenting an interface with the other components. To simplify the problem and take into account the relative importance of physical phenomena, the only interface considered is that between liquid and gas, fluid/solid interfaces being neglected.

2.2. Ratings#

We assume that the pores of the solid are occupied by at most two components, each coexisting in a maximum of two phases, one liquid and the other gaseous. The quantities \(X\) associated with the phase \(j\) (\(j\mathrm{=}\mathrm{1,2}\)) of the fluid \(i\) will be noted: \({X}_{\text{ij}}\). When there are two components in addition to the solid, they are a liquid (typically water) and a gas (typically dry air), knowing that the liquid can be present in gaseous form (vapor) in the gas mixture and that air can be present in dissolved form in water. When there is only one component besides the solid, it can be a liquid or a gas. Hereinafter, we will talk about air for the gaseous component, but it can be any other component (hydrogen, \({\text{CO}}_{2}\) etc.).

The porous medium at the current moment is noted \(\Omega\), its border \(\partial \Omega\). It was noted \({\Omega }_{0},\partial {\Omega }_{0}\) at the initial point in time.

The environment is defined by:

  • parameters (position vector \(x\), time \(t\)),

  • variables (movements, pressures, temperature),

  • intrinsic quantities (stresses and deformations, mass inputs, mass, heat, enthalpies, hydraulic and thermal flows, etc.).

The general assumptions made are as follows:

  • hypothesis of small trips,

  • reversible thermodynamic changes (not necessarily for mechanics),

  • isotropic, transverse isotropic (3D), orthotropic (2D) 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. Descriptive variables of the environment#

These are the variables whose knowledge as a function of time and place makes it possible to fully know the state of the environment. These variables are divided into two categories:

  • geometric variables,

  • thermodynamic state variables.

2.2.1.1. Geometric variables#

In everything that follows, we adopt a Lagrangian representation with respect to the skeleton (in the sense of [1]) and the coordinates \(\mathrm{x}\text{=}{\mathrm{x}}_{\mathrm{s}}(t)\) are those of a material point attached to the skeleton. All spatial derivation operators are defined with respect to these coordinates.

The movements of the skeleton are noted \(\mathrm{u}(\mathrm{x},t)\mathrm{=}(\begin{array}{c}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array})\).

2.2.1.2. Thermodynamic state variables#

In general, the following indices are used:

\(w\) for liquid water,

\(\text{ad}\) for dissolved air,

\(\text{as}\) for dry air,

\(\text{vp}\) for water vapor.

The thermodynamic variables are:

  • the pressures of the constituents: \({p}_{w}(x,t)\), \({p}_{\text{ad}}(x,t)\),, \({p}_{\text{vp}}(x,t)\), \({p}_{\text{as}}(x,t)\),

  • the temperature of the medium: \(T(x,t)\).

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.

For the so-called saturated case (a single air or water component) we chose the pressure of this single component.

For the so-called unsaturated case (presence of air and water), we chose as independent variables:

  • the total pressure of the gas: \({p}_{\text{gz}}(x,t)\text{=}{p}_{\text{vp}}\text{+}{p}_{\text{as}}\),

  • capillary pressure: \({p}_{c}(x,t)={p}_{\text{gz}}-{p}_{\text{lq}}={p}_{\text{gz}}-{p}_{w}-{p}_{\text{ad}}\).

These pressures have a very strong physical interpretation, total gas pressure for obvious reasons, and capillary pressure, also called suction, because capillary phenomena are very important in the modeling presented here. It would also have been possible to choose the vapour pressure or the degree of relative humidity (relationship between vapour pressure and the saturation vapour pressure at the same temperature) that was physically accessible. The modeling then becomes more complex and in any case, capillary pressure, gas pressure and degree of relative humidity (ratio between vapor pressure and saturated vapor pressure) HR are linked by Kelvin’s law.

For the particular case of the so-called “LIQU_GAZ_ATM” behavior, the so-called Richards hypothesis is made: the pores are not saturated by the liquid, but the gas pressure is assumed to be constant and the only pressure variable is the liquid pressure.

2.2.1.3. Descriptive fields of the environment#

The main unknowns, which are also the nodal unknowns (noted \(U(x,t)\) in this document) are:

  • 2 or 3 (depending on the space dimension) \({u}_{x}(x,t),{u}_{y}(x,t),{u}_{z}(x,t)\) movements for KIT_HM, KIT_HHM, KIT_THM, KIT_THHM models,

  • temperature \(T(x,t)\) for KIT_THH, KIT_THM, KIT_THHM models,

  • two \({p}_{1}(\mathrm{x},t),{p}_{2}(\mathrm{x},t)\) pressures (which are \({p}_{c}(x,t)\), \({p}_{\text{gz}}(x,t)\) in the case studied) for the models KIT_HHM, KIT_THH, KIT_THHM,

  • a pressure \({p}_{1}(x,t)\) (which is \({p}_{w}(x,t)\) or \({p}_{\text{gz}}(x,t)\) depending on whether the medium is saturated by a liquid or a gas) for the models KIT_H, KIT_HM, KIT_THM.

2.2.2. Particle derivatives#

This paragraph partly repeats the paragraph « particle derivatives, volume and mass densities » from document [R7.01.10]. Our description of the environment is Lagrangian in relation to the skeleton.

Let \(a\) be any field on \(\Omega\), let \({x}_{s}(t)\) be the coordinate of a point attached to the skeleton that we are following in its movement and let \({x}_{\mathrm{fl}}(t)\) be the coordinate of a point attached to the fluid. We note \(\dot{a}\text{=}\frac{{d}^{S}a}{\mathrm{dt}}\) the time derivative in skeletal movement:

\(\dot{a}=\frac{{d}^{S}a}{\mathrm{dt}}=\underset{\Delta t\to \text{0}}{\mathrm{lim}}\frac{a(x(t\text{+}\Delta t),(t\text{+}\Delta t)-a(x(t),t))}{\Delta t}\)

\(\dot{a}\) is called a particulate derivative and is often referred to as \(\frac{\mathrm{da}}{\mathrm{dt}}\). We prefer to use a notation that recalls that the configuration used to locate a particle is that of the skeleton with respect to which a fluid particle has a relative speed. For a fluid particle, the identification \({x}_{s}(t)\) is random, that is to say that the fluid particle that occupies the position \({x}_{s}(t)\) at the instant \(t\) is not the same as the one that occupies the position \({\Omega }^{0}(1+{\varepsilon }_{V})(1-\varphi )={\Omega }^{0}(1-{\varphi }^{0})(1+{\varepsilon }_{\text{Vs}})\) at another moment \(t\text{'}\).

2.2.3. Sizes#

The equilibrium equations are:

  • the conservation of the quantity of movement for mechanics,

  • the conservation of fluid masses for hydraulics,

  • the conservation of energy for thermal purposes.

The writing of these equations is given in document [R7.01.10] [4], which also defines what we generally call a law of behavior THM and gives the definitions of generalized stresses and deformations. This document uses these definitions. Equilibrium equations involve generalized constraints directly.

Generalized stresses are linked to generalized deformations by laws of behavior. Generalized deformations are calculated directly from state variables and their spatial temporal gradients.

The laws of behavior can use ancillary quantities, often arranged in internal variables. Here, under the term of quantity, we group together stresses, deformations and ancillary quantities.

2.2.3.1. Characteristic quantities of a heterogeneous environment#

  • Eulerian porosity: \(\phi\).

If we note \({\Omega }_{(\phi )}\) the part of volume \(\Omega\) occupied by the voids in the current configuration, we have:

\(\varphi =\frac{{\Omega }_{(\varphi )}}{\Omega }\)

The definition of porosity is therefore that of Eulerian porosity.

  • Liquid saturation: \({S}_{\text{lq}}\)

If we write \({\Omega }_{\text{lq}}\) the total volume occupied by the liquid, in the current configuration, we have by definition:

\({S}_{\text{lq}}=\frac{{\Omega }_{\text{lq}}}{{\Omega }_{\phi }}\)

This saturation is therefore finally a proportion varying between 0 and 1.

  • The Eulerian densities of water \({\mathrm{\rho }}_{w}\), of dissolved air \({\mathrm{\rho }}_{\text{ad}}\), of dissolved air, of dry air \({\mathrm{\rho }}_{\text{as}}\), of steam \({\mathrm{\rho }}_{\text{vp}}\), of gas \({\mathrm{\rho }}_{\text{gz}}\).

If we note \({\mathrm{\gamma }}_{w}\) (resp \({\gamma }_{\text{ad}}\), \({\gamma }_{\text{as}}\), \({\gamma }_{\text{vp}}\)) the bodies of water (respectively dissolved air, dry air and steam) contained in a volume of the \(\Omega\) skeleton in the current configuration, we have by definition:

\(\begin{array}{}{\gamma }_{w}={\int }_{\Omega }{\rho }_{w}{S}_{\text{lq}}\varphi d\Omega {\gamma }_{\text{ad}}={\int }_{\Omega }{\rho }_{\text{ad}}{S}_{\text{lq}}\varphi d\Omega \\ {\gamma }_{\text{as}}={\int }_{\Omega }{\rho }_{\text{as}}(1-{S}_{\text{lq}})\varphi d\Omega {\gamma }_{\text{vp}}={\int }_{\Omega }{\rho }_{\text{vp}}(1-{S}_{\text{lq}})\varphi d\Omega \end{array}\)

The density of the gas mixture is simply the sum of the densities of dry air and steam:

\({\rho }_{\mathrm{gz}}\text{=}{\rho }_{\mathrm{as}}\text{+}{\rho }_{\mathrm{vp}}\)

Likewise for the liquid mixture:

\({\mathrm{\rho }}_{\text{lq}}={\mathrm{\rho }}_{w}+{\mathrm{\rho }}_{\text{ad}}\)

We note \({\mathrm{\rho }}_{w}^{0},{\mathrm{\rho }}_{\text{ad}}^{0},{\mathrm{\rho }}_{\text{vp}}^{0},{\mathrm{\rho }}_{\text{as}}^{0}\) the initial values of the densities.

  • The homogenized Lagrangian density: \(r\).

At the current moment the mass of volume \(\Omega\), \({M}_{\Omega }\), is given by: \({M}_{\Omega }={\int }_{{\Omega }_{0}}\text{rd}{\Omega }_{0}\).

2.2.3.2. Mechanical quantities#

  • The \(\varepsilon (u)(x,t)\text{=}\frac{1}{2}(\nabla u+{}^{T}\text{}\nabla u)\) deformation tensor.

Note \({\epsilon }_{V}=\text{tr}(\varepsilon )\).

  • The stress tensor that is exerted on the porous medium: \(\sigma\).

This tensor is broken down into an effective stress tensor plus a pressure stress tensor, \(\sigma \text{=}\sigma \text{'}+{\sigma }_{p}\) and \({\sigma }_{p}\) are components of generalized stresses. This division is finally quite arbitrary, but still corresponds to a fairly commonly accepted hypothesis, at least for environments saturated with liquid.

2.2.3.3. Hydraulic quantities#

  • Mass contributions of components \({m}_{w},{m}_{\text{ad}},{m}_{\text{vp}},{m}_{\text{as}}\) (unit: kilogram per cubic meter).

They represent the mass of fluid supplied between the initial and current moments. They are part of the generalized constraints.

  • Hydraulic flows \({M}_{w},{M}_{\text{ad}},{M}_{\text{vp}},{M}_{\text{as}}\) (unit: kilogram/second/square meter).

We could very well not give any more precise definition of mass inputs and flows, considering that their definition boils down to verifying the hydraulic equilibrium equations:

\(\{\begin{array}{}\dot{{m}_{w}}+{\dot{m}}_{\text{vp}}+\text{Div}({M}_{w}+{M}_{\text{vp}})=0\\ \dot{{m}_{\text{as}}}+{\dot{m}}_{\text{ad}}\text{+}\text{Div}({M}_{\text{as}}\text{+}{M}_{\text{ad}})=0\end{array}\) eq 2.2.3.3-1

However, we are going to specify the physical meaning of these quantities, knowing that what we are writing now is already a law of behavior.

The speeds of the components are measured in a fixed frame of reference in space and time.

We note \({v}_{w}\) the speed of water, \({v}_{\mathrm{ad}}\) that of dissolved air, that of dissolved air, \({v}_{\mathrm{vp}}\) that of steam, \({v}_{\mathrm{as}}\) that of dry air, and \({v}_{S}=\frac{du}{\mathrm{dt}}\) that of the skeleton.

Mass inputs are defined by:

\(\begin{array}{}{m}_{w}\text{=}{\mathrm{\rho }}_{w}(1\text{+}{\mathrm{\varepsilon }}_{V})\phi {S}_{\text{lq}}\text{-}{\mathrm{\rho }}_{w}^{0}{\phi }^{0}{S}_{\text{lq}}^{0}\\ {m}_{\text{ad}}\text{=}{\mathrm{\rho }}_{\text{ad}}(1\text{+}{\mathrm{\varepsilon }}_{V})\phi {S}_{\text{lq}}\text{-}{\mathrm{\rho }}_{\text{ad}}^{0}{\phi }^{0}{S}_{\text{lq}}^{0}\\ {m}_{\text{as}}\text{=}{\mathrm{\rho }}_{\text{as}}(1\text{+}{\mathrm{\varepsilon }}_{V})\phi (1\text{-}{S}_{\text{lq}})-{\mathrm{\rho }}_{\text{as}}^{0}{\phi }^{0}(1\text{-}{S}_{\text{lq}}^{0})\\ {m}_{\text{vp}}\text{=}{\mathrm{\rho }}_{\text{vp}}(1\text{+}{\mathrm{\varepsilon }}_{V})\phi (1\text{-}{S}_{\text{lq}})-{\mathrm{\rho }}_{\text{vp}}^{0}{\phi }^{0}(1\text{-}{S}_{\text{lq}}^{0})\end{array}\) eq 2.2.3.3-2

Mass flows are defined by:

\(\begin{array}{}{M}_{w}={\rho }_{w}\varphi {S}_{l}({v}_{w}-{v}_{s})\\ {M}_{\text{ad}}={\rho }_{\text{ad}}\varphi {S}_{l}({v}_{\text{ad}}-{v}_{s})\\ {M}_{\text{as}}={\rho }_{\text{as}}\varphi (1-{S}_{l})({v}_{\text{as}}-{v}_{s})\\ {M}_{\text{vp}}={\rho }_{\text{vp}}\varphi (1-{S}_{l})({v}_{\text{vp}}-{v}_{s})\end{array}\) eq 2.2.3.3-3

Mass inputs make it possible to define the overall density seen in relation to the reference configuration:

\(r\text{=}{r}_{0}\text{+}{m}_{w}\text{+}{m}_{\text{ad}}\text{+}{m}_{\text{vp}}\text{+}{m}_{\text{as}}\) eq 2.2.3.3-4

where \({r}_{0}\) refers to the density homogenized at the initial instant.

Other intermediate hydraulic quantities are introduced:

  • concentration of the vapor in the gas: \({C}_{\text{vp}}=\frac{{p}_{\text{vp}}}{{p}_{\text{gz}}}\),

  • gas flow: \(\frac{{M}_{\text{gz}}}{{\rho }_{\text{gz}}}=(1-{C}_{\text{vp}})\frac{{M}_{\text{as}}}{{\rho }_{\text{as}}}+{C}_{\text{vp}}\frac{{M}_{\text{vp}}}{{\rho }_{\text{vp}}}\). This equation states that the gas speed is obtained by averaging (weighted sum) the velocities of the various gases as a function of their concentration,

  • vapour pressure \({p}_{\text{vp}}\).

2.2.3.4. Thermal quantities#

  • non-convected heat \({Q}^{\text{'}}\) (see later) (unit: Joule),

  • the mass enthalpies of the components \({h}_{\text{ij}}^{m}\) (\({h}_{w}^{m},{h}_{\text{ad}}^{m},{h}_{\text{vp}}^{m},{h}_{\text{as}}^{m}\)) (unit: Joule/Kelvin),

  • heat flow: \(q\) (unit: \(J\mathrm{/}s\mathrm{/}{m}^{2}\)).

All these quantities belong to the generalized constraints in the sense of document [R7.01.10] [4].

2.2.4. External data#

  • the mass force \({F}^{m}\) (in practice gravity),

  • heat sources \(\mathrm{\Theta }\),

  • boundary conditions relating either to imposed variables or to imposed flows.