2. Presentation of the problem: hypotheses, 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 connected pores. Inside these pores there are at most two components present at most in two phases. The system is considered to be closed.

2.2. Ratings#

The quantities associated with a component \(c\) present in a \(p\) phase are noted \({X}_{c}^{p}\). The index of component \(c\) can vary from \(1\) to \(2\) and that of the phase as well. These components can be (and will be indicated as such if necessary):

\(w\) for liquid water,

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

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

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

The porous medium at the current moment is noted \(\mathrm{\Omega }\), its border \(\partial \mathrm{\Omega }\), and it is noted \({\mathrm{\Omega }}_{0},\partial {\mathrm{\Omega }}_{0}\) at the initial moment.

\(n\) refers to the normal at a point from \(\partial \Omega\), which is the image of the normal \({n}_{0}\) to \(\partial {\Omega }_{0}\). We’ll denote \(d(\partial \Omega )\) (respectively \(d(\partial {\Omega }_{0})\)) as the surface element of \(\partial \Omega\) (respectively \(\partial {\Omega }_{0}\)).

The environment is defined by:

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

  • variables (movements, pressures, temperature),

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

For the solid phase, the hypothesis of small displacements is made.

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 [bib1]) and the coordinates \(\mathrm{x}\mathrm{=}{\mathrm{x}}_{s}(t)\) 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 \(u(x,t)=(\begin{array}{c}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array})\).

2.2.1.2. Thermodynamic state variables#

The thermodynamic variables are:

  • the pressures of the components: given that we consider that there are at most two components, there will be at most two equations for conservation of mass, and therefore by duality at most two pressure variables,

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

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 \({p}_{1}(x,t),{p}_{2}(x,t)\) presses for KIT_HHM, KIT_THH, KIT_THHM models,

  • \({p}_{1}(x,t)\) pressure for KIT_HM, KIT_THM models.

2.2.2. 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.

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.

Behavioral laws can use ancillary quantities, often arranged in internal variables. These quantities are not described in this document, which does not deal with laws of behavior strictly speaking.

2.2.2.1. Characteristic quantities of a heterogeneous environment#

  • Eulerian porosity: \(\phi\).

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

\(\varphi =\frac{{\Omega }_{\varphi }}{\Omega }\) eq 2.2.2.1-1

The definition of porosity \(\varphi\) is therefore that of Eulerian porosity.

  • The saturation of phase \(p\): \({S}^{p}\).

If we write \({\Omega }^{p}\) the total volume occupied by phase \(p\), in the current configuration, we have by definition:

\({S}^{p}=\frac{{\Omega }^{p}}{{\Omega }_{j}}\) eq 2.2.2.1-2

This saturation is therefore finally a proportion varying between \(0\) and \(1\). In the balance equations, it is clear that it is the product of porosity by saturation \(\phi {S}^{p}\) that will intervene. One can therefore legitimately ask why it is not this quantity that is taken as unknown. The answer comes from the fact that it is the \({S}^{p}\) saturation that intervenes more simply in the laws of behavior.

  • The Eulerian density of the constituent \(c\) in the \(p\) phase: \({\rho }_{c}^{p}\).

If we denote \({M}_{c}^{p}\) the mass of the phase \(p\) of the constituent \(c\), in the volume \(\Omega\) of the skeleton in the current configuration, we have by definition:

\({M}_{c}^{p}={\int }_{{\Omega }^{p}}{{\rho }_{c}^{p}d\Omega }^{p}={\int }_{{\Omega }_{\phi }}{\rho }_{c}^{p}{S}^{p}d{\Omega }_{\phi }={\int }_{\Omega }{\rho }_{c}^{p}{S}^{p}\phid \Omega\) eq 2.2.2.1-3

The density of phase \(p\) is simply the sum of the densities of its components:

\({\rho }^{p}=\sum _{c}{\rho }_{c}^{p}\)

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

At the current moment, the mass of volume \(\Omega\), \({M}_{\Omega }\) is given by:

\({M}_{\Omega }={\int }_{{\Omega }_{0}}rd{\Omega }_{0}\) eq 2.2.2.1-4

2.2.2.2. Mechanical quantities#

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

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

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

2.2.2.3. Hydraulic quantities#

  • Mass contributions of components \({m}_{c}^{p}\) (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.

\({m}_{c}^{p}=J{\rho }_{c}^{p}\phi {S}^{p}-{\rho }_{c}^{{p}_{0}}{\phi }_{0}{S}_{0}^{p}\) eq 2.2.3-1

Mass inputs make it possible to define the overall density seen in relation to the reference configuration: \(r(x,t)={r}_{0}+{m}_{\mathrm{lq}}(x,t)+{m}_{\mathrm{vp}}(x,t)+{m}_{\mathrm{as}}(x,t)\), where \({r}_{0}\) refers to the density homogenized in the initial state.

  • Hydraulic flows:

\({w}_{c}^{p}\) (unit: kilogram/second/square meter) in Eulerian representation

\({M}_{c}^{p}\) (unit: kilogram/second/square meter) in Lagrangian representation

We note \({v}_{c}^{p}\) the speed of the \(c\) component in the \(p\) phase, \(J\) the Jacobian of material transformation and \({v}_{S}\text{=}\frac{du}{\mathrm{dt}}\) the speed of the skeleton. \({\rho }_{c0}^{p},{\phi }_{0},{S}_{0}^{p}\) refer to densities, porosity, and saturations at the initial point in time. By definition:

\({w}_{c}^{p}={\rho }_{c}^{p}\phi {S}^{p}({v}_{c}^{p}-{v}_{s})\) eq 2.2.2.3-2

The Lagrangian form of

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Noted

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is obtained by writing:

\({M}_{c}^{p}\text{.}{n}_{0}d(\partial {\Omega }_{0})={w}_{c}^{p}\text{.}nd(\partial \Omega )\) eq 2.2.2.3-3

Variables \({m}_{1},{M}_{1}\) and \({m}_{2},{M}_{2}\) each refer to a conservative mass constituent.

As a matter of principle, we set out:

\(\begin{array}{}{m}_{1}={m}_{1}^{1}+{m}_{1}^{2};{M}_{1}={M}_{1}^{1}+{M}_{1}^{2}\\ {m}_{2}={m}_{2}^{1}+{m}_{2}^{2};{M}_{2}={M}_{2}^{1}+{M}_{2}^{2}\end{array}\)

What we will write:

\(\begin{array}{}{m}_{\mathrm{constituant}}=\sum _{\begin{array}{}\mathrm{nb}\mathrm{phase}\mathrm{du}\\ \mathrm{constituant}\end{array}}{m}_{\mathrm{constituant}}^{\mathrm{phase}}\\ {M}_{\mathrm{constituant}}=\sum _{\begin{array}{}\mathrm{nb}\mathrm{phase}\mathrm{du}\\ \mathrm{constituant}\end{array}}{M}_{\mathrm{constituant}}^{\mathrm{phase}}\end{array}\)

In applications, for example, we could have:

  • 2 components: air and water,

  • 2 phases for water,

  • 1 phase for the air.

We would then have: \({m}_{1}^{1}\text{et}{M}_{1}^{1}\): mass supply and liquid water flow

\({m}_{1}^{2}\text{et}{M}_{1}^{2}\): mass supply and steam flow

\({m}_{2}^{1}\text{et}{M}_{2}^{1}\): mass input and dry air flow

\({m}_{2}^{2}\text{et}{M}_{2}^{2}\): nonexistent

  • Pressures:

Since we consider that there can be two components other than the solid, there are two mass conservation equations, and therefore two associated multipliers, i.e. two pressures \({p}_{1}\text{et}{p}_{2}\). No assumptions are made about what these two pressures \({p}_{1}\text{et}{p}_{2}\) mean. It will depend on the laws of behavior and how to write them. For example, you can choose:

\(\begin{array}{}{p}_{1}=\text{pression capillaire}(\text{p}(\text{gaz})-\text{p}(\text{liquide}))\\ {p}_{2}=\text{pression de gaz}(\text{vapeur + air})\end{array}\)

2.2.2.4. Thermal quantities#

  • Non-convected heat \(Q\text{'}\) (see below) (unit: Joule),

  • The mass enthalpies of the components \({{h}^{m}}_{c}^{p}\) (unit: joule/kelvin/kilogram),

  • Heat flow: \(q\) (unit: J/s/square meter).

2.2.3. External data#

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

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

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

2.3. Particle derivatives, volume and mass densities#

Our description of the environment is Lagrangian in relation to the skeleton. We will find in [bib1] a definition of the concept of skeleton: « the matrix (solid part+occluded porosity) constituting the material part of the skeleton and the connected porous space of the elementary volume in question constitute the material point of the skeleton or particle of 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 \(\dot{a}=\frac{{d}^{s}a}{\mathrm{dt}}\) the time derivative in skeletal movement:

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

\(\dot{a}\) is called a particulate derivative and is often referred to as \(\frac{\mathrm{da}}{\mathrm{dt}}\) (for example in [bib1]). 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 \({x}_{S}(t\text{'})\) at another moment \(t\text{'}\).

Let \(A=\underset{\Omega }{\int }ad\Omega\) then be a quantity linked to a volumic density \(a\), which density is itself carried in part by solid grains and by fluids. Let \({{a}^{m}}_{c}^{p}\) be the mass density of \(a\) carried by the fluid phase \(p\) of the constituent \(c\) and be \({a}_{s}\) the volumic density of \(a\) linked to solid grains. All these definitions are finally equivalent to writing:

\(A={\int }_{\Omega }\mathrm{ad}\Omega ={A}_{s}+{A}_{\mathrm{fl}}={\int }_{\Omega }{a}_{s}d\Omega +{\int }_{\Omega }{a}_{\mathrm{fl}}d\Omega ={\int }_{\Omega }({a}_{s}+\sum _{p,c}{\rho }_{c}^{p}{\mathrm{jS}}^{p}{{a}^{m}}_{c}^{p})d\Omega\) eq 2.3-1

When following [bib1], we note \(\frac{{d}^{\mathrm{fl}}{A}_{\mathrm{fl}}}{\mathrm{dt}}\) the derivative of \({A}_{\mathrm{fl}}\) if we follow \(\Omega\) in fluid motion and \(\frac{{d}^{s}{A}_{s}}{\mathrm{dt}}\) the derivative of \({A}_{s}\) if we follow \(\Omega\) in skeletal motion.

We then define:

\(\frac{\mathrm{DA}}{\mathrm{Dt}}=\frac{{d}^{s}{A}_{s}}{\mathrm{dt}}+\frac{{d}^{\mathrm{fl}}{A}_{\mathrm{fl}}}{\mathrm{dt}}=\frac{{d}^{s}}{\mathrm{dt}}\int {}_{\Omega }\text{}{a}_{s}d\Omega +\frac{{d}^{\mathrm{fl}}}{\mathrm{dt}}\int {}_{\Omega }\text{}\sum _{p,c}{\rho }_{c}^{p}\phi {S}^{p}{a}^{{m}_{c}^{p}}d\Omega\) eq 2.3-2

Density \({{a}^{m}}_{c}^{p}\) is transported with a relative speed of \(({v}_{c}^{p}-{v}_{s})\) compared to the skeleton. Given the definition of \(\dot{a}=\frac{{d}^{s}a}{\mathrm{dt}}\), and the definition \({w}_{c}^{p}={\rho }_{c}^{p}\phi {S}^{p}({v}_{c}^{p}-{v}_{s})\), it is easy to see that the total derivative of \(A\) with respect to time is finally written as:

\(\frac{\mathrm{DA}}{\mathrm{Dt}}=\int {}_{\Omega }\text{}(\dot{a}+\sum _{p,c}\text{Div}({{a}^{m}}_{c}^{p}{w}_{c}^{p}))d\Omega\) eq 2.3-3

Note:

Insofar as we hypothesized the small movements of the skeleton, \(\dot{a}=\frac{{d}^{s}a}{\mathrm{dt}}\) may be confused with the partial derivative with respect to time \(\frac{\partial a}{\partial t}\) and \({v}_{s}\) may be considered zero. Likewise, in the rest of the note we will confuse the Lagrangian and Eulerian representations of flows, \({M}_{c}^{p}\) and \({w}_{c}^{p}\) .