4. Calculation of generalized stresses#

In this chapter, we specify how the relationships described in Chapter 3 are integrated. Even more precisely, we give the expressions of generalized constraints in the sense of document [R7.01.10] [4] when the behavioral laws THM are called for option RAPH_MECA in the sense of document [R5.03.01] [3]. In order for this document to follow the programming order more closely, we will consider two cases: the case without dissolved air and the case with.

The generalized constraints are:

\({\sigma }^{\text{'}},{\sigma }_{p};{m}_{{w}_{}},{\mathrm{M}}_{{\mathrm{w}}_{}},{h}_{w}^{m};{m}_{{\text{vp}}_{}},{\mathrm{M}}_{\text{vp}},{h}_{\text{vp}}^{m};{m}_{{\text{as}}_{}},{\mathrm{M}}_{{\text{as}}_{}},{h}_{\text{as}}^{m};{m}_{\text{ad}},{\mathrm{M}}_{\text{ad}},{h}_{\text{ad}}^{m};Q\text{'},\mathrm{q}\)

The generalized deformations, from which the generalized stresses are calculated are:

\(u,\varepsilon (u);{p}_{c},\nabla {p}_{c};{p}_{\text{gz}},\nabla {p}_{\text{gz}};T,\nabla T\)

The internal variables that we selected are:

In the case without steam:

\(\phi ,{\rho }_{w},{S}_{\text{lq}}\)

In the case of steam and without dissolved air:

\(\varphi ,{\rho }_{w},{p}_{\text{vp}},{S}_{\text{lq}}\)

In the case of dissolved steam and air:

\(\phi ,{\rho }_{w},{p}_{\text{vp}},{p}_{\text{ad}},{S}_{\text{lq}}\)

In this chapter, we adopt the usual Aster notations, namely the + indices for the values of the quantities at the end of the time step and the indices - for the quantities at the beginning of the time step.

Thus, the known quantities are:

the stresses, generalized deformations and internal variables at the beginning of the time step:
\({\sigma \text{'}}^{\text{-}},{\sigma }_{p}^{\text{-}};{m}_{w}^{\text{-}},{\mathrm{M}}_{\mathrm{w}}^{\text{-}},{h}_{w}^{{m}^{\text{-}}};{m}_{\text{vp}}^{\text{-}},{\mathrm{M}}_{\text{vp}}^{\text{-}},{h}_{\text{vp}}^{{m}^{\text{-}}};{m}_{\text{as}}^{\text{-}},{\mathrm{M}}_{\text{as}}^{\text{-}},{h}_{\text{as}}^{{m}^{\text{-}}};{m}_{\text{ad}}^{\text{-}},{\mathrm{M}}_{\text{ad}}^{\text{-}},{h}_{\text{ad}}^{{m}^{\text{-}}};Q{\text{'}}^{\text{-}},{\mathrm{q}}^{\text{-}}\)
\({u}^{\text{-}},\varepsilon ({u}^{\text{-}});{p}_{c}^{\text{-}},\nabla {p}_{c}^{\text{-}};{p}_{\text{gz}}^{\text{-}},\nabla {p}_{\text{gz}}^{\text{-}};{T}^{\text{-}},\nabla {T}^{\text{-}}\)
\({\phi }^{\text{-}},{\rho }_{w}^{\text{-}},{p}_{\text{vp}}^{\text{-}},{p}_{\text{ad}}^{\text{-}}\)
the generalized deformations at the end of the time step:
\({\mathrm{u}}^{\text{+}},\varepsilon ({\mathrm{u}}^{\text{+}});{p}_{c}^{\text{+}},\mathrm{\nabla }{p}_{c}^{\text{+}};{p}_{\text{gz}}^{\text{+}},\mathrm{\nabla }{p}_{\text{gz}}^{\text{+}};{T}^{\text{+}},\mathrm{\nabla }{T}^{\text{+}}\)
Unknown quantities are the constraints, and internal variables at the end of the time step:
\({\sigma \text{'}}^{\text{+}},{\sigma }_{p}^{\text{+}};{m}_{w}^{\text{+}},{\mathrm{M}}_{\mathrm{w}}^{\text{+}},{h}_{w}^{{m}^{\text{+}}};{m}_{\text{vp}}^{\text{+}},{\mathrm{M}}_{\text{vp}}^{\text{+}},{h}_{\text{vp}}^{{m}^{\text{+}}};{m}_{\text{as}}^{\text{+}},{\mathrm{M}}_{\text{as}}^{\text{+}},{h}_{\text{as}}^{{m}^{\text{+}}};{m}_{\text{ad}}^{\text{+}},{\mathrm{M}}_{\text{ad}}^{\text{+}},{h}_{\text{ad}}^{{m}^{\text{+}}};Q{\text{'}}^{\text{+}},{\mathrm{q}}^{\text{+}}\)
\({\phi }^{\text{+}},{\rho }_{w}^{\text{+}},{p}_{\text{vp}}^{\text{+}},{p}_{\text{ad}}^{\text{+}}\)

4.1. Case without dissolved air#

4.1.1. Calculation of the porosity and the density of the fluid#

4.1.1.1. Porosity calculation: isotropic case#

The first thing to do, of course, is to calculate the saturation at the end of time step \({S}_{\text{lq}}^{\text{+}}\text{=}{S}_{\text{lq}}({p}_{c}^{\text{+}})\). The porosity is found by integrating the equation [éq 3.2.1-1] over the time step.

We then obtain:

\(\text{ln}\left\{\frac{b\text{-}{\varphi }^{\text{+}}}{b\text{-}{\varphi }^{\text{-}}}\right\}\text{=}\left\{\text{-}({\varepsilon }_{v}^{\text{+}}\text{-}{\varepsilon }_{v}^{\text{-}})\text{+}3{\alpha }_{0}({T}^{\text{+}}\text{-}{T}^{\text{-}})\text{-}\frac{({p}_{\mathrm{gz}}^{\text{+}}\text{-}{p}_{\mathrm{gz}}^{\text{-}})\text{-}{S}_{\mathrm{lq}}^{\text{+}}({p}_{c}^{\text{+}}\text{-}{p}_{c}^{\text{-}})}{{K}_{S}}\right\}\) eq 4.1.1-1

4.1.1.2. Calculation of porosity: transverse isotropic case#

We then obtain:

\(({\varphi }^{\text{+}}\text{-}{\varphi }^{\text{-}})\text{=}B:({\varepsilon }^{\text{+}}\text{-}{\varepsilon }^{\text{-}})\text{-}{\varphi }^{\text{-}}({\varepsilon }_{\text{v}}^{\text{+}}\text{-}{\varepsilon }_{\text{v}}^{\text{-}})\text{-}3{\alpha }_{\varphi }({T}^{\text{+}}\text{-}{T}^{\text{-}})\text{+}\frac{({p}_{\text{gz}}^{\text{+}}\text{-}{p}_{\text{gz}}^{\text{-}})\text{-}{S}_{\text{lq}}^{\text{+}}({p}_{\text{c}}^{\text{+}}\text{-}{p}_{\text{c}}^{\text{-}})}{{M}_{\varphi }}\) eq 4.1.1-2

4.1.1.3. Calculation of the density of the fluid#

The density of the liquid is found by integrating the equation [éq 3.2.3.1-1] over the time step.

This results in:

\(\text{ln}\left(\frac{{\rho }_{w}^{\text{+}}}{{\rho }_{w}^{\text{-}}}\right)\text{=}\frac{{p}_{\text{gz}}^{\text{+}}\text{-}{p}_{\text{gz}}^{\text{-}}\text{-}{p}_{c}^{\text{+}}\text{+}{p}_{c}^{\text{-}}}{{K}_{w}}\text{-}3{\alpha }_{w}\left({T}^{\text{+}}\text{-}{T}^{\text{-}}\right)\) eq 4.1.1-3

4.1.2. Calculation of expansion coefficients#

It is important to note that the differential thermal expansion coefficient is recalculated from the porosity evaluated at the end of the time step. Taking this into account, we pose:

\({\alpha }_{\phi }^{\text{+}}\text{=}\frac{(B\text{-}{\phi }^{\text{+}}\delta )\mathrm{:}{\alpha }_{0}}{3}\) eq 4.1.2-1

Note:

In the isotropic case [éq 4.1.2-1] becomes:

\({\alpha }_{\phi }^{\text{+}}\text{=}(b\text{-}{\phi }^{\text{+}}){\alpha }_{0}\)

It is then a simple application of the formulas [éq 3.2.4.3-2] and [éq 3.2.4.3-3], which are evaluated at the end of the time step:

\({\alpha }_{\mathit{vp}}^{{m}^{\text{+}}}\text{=}{\alpha }_{\mathit{as}}^{{m}^{\text{+}}}\text{=}{\alpha }_{\mathit{gz}}^{{m}^{\text{+}}}\text{=}(1\text{-}{S}_{\mathit{lq}}^{\text{+}}){\alpha }_{\phi }^{\text{+}}\text{+}\frac{{\phi }^{\text{+}}(1\text{-}{S}_{\mathit{lq}}^{\text{+}})}{3{T}^{\text{+}}}\) eq 4.1.2-2

\({\alpha }_{w}^{{m}^{\text{+}}}\text{=}{S}_{\text{lq}}^{\text{+}}{\alpha }_{\phi }^{\text{+}}\text{+}{\alpha }_{\text{lq}}{\phi }^{\text{+}}{S}_{\text{lq}}^{\text{+}}\) eq 4.1.2-3

4.1.3. Calculation of fluid enthalpies#

The fluid enthalpies are calculated by integrating the equations [éq 3.2.4.1-1], [éq 3.2.4.2-1], [éq3.2.4.2-2].

\({h}_{w}^{{m}^{\text{+}}}={h}_{w}^{{m}^{\text{-}}}+{C}_{w}^{p}({T}^{\text{+}}-{T}^{\text{-}})+\frac{(1-3{\alpha }_{w}{T}^{\text{+}})}{{\rho }_{w}^{\text{+}}}({p}_{\text{gz}}^{\text{+}}-{p}_{\text{gz}}^{\text{-}}-{p}_{c}^{\text{+}}\text{+}{p}_{c}^{\text{-}})\) eq 4.1.3-1

\({h}_{\text{vp}}^{{m}^{\text{+}}}\text{=}{h}_{\text{vp}}^{{m}^{\text{-}}}\text{+}{C}_{\text{vp}}^{p}({T}^{\text{+}}\text{-}{T}^{\text{-}})\) eq 4.1.3-2

\({h}_{\text{as}}^{{m}^{\text{+}}}\text{=}{h}_{\text{as}}^{{m}^{\text{-}}}\text{+}{C}_{\text{as}}^{p}({T}^{\text{+}}\text{-}{T}^{\text{-}})\) eq 4.1.3-3

4.1.4. Vapor and air pressures#

Starting from the relationship [éq 3.2.6-4] in which we carry the law of behavior of ideal gases [éq3.2.3.2-1], we find \(\frac{{\text{dp}}_{\text{vp}}}{{p}_{\text{vp}}}=\frac{{M}_{\text{vp}}^{\text{ol}}}{\text{RT}}(\frac{1}{{\rho }_{w}}{\text{dp}}_{\text{gz}}-\frac{1}{{\rho }_{w}}{\text{dp}}_{c}+({h}_{\text{vp}}^{m}-{h}_{w}^{m})\frac{\text{dT}}{T})\) which we integrate by a path first at constant temperature (we then consider the density of water constant), then from \({T}^{\text{-}}\) to \({T}^{\text{+}}\) at constant pressures.

\(\text{ln}(\frac{{p}_{\text{vp}}^{\text{+}}}{{p}_{\text{vp}}^{\text{-}}})=\frac{{M}_{\text{vp}}^{\text{ol}}}{{\text{RT}}^{\text{+}}}\frac{1}{{\rho }_{w}^{\text{+}}}\left[({p}_{{\text{gz}}^{\text{+}}}-{p}_{\text{gz}}^{\text{-}})-({p}_{{c}^{\text{+}}}-{p}_{c}^{\text{-}})\right]+\frac{{M}_{\text{vp}}^{\text{ol}}}{R}{\int }_{{T}^{\text{-}}}^{{T}^{\text{+}}}({h}_{\text{vp}}^{m}-{h}_{w}^{m})\frac{\text{dT}}{{T}^{2}}\)

The first term corresponds to the path at constant temperature, the second to the path at constant pressures. Using the definitions [éq 3.2.4.1-1] and [éq 3.2.4.2-1] of enthalpies, we see that for an evolution at constant pressures:

\(\frac{{h}_{\text{vp}}^{m}-{h}_{w}^{m}}{{T}^{2}}=\frac{{h}_{\text{vp}}^{{m}^{\text{-}}}-{h}_{w}^{{m}^{\text{-}}}}{{T}^{2}}+\frac{({C}_{\text{vp}}^{p}-{C}_{w}^{p})(T-{T}^{\text{-}})}{{T}^{2}}\)

Therefore, for such a path, we have:

\({\int }_{{T}^{\text{-}}}^{{T}^{\text{+}}}({h}_{\text{vp}}^{m}-{h}_{w}^{m})\frac{\text{dT}}{{T}^{2}}=({h}_{\text{vp}}^{{m}^{\text{-}}}-{h}_{w}^{{m}^{\text{-}}})(\frac{1}{{T}^{\text{-}}}-\frac{1}{{T}^{\text{+}}})\text{+}({C}_{\text{vp}}^{p}-{C}_{w}^{p})(\text{ln}(\frac{{T}^{\text{+}}}{{T}^{\text{-}}})+{T}^{\text{-}}(\frac{1}{{T}^{\text{+}}}-\frac{1}{{T}^{\text{-}}}))\)

Or finally:

\(\begin{array}{}\text{ln}(\frac{{p}_{\text{vp}}^{\text{+}}}{{p}_{\text{vp}}^{\text{-}}})=\frac{{M}_{\text{vp}}^{\text{ol}}}{{\text{RT}}^{\text{+}}}\frac{1}{{\rho }_{w}^{\text{+}}}\left[({p}_{{\text{gz}}^{\text{+}}}-{p}_{\text{gz}}^{\text{-}})-({p}_{{c}^{\text{+}}}-{p}_{c}^{\text{-}})\right]\text{+}\\ \frac{{M}_{\text{vp}}^{\text{ol}}}{R}({h}_{\text{vp}}^{{m}^{\text{-}}}-{h}_{w}^{{m}^{\text{-}}})(\frac{1}{{T}^{\text{-}}}-\frac{1}{{T}^{\text{+}}})\text{+}\frac{{M}_{\text{vp}}^{\text{ol}}}{R}({C}_{\text{vp}}^{p}-{C}_{w}^{p})(\text{ln}(\frac{{T}^{\text{+}}}{{T}^{\text{-}}})+\frac{{T}^{\text{-}}}{{T}^{\text{+}}}-1)\end{array}\) eq 4.1.4-1

We can then calculate the densities of steam and air by the relationships [éq3.2.3.2-1] and [éq3.2.3.2-2]:

\({\rho }_{\text{vp}}^{\text{+}}\text{=}\frac{{M}_{\text{vp}}^{\text{ol}}}{R}\frac{{p}_{\text{vp}}^{\text{+}}}{{T}^{\text{+}}}\) eq 4.1.4-2

\({\rho }_{\text{as}}^{\text{+}}=\frac{{M}_{\text{as}}^{\text{ol}}}{R}\frac{({p}_{\text{gz}}^{+}-{p}_{\text{vp}}^{+})}{{T}^{+}}\) eq 4.1.4-3

4.1.5. Calculation of mass inputs#

The equations [éq 3.2.2-1] give zero mass inputs at time 0. We write the equations [éq 3.2.2-1] incrementally:

\(\begin{array}{}{m}_{w}^{\text{+}}\text{=}{m}_{w}^{\text{-}}\text{+}{\rho }_{w}^{\text{+}}(1\text{+}{\varepsilon }_{V}^{\text{+}}){\varphi }^{\text{+}}{S}_{\text{lq}}^{\text{+}}\text{-}{\rho }_{w}^{\text{-}}(1\text{+}{\varepsilon }_{V}^{\text{-}}){\varphi }^{\text{-}}{S}_{\text{lq}}^{\text{-}}\\ {m}_{\text{as}}^{\text{+}}\text{=}{m}_{\text{as}}^{\text{-}}\text{+}{\rho }_{\text{as}}^{\text{+}}(1\text{+}{\varepsilon }_{V}^{\text{+}}){\varphi }^{\text{+}}(1\text{-}{S}_{\text{lq}}^{\text{+}})\text{-}{\rho }_{\text{as}}^{\text{-}}(1\text{+}{\varepsilon }_{V}^{-\text{}}){\varphi }^{\text{-}}(1\text{-}{S}_{\text{lq}}^{\text{-}})\\ {m}_{\text{vp}}^{\text{+}}\text{=}{m}_{\text{vp}}^{\text{-}}\text{+}{\rho }_{\text{vp}}^{\text{+}}(1\text{+}{\varepsilon }_{V}^{\text{+}}){\varphi }^{\text{+}}(1\text{-}{S}_{\text{lq}}^{\text{+}})\text{-}{\rho }_{\text{vp}}^{\text{-}}(1+{\varepsilon }_{V}^{\text{-}}){\varphi }^{\text{-}}(1\text{-}{S}_{\text{lq}}^{\text{-}})\end{array}\) eq 4.1.5-1

4.1.6. Calculation of heat capacity and heat Q”#

We now have all the elements to apply the formula [éq 3.2.4.3-5] to the end of the time step:

\({C}_{\sigma }^{{0}^{\text{+}}}\text{=}(1\text{-}{\phi }^{\text{+}}){\rho }_{s}{C}_{\sigma }^{s}\text{+}{S}_{\text{lq}}^{\text{+}}{\phi }^{\text{+}}{\rho }_{w}^{\text{+}}{C}_{w}^{p}\text{+}(1\text{-}{S}_{\text{lq}}^{\text{+}}){\phi }^{\text{+}}({\rho }_{\text{vp}}^{\text{+}}{C}_{\text{vp}}^{p}\text{+}{\rho }_{\text{as}}^{\text{+}}{C}_{\text{as}}^{p})\) eq 4.1.6-1

Of course we use [éq 3.2.4.3-4] which gives:

\({C}_{\varepsilon }^{0\text{+}}\text{=}{C}_{\sigma }^{0\text{+}}\text{-}{T}^{\text{+}}({C}_{0}\mathrm{:}{\alpha }_{0})\mathrm{:}{\alpha }_{0}\) eq 4.1.6-2

Note:

In the isotropic case [éq 4.1.6-2] becomes:

\({C}_{\varepsilon }^{0\text{+}}\text{=}{C}_{\sigma }^{0\text{+}}\text{-}9{T}^{\text{+}}{K}_{0}{\alpha }_{0}^{2}\)

Although the heat variation \(\delta {Q}^{\text{'}}\) is not a total differential, it is nevertheless legal to integrate it over the time step and we obtain by integrating [éq 3.2.4.3-1].

\(Q{\text{'}}^{\text{+}}\text{=}Q{\text{'}}^{\text{-}}\text{+}({C}_{0}\text{:}{\alpha }_{0})\text{:}({\varepsilon }^{\text{+}}\text{-}{\varepsilon }^{\text{-}}){T}^{}\text{+}3{\alpha }_{\text{lq}}^{{m}^{\text{+}}}{T}^{}({p}_{c}^{\text{+}}\text{-}{p}_{{c}_{}}^{\text{-}})\text{-}(3{\alpha }_{\text{gz}}^{{m}^{\text{+}}}+3{\alpha }_{\text{lq}}^{{m}^{\text{+}}}){T}^{}({p}_{{\text{gz}}_{}}^{\text{+}}\text{-}{p}_{{\text{gz}}_{}}^{\text{-}})+{C}_{\varepsilon }^{{0}^{\text{+}}}({T}^{\text{+}}\text{-}{T}^{\text{-}})\) eq 4.1.6-3

where we noted: \({T}^{}\text{=}\frac{{T}^{\text{+}}\text{+}{T}^{\text{-}}}{2}\). Here we have chosen a « middle point » formula for the temperature variable.

Note:

In the isotropic case [éq 4.1.6-3] becomes:

\(Q{\text{'}}^{\text{+}}\text{=}Q{\text{'}}^{\text{-}}\text{+}(3{K}_{0}{\alpha }_{0})({\varepsilon }_{V}^{\text{+}}\text{-}{\varepsilon }_{V}^{\text{-}}){T}^{}\text{+}3{\alpha }_{\text{lq}}^{{m}^{\text{+}}}{T}^{}({p}_{c}^{\text{+}}\text{-}{p}_{{c}_{}}^{\text{-}})\text{-}(3{\alpha }_{\text{gz}}^{{m}^{\text{+}}}+3{\alpha }_{\text{lq}}^{{m}^{\text{+}}}){T}^{}({p}_{{\text{gz}}_{}}^{\text{+}}\text{-}{p}_{{\text{gz}}_{}}^{\text{-}})+{C}_{\varepsilon }^{{0}^{\text{+}}}({T}^{\text{+}}\text{-}{T}^{\text{-}})\)

4.1.7. Calculation of mechanical stresses#

The calculation of the effective stresses is done by invoking the incremental laws of mechanics chosen by the user. We integrate on the [éq 3.2.8-2] time step and we have:

\({\sigma }_{p}^{\text{+}}\text{=}{\sigma }_{p}^{\text{-}}\text{-}B({p}_{\mathit{gz}}^{\text{+}}\text{-}{p}_{\mathit{gz}}^{\text{-}})\text{+}B{S}_{\mathit{lq}}^{\text{+}}({p}_{c}^{\text{+}}\text{-}{p}_{c}^{\text{-}})\) eq 4.1.7-1

In the isotropic case we have \(B\text{=}b\text{.}1\), \({\sigma }_{p}^{\text{+}}\text{=}{\sigma }_{p}^{\text{+}}\text{.}1\) and \({\sigma }_{p}^{\text{-}}\text{=}{\sigma }_{p}^{\text{-}}\text{.}1\)

4.1.8. Calculation of water and heat flows#

It is of course necessary to calculate all the diffusion coefficients:

Fick’s coefficient \({F}^{\text{+}}\text{=}F({T}^{\text{+}},{p}_{c}^{\text{+}},{p}_{\text{gz}}^{\text{+}})\)

The \({\lambda }^{T\text{+}}\text{=}{\lambda }_{\phi }^{T}({\varphi }^{\text{+}})\text{.}{\lambda }_{S}^{T}({S}_{\text{lq}}^{+})\text{.}{\lambda }_{T}^{T}({T}^{\text{+}})+{\lambda }_{\text{cte}}^{T}\) thermal diffusivity tensor

Hydraulic permeability and conductivity tensors: \({\lambda }_{\text{lq}}^{{H}^{\text{+}}}\text{=}\frac{{K}^{\text{int}}({\varphi }^{\text{+}})\text{.}{k}_{w}^{\text{rel}}({S}_{\text{lq}}^{\text{+}})}{{\mu }_{w}({T}^{\text{+}})}{\lambda }_{\text{gz}}^{{H}^{\text{+}}}\text{=}\frac{{K}^{\text{int}}({\varphi }^{\text{+}})\text{.}{k}_{\text{gz}}^{\text{rel}}({S}_{\text{lq}}^{\text{+}},{p}_{\text{gz}}^{\text{+}})}{{\mu }_{\text{gz}}({T}^{\text{+}})}\)

In the isotropic case, \({K}^{\text{int}}\text{=}{K}^{\text{int}}\text{.}1\), \({\lambda }^{T}\text{=}{\lambda }^{T}\text{.}1\),, \({\lambda }_{T}^{T}(T)\text{=}{\lambda }_{T}^{T}(T)\text{.}1\), and \({\lambda }_{\mathit{cte}}^{T}\text{=}{\lambda }_{\mathit{cte}}^{T}\text{.}1\)

Vapour concentration: \({C}_{\text{vp}}^{\text{+}}\text{=}\frac{{p}_{\text{vp}}^{\text{+}}}{{p}_{\text{vp}}^{\text{+}}}\)

All that remains is to apply the formulas [éq 3.2.5.1-1], [éq 3.2.5.2-15], [], [éq 3.2.5.2-16] and [éq3.2.5.2-17] to find:

\({\mathrm{q}}^{\text{+}}\text{=}\text{-}{\lambda }^{{T}^{\text{+}}}\mathrm{\nabla }{T}^{\text{+}}\) eq 4.1.8-1

\(\frac{{\mathrm{M}}_{\text{as}}^{\text{+}}}{{\rho }_{\text{as}}^{\text{+}}}\text{=}{\lambda }_{\text{gz}}^{{H}^{\text{+}}}\left[\text{-}\mathrm{\nabla }{p}_{\text{gz}}^{\text{+}}\text{+}({\rho }_{{\text{as}}_{}}^{\text{+}}\text{+}{\rho }_{\text{vp}}^{\text{+}}){\mathrm{F}}^{\mathrm{m}}\right]\text{+}{C}_{\text{vp}}^{\text{+}}{F}_{\text{vp}}^{\text{+}}\mathrm{\nabla }{C}_{\text{vp}}^{\text{+}}\) eq 4.1.8-2

\(\frac{{\mathrm{M}}_{\text{vp}}^{\text{+}}}{{\rho }_{\text{vp}}^{\text{+}}}\text{=}{\lambda }_{\text{gz}}^{{H}^{\text{+}}}\left[\text{-}\mathrm{\nabla }{p}_{\text{gz}}^{\text{+}}\text{+}({\rho }_{\text{as}}^{\text{+}}\text{+}{\rho }_{\text{vp}}^{\text{+}}){\mathrm{F}}^{\mathrm{m}}\right]\text{-}(1\text{-}{C}_{\text{vp}}^{\text{+}}){F}_{\text{vp}}^{\text{+}}\mathrm{\nabla }{C}_{\text{vp}}^{\text{+}}\) eq 4.1.8-3

\(\frac{{\mathrm{M}}_{w}^{\text{+}}}{{\rho }_{w}^{\text{+}}}\text{=}{\lambda }_{\text{lq}}^{{H}^{\text{+}}}\left[\text{-}\mathrm{\nabla }{p}_{\text{lq}}^{\text{+}}\text{+}{\rho }_{w}^{\text{+}}{F}^{m}\right]\) eq 4.1.8-4

In isotropic cases, \({\lambda }_{{\text{lq}}_{}}^{H}\text{=}{\lambda }_{{\text{lq}}_{}}^{H}\text{.}1\) and \({\lambda }_{{\text{gz}}_{}}^{H}\text{=}{\lambda }_{{\text{gz}}_{}}^{H}\text{.}1\).

4.2. Case with dissolved air#

4.2.1. Porosity calculation#

4.2.1.1. Porosity calculation: isotropic case#

In the same way, the first thing we need to do is calculate the saturation at the end of time step \({S}_{\text{lq}}^{\text{+}}\text{=}{S}_{\text{lq}}\left({p}_{c}^{\text{+}}\right)\). The porosity is found by integrating the equation [éq 3.2.1-1] over the time step. We therefore recall that:

\(\text{ln}\left\{\frac{b\text{-}{\phi }^{\text{+}}}{b\text{-}{\phi }^{\text{-}}}\right\}\text{=}\left\{\text{-}({\varepsilon }_{v}^{\text{+}}\text{-}{\varepsilon }_{v}^{\text{-}})\text{+}3{\alpha }_{0}({T}^{\text{+}}\text{-}{T}^{\text{-}})\text{-}\frac{({p}_{\mathrm{gz}}^{\text{+}}\text{-}{p}_{\mathrm{gz}}^{\text{-}})\text{-}{S}_{\mathrm{lq}}^{\text{+}}({p}_{c}^{\text{+}}\text{-}{p}_{c}^{\text{-}})}{{K}_{S}}\right\}\)

4.2.1.2. Calculation of porosity: transverse isotropic case#

In the same way, the first thing we need to do is calculate the saturation at the end of time step \({S}_{\text{lq}}^{\text{+}}\text{=}{S}_{\text{lq}}({p}_{c}^{\text{+}})\). The porosity is found by integrating the equation [éq 3.2.1-1] over the time step. We therefore recall that:

\(({\varphi }^{\text{+}}\text{-}{\varphi }^{\text{-}})\text{=}B\mathrm{:}({\varepsilon }^{\text{+}}\text{-}{\varepsilon }^{\text{-}})\text{-}{\varphi }^{\text{-}}({\varepsilon }_{\text{v}}^{\text{+}}\text{-}{\varepsilon }_{\text{v}}^{\text{-}})\text{-}3{\alpha }_{\varphi }({T}^{\text{+}}\text{-}{T}^{\text{-}})\text{+}\frac{({p}_{\text{gz}}^{\text{+}}\text{-}{p}_{\text{gz}}^{\text{-}})\text{-}{S}_{\text{lq}}^{\text{+}}({p}_{\text{c}}^{\text{+}}\text{-}{p}_{\text{c}}^{\text{-}})}{{M}_{\varphi }}\)

4.2.2. Calculation of expansion coefficients#

In the same way, the differential thermal expansion coefficient is recalculated from the porosity evaluated at the end of the time step. Taking this into account, we pose:

\({\alpha }_{\varphi }^{\text{+}}\text{=}\frac{(B\text{-}{\varphi }^{\text{+}}\delta )\mathrm{:}{\alpha }_{0}}{3}\) eq 4.2.2-1

Note:

In the isotropic case [éq 4.2.2-1] becomes:

\({\alpha }_{\varphi }^{\text{+}}\text{=}(b\text{-}{\varphi }^{\text{+}}){\alpha }_{0}\)

It is then a simple application of the formulas [éq 3.2.4.3-2] and [éq 3.2.4.3-3], which are evaluated at the end of the time step:

\({\alpha }_{\text{vp}}^{{m}^{\text{+}}}\text{=}{a}_{\text{as}}^{m\text{+}}\text{=}{a}_{\text{gz}}^{m\text{+}}\text{=}\left(1\text{-}{S}_{\text{lq}}^{\text{+}}\right){\alpha }_{\phi }^{\text{+}}\text{+}\frac{{\phi }^{\text{+}}\left(1\text{-}{S}_{\text{lq}}^{\text{+}}\right)}{{\mathrm{3T}}^{\text{+}}}\) eq 4.2.2-2

\({\alpha }_{w}^{{m}^{\text{+}}}\text{=}{S}_{\text{lq}}^{\text{+}}{\alpha }_{\phi }^{\text{+}}\text{+}{\alpha }_{\text{lq}}{\phi }^{\text{+}}{S}_{\text{lq}}^{\text{+}}\) eq 4.2.2-3

\({\alpha }_{\text{ad}}^{{m}^{\text{+}}}\text{=}{S}_{\text{lq}}^{\text{+}}{\alpha }_{\phi }^{\text{+}}\text{+}\frac{{\phi }^{\text{+}}{S}_{\text{lq}}^{\text{+}}}{{\mathrm{3T}}^{\text{+}}}\) eq 4.2.2-4

4.2.3. Calculation of vapour, dissolved and dry air pressures and densities#

Starting from the relationship [éq 3.2.6-4] in which we carry the law of behavior of ideal gases [éq3.2.3.2-1], we find:

\(\frac{{\text{dp}}_{\text{vp}}}{{p}_{\text{vp}}}=\frac{{M}_{\text{vp}}^{\text{ol}}}{\text{RT}}(\frac{1}{{\rho }_{w}}{\text{dp}}_{w}\text{+}({h}_{\text{vp}}^{m}-{h}_{w}^{m})\frac{\text{dT}}{T})\) eq 4.2.3-1

Unlike the case without dissolved air \({p}_{w}\) is no longer known:

\({p}_{w}={p}_{\text{lq}}-{p}_{\text{ad}}={p}_{\text{gz}}-{p}_{c}-\frac{\text{RT}}{{K}_{H}}{p}_{\text{as}}={p}_{\text{gz}}-{p}_{c}-\frac{\text{RT}}{{K}_{H}}({p}_{\text{gz}}-{p}_{\text{vp}})\)

So:

\({\text{dp}}_{w}={\text{dp}}_{\text{gz}}-{\text{dp}}_{c}-\frac{\text{RT}}{{K}_{H}}({\text{dp}}_{\text{gz}}-{\text{dp}}_{\text{vp}})-\frac{R}{{K}_{H}}({p}_{\text{gz}}-{p}_{\text{vp}})\text{dT}\) eq 4.2.3-2

We integrate [éq 4.2.3.1] by including [éq 4.2.3.2] by a path first at constant temperature (we then consider the density of water constant), then from \({T}^{\text{-}}\) to \({T}^{\text{+}}\) at constant pressures. In the end we get:

\(\begin{array}{}\text{ln}(\frac{{p}_{\text{vp}}^{\text{+}}}{{p}_{\text{vp}}^{\text{-}}})=\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{w}^{\text{-}}}(\frac{1}{{\text{RT}}^{\text{+}}}-\frac{1}{{K}_{H}})({p}_{{\text{gz}}^{\text{+}}}-{p}_{\text{gz}}^{\text{-}})\text{+}\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{w}^{\text{-}}{K}_{H}}({p}_{{\text{vp}}^{\text{+}}}-{p}_{\text{vp}}^{\text{-}})-\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{w}^{\text{-}}{\text{RT}}^{\text{+}}}({p}_{{c}^{\text{+}}}-{p}_{c}^{\text{-}})\text{+}\\ \frac{{M}_{\text{vp}}^{\text{ol}}R}{{\rho }_{w}^{\text{-}}{K}_{H}}({p}_{{\text{vp}}^{\text{+}}}-{p}_{\text{gz}}^{\text{+}})\text{ln}(\frac{{T}^{\text{+}}}{{T}^{\text{-}}})\text{+}\frac{{M}_{\text{vp}}^{\text{ol}}}{R}{\int }_{{T}^{\text{-}}}^{{T}^{\text{+}}}({h}_{\text{vp}}^{m}\text{}-{h}_{w}^{m})\frac{\text{dT}}{{T}^{2}}\end{array}\) eq 4.2.3-3

Contrary to the previous case, here we have a non-linear equation to solve. To do this, we will use a corrector-predictor method. We ask \({\tilde{p}}_{\text{vp}}\) such that:

\(\begin{array}{}\text{ln}(\frac{{\tilde{p}}_{\text{vp}}}{{p}_{\text{vp}}^{\text{-}}})=\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{w}^{\text{-}}}(\frac{1}{{\text{RT}}^{\text{+}}}-\frac{1}{{K}_{H}})({p}_{{\text{gz}}^{\text{+}}}-{p}_{\text{gz}}^{\text{-}})-\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{w}^{\text{-}}{\text{RT}}^{\text{+}}}({p}_{{c}^{\text{+}}}-{p}_{c}^{\text{-}})\\ \text{+}\frac{{M}_{\text{vp}}^{\text{ol}}}{R}{\int }_{{T}^{\text{-}}}^{{T}^{\text{+}}}({h}_{\text{vp}}^{m}-{h}_{w}^{m})\frac{\text{dT}}{{T}^{2}}\end{array}\) eq 4.2.3-4

And so

\({\tilde{p}}_{\text{vp}}={p}_{\text{vp}}^{\text{-}}\text{.}\text{exp}(\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{{}_{w}}^{\text{-}}}(\frac{1}{{\text{RT}}^{\text{+}}}-\frac{1}{{K}_{H}})({p}_{{\text{gz}}^{\text{+}}}-{p}_{\text{gz}}^{\text{-}})-\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{{}_{w}}^{\text{-}}{\text{RT}}^{\text{+}}}({p}_{{c}^{\text{+}}}-{p}_{c}^{\text{-}})+{\int }_{{T}^{\text{-}}}^{{T}^{\text{+}}}({h}_{\text{vp}}^{m}-{h}_{w}^{m})\frac{\text{dT}}{{T}^{2}})\) eq 4.2.3-5

In addition, as in section [§4.1.4], we recall that:

\({\int }_{{T}^{\text{-}}}^{{T}^{\text{+}}}({h}_{\text{vp}}^{m}\text{-}{h}_{w}^{m})\frac{\text{dT}}{{T}^{2}}\text{=}({h}_{\text{vp}}^{{m}^{\text{-}}}\text{-}{h}_{w}^{{m}^{\text{-}}})(\frac{1}{{T}^{\text{-}}}\text{-}\frac{1}{{T}^{\text{+}}})\text{+}({C}_{\text{vp}}^{p}\text{-}{C}_{w}^{p})(\text{ln}(\frac{{T}^{\text{+}}}{{T}^{\text{-}}})\text{+}{T}^{\text{-}}(\frac{1}{{T}^{\text{+}}}\text{-}\frac{1}{{T}^{\text{-}}}))\)

Like \(\text{ln}(\frac{{p}_{\text{vp}}^{\text{+}}}{{p}_{\text{vp}}^{\text{-}}})\text{=}\text{ln}(\frac{{\tilde{p}}_{\text{vp}}}{{p}_{\text{vp}}^{\text{-}}})\text{+}\text{ln}(\frac{{p}_{\text{vp}}^{\text{+}}}{{\tilde{p}}_{\text{vp}}})\) and only by D.L \(\text{ln}(\frac{{p}_{\text{vp}}^{\text{+}}}{{\tilde{p}}_{\text{vp}}})=\text{ln}(1+\frac{{p}_{\text{vp}}^{\text{+}}-{\tilde{p}}_{\text{vp}}}{{\tilde{p}}_{\text{vp}}})\approx \frac{{p}_{\text{vp}}^{\text{+}}}{{\tilde{p}}_{\text{vp}}}-1\),

\({p}_{\text{vp}}^{+}\) will therefore be given by the following linear expression:

\(\frac{{p}_{\text{vp}}^{\text{+}}}{{\tilde{p}}_{\text{vp}}}=1+\frac{{M}_{\text{vp}}^{\text{ol}}}{{\rho }_{w}^{\text{-}}{K}_{H}}({p}_{\text{vp}}^{\text{+}}-{p}_{\text{vp}}^{\text{-}})+\frac{{M}_{\text{vp}}^{\text{ol}}R}{{\rho }_{w}^{\text{-}}{K}_{H}}({p}_{\text{vp}}^{\text{+}}-{p}_{\text{gz}}^{\text{-}})\text{ln}(\frac{{T}^{\text{+}}}{{T}^{\text{-}}})\) eq 4.2.3-6

From where

\({p}_{\text{vp}}^{\text{+}}=\frac{({\rho }_{w}^{\text{-}}{K}_{H}-{M}_{\text{vp}}^{\text{ol}}({p}_{\text{vp}}^{\text{-}}+{p}_{\text{gz}}^{\text{-}}R\text{ln}(\frac{{T}^{\text{+}}}{{T}^{\text{-}}})))}{(\frac{{\rho }_{w}^{\text{-}}{K}_{H}}{{\tilde{p}}_{\text{vp}}}-{M}_{\text{vp}}^{\text{ol}}(1+R\text{ln}(\frac{{T}^{\text{+}}}{{T}^{\text{-}}})))}\) eq 4.2.3-7

From there, the other pressures are easily calculated:

\({p}_{\text{as}}^{\text{+}}={p}_{\text{gz}}^{\text{+}}-{p}_{\text{vp}}^{\text{+}}\)

\({p}_{\text{ad}}^{\text{+}}\text{=}\frac{{p}_{\text{as}}^{\text{+}}}{{K}_{H}}{\text{RT}}^{\text{+}}\)

\({p}_{w}^{\text{+}}={p}_{\text{gz}}^{\text{+}}-{p}_{c}^{\text{+}}-{p}_{\text{ad}}^{\text{+}}\)

We can then calculate the densities of steam and air by the relationships [éq3.2.3.2-1], [éq3.2.3.2-2] and [éq 3.2.7-3]:

\({\rho }_{\text{vp}}^{\text{+}}\text{=}\frac{{M}_{\text{vp}}^{\text{ol}}}{R}\frac{{p}_{\text{vp}}^{\text{+}}}{{T}^{\text{+}}}\) eq 4.2.3-8

\({\rho }_{\text{as}}^{\text{+}}\text{=}\frac{{M}_{\text{as}}^{\text{ol}}}{R}\frac{({p}_{\text{gz}}^{\text{+}}-{p}_{\text{vp}}^{\text{+}})}{{T}^{\text{+}}}\) eq 4.2.3-9

\({\rho }_{\text{ad}}^{\text{+}}\text{=}\frac{{p}_{\text{ad}}^{\text{+}}{M}_{\text{as}}^{\text{ol}}}{{\text{RT}}^{\text{+}}}\) eq 4.2.3-10

The density of water is found by integrating the equation [éq 3.2.3.1-1] over the time step.

This results in:

\(\text{ln}(\frac{{\rho }_{w}^{\text{+}}}{{\rho }_{w}^{\text{-}}})=\frac{{p}_{\text{gz}}^{\text{+}}-{p}_{\text{gz}}^{\text{-}}-{p}_{c}^{\text{+}}+{p}_{c}^{\text{-}}-{p}_{\text{ad}}^{\text{+}}+{p}_{\text{ad}}^{\text{-}}}{{K}_{w}}-3{\alpha }_{w}({T}^{\text{+}}-{T}^{\text{-}})\) eq 4.2.3-11

4.2.4. Calculation of fluid enthalpies#

The fluid enthalpies are calculated by integrating the equations [éq 3.2.4.1-1], [éq 3.2.4.1-3], [éq3.2.4.2-1], [éq3.2.4.2-2].

\({h}_{\text{ad}}^{{m}^{\text{+}}}={h}_{\text{ad}}^{{m}^{\text{-}}}+{C}_{\text{ad}}^{p}({T}^{\text{+}}-{T}^{\text{-}})\) eq 4.2.4-2

\({h}_{\text{vp}}^{{m}^{\text{+}}}={h}_{\text{vp}}^{{m}^{\text{-}}}+{C}_{\text{vp}}^{p}({T}^{\text{+}}-{T}^{\text{-}})\) eq 4.2.4-3

\({h}_{\text{as}}^{{m}^{\text{+}}}={h}_{\text{as}}^{{m}^{\text{-}}}+{C}_{\text{as}}^{p}({T}^{\text{+}}-{T}^{\text{-}})\) eq 4.2.4-4

4.2.5. Calculation of mass inputs#

The equations [éq 3.2.2-1] give zero mass inputs at time 0. We write the equations [éq 3.2.2-1] incrementally:

\(\begin{array}{}{m}_{w}^{\text{+}}\text{=}{m}_{w}^{\text{-}}\text{+}{\rho }_{w}^{\text{+}}(1\text{+}{\varepsilon }_{V}^{\text{+}}){\phi }^{\text{+}}{S}_{\text{lq}}^{\text{+}}\text{-}{\rho }_{w}^{\text{-}}(1\text{+}{\varepsilon }_{V}^{\text{-}}){\phi }^{\text{-}}{S}_{\text{lq}}^{\text{-}}\\ {m}_{\text{ad}}^{\text{+}}\text{=}{m}_{\text{ad}}^{\text{-}}\text{+}{\rho }_{\text{ad}}^{\text{+}}(1\text{+}{\varepsilon }_{V}^{\text{+}}){\phi }^{\text{+}}{S}_{\text{lq}}^{\text{+}}\text{-}{\mathrm{\rho }}_{\text{ad}}^{\text{-}}(1\text{+}{\varepsilon }_{V}^{\text{-}}){\phi }^{\text{-}}{S}_{\text{lq}}^{\text{-}}\\ {m}_{\text{as}}^{\text{+}}\text{=}{m}_{\text{as}}^{\text{-}}\text{+}{\rho }_{\text{as}}^{\text{+}}(1\text{+}{\varepsilon }_{V}^{\text{+}}){\phi }^{\text{+}}(1\text{-}{S}_{\text{lq}}^{\text{+}})\text{-}{\rho }_{\text{as}}^{\text{-}}(1\text{+}{\varepsilon }_{V}^{\text{-}}){\phi }^{\text{-}}(1\text{-}{S}_{\text{lq}}^{\text{-}})\\ {m}_{\text{vp}}^{\text{+}}\text{=}{m}_{\text{vp}}^{\text{-}}\text{+}{\rho }_{\text{vp}}^{\text{+}}(1\text{+}{\varepsilon }_{V}^{\text{+}}){\phi }^{\text{+}}(1\text{-}{S}_{\text{lq}}^{\text{+}})\text{-}{\rho }_{\text{vp}}^{\text{-}}(1\text{+}{\varepsilon }_{V}^{\text{-}}){\phi }^{\text{-}}(1\text{-}{S}_{\text{lq}}^{\text{-}})\end{array}\) eq 4.2.5-1

4.2.6. Calculation of heat capacity and heat Q”#

We now have all the elements to apply the formula [éq 3.2.4.3-5] to the end of the time step:

\({C}_{\sigma }^{{0}^{\text{+}}}\text{=}(1\text{-}{\phi }^{\text{+}}){\rho }_{s}{C}_{\sigma }^{s}\text{+}{S}_{\text{lq}}^{\text{+}}{\phi }^{\text{+}}({\rho }_{w}^{\text{+}}{C}_{w}^{p}\text{+}{\rho }_{\text{ad}}^{\text{+}}{C}_{\text{ad}}^{p})\text{+}(1\text{-}{S}_{\text{lq}}^{\text{+}}){\phi }^{\text{+}}({\rho }_{\text{vp}}^{\text{+}}{C}_{\text{vp}}^{p}\text{+}{\rho }_{\text{as}}^{\text{+}}{C}_{\text{as}}^{p})\) eq 4.2.6-1

Of course we use [éq 3.2.4.3-4] which gives:

\({C}_{\varepsilon }^{0\text{+}}\text{=}{C}_{\sigma }^{0\text{+}}\text{-}{T}^{\text{+}}({C}_{0}\mathrm{:}{\alpha }_{0})\mathrm{:}{\alpha }_{0}\) eq 4.2.6-2

Note:

In the isotropic case [éq 4.2.6-2] becomes:

\({C}_{\varepsilon }^{0\text{+}}\text{=}{C}_{\sigma }^{0\text{+}}\text{-}9{T}^{\text{+}}{K}_{0}{\alpha }_{0}^{2}\)

Although the heat variation \(\delta {Q}^{\text{'}}\) is not a total differential, it is nevertheless legal to integrate it over the time step and we obtain by integrating [éq 3.2.4.3-1].

where we noted: \({T}^{}\text{=}\frac{{T}^{\text{+}}\text{+}{T}^{\text{-}}}{2}\). Here we have chosen a « middle point » formula for the temperature variable.

Note:

In the isotropic case [éq 4.2.6-3] becomes:

4.2.7. Calculation of mechanical stresses#

The calculation of the effective stresses is done by invoking the incremental laws of mechanics chosen by the user. We integrate on the [éq 3.2.8-2] time step and we have:

In the isotropic case we have \(B\text{=}b\text{.}1\), \({\sigma }_{p}^{\text{+}}\text{=}{\sigma }_{p}^{\text{+}}\text{.}1\) and \({\sigma }_{p}^{\text{-}}\text{=}{\sigma }_{p}^{\text{-}}\text{.}1\)

4.2.8. Calculation of water and heat flows#

It is of course necessary to calculate all the diffusion coefficients:

Fick coefficients \({F}_{\text{vp}}^{\text{+}}({P}_{\text{vp}}^{\text{+}},{P}_{\text{gz}}^{\text{+}},{T}^{\text{+}},{S}^{\text{+}})\) and \({F}_{\text{ad}}^{\text{+}}({P}_{\text{ad}}^{\text{+}},{P}_{\text{lq}}^{\text{+}},{T}^{\text{+}},{S}^{\text{+}})\)

The \({\lambda }^{T\text{+}}\text{=}{\lambda }_{\varphi }^{T}({\varphi }^{\text{+}})\text{.}{\lambda }_{S}^{T}({S}_{\text{lq}}^{\text{+}})\text{.}{\lambda }_{T}^{T}({T}^{\text{+}})\text{+}{\lambda }_{\text{cte}}^{T}\) thermal diffusivity tensor

Permeability and hydraulic conductivity tensors:

\({\lambda }_{\text{lq}}^{{H}^{\text{+}}}\text{=}\frac{{K}^{\text{int}}({\varphi }^{\text{+}}\text{})\text{.}{k}_{w}^{\text{rel}}({S}_{\text{lq}}^{\text{+}})}{{\mu }_{w}\text{}({T}^{\text{+}})}{\lambda }_{\text{gz}}^{{H}^{\text{+}}}\text{=}\frac{{K}^{\text{int}}({\varphi }^{\text{+}}\text{})\text{.}{k}_{\text{gz}}^{\text{rel}}({S}_{\text{lq}}^{\text{+}},{p}_{{\text{gz}}_{}}^{\text{+}})}{{\mu }_{\text{gz}}\text{}({T}^{\text{+}})}\)

Dissolved vapour and air concentrations: \({C}_{\text{vp}}^{\text{+}}\text{=}\frac{{p}_{\text{vp}}^{\text{+}}}{{p}_{\text{gz}}^{\text{+}}}\) and \({C}_{\text{ad}}^{\text{+}}\text{=}{\rho }_{\text{ad}}^{\text{+}}\)

All that remains is to apply the formulas [éq 3.2.5.1-1], [éq 3.2.5.2-15], [], [], [], [éq 3.2.5.2-16], and [éq3.2.5.2-18] to find: éq3.2.5.2-17

\({\mathrm{q}}^{\text{+}}\text{=}\text{-}{\lambda }^{{T}^{\text{+}}}\mathrm{\nabla }{T}^{\text{+}}\) eq 4.2.8-1

\(\frac{{M}_{\text{as}}^{\text{+}}}{{\rho }_{\text{as}}^{\text{+}}}\text{=}{\lambda }_{\text{gz}}^{{H}^{\text{+}}}\left[\text{-}\mathrm{\nabla }{p}_{{\text{gz}}_{}}^{\text{+}}\text{+}({\rho }_{{\text{as}}_{}}^{\text{+}}\text{+}{\rho }_{{\text{vp}}_{}}^{\text{+}}){F}^{m}\right]\text{+}{C}_{\text{vp}}^{\text{+}}{F}_{\text{vp}}^{\text{+}}\mathrm{\nabla }{C}_{\text{vp}}^{\text{+}}\) eq 4.2.8-2

\(\frac{{\mathrm{M}}_{\text{vp}}^{\text{+}}}{{\rho }_{\text{vp}}^{\text{+}}}\text{=}{\lambda }_{\text{gz}}^{{H}^{\text{+}}}\left[\text{-}\mathrm{\nabla }{p}_{{\text{gz}}_{}}^{\text{+}}\text{+}({\rho }_{{\text{as}}_{}}^{\text{+}}\text{+}{\rho }_{{\text{vp}}_{}}^{\text{+}}){\mathrm{F}}^{m}\right]\text{-}(1\text{-}{C}_{\text{vp}}^{\text{+}}){F}_{\text{vp}}^{\text{+}}\mathrm{\nabla }{C}_{\text{vp}}^{\text{+}}\) eq 4.2.8-3

\({\mathrm{M}}_{\text{ad}}^{\text{+}}\text{=}{\rho }_{\text{ad}}^{\text{+}}{\lambda }_{\text{lq}}^{H}\left[\text{-}\mathrm{\nabla }{p}_{\text{lq}}\text{+}({\rho }_{w}^{\text{+}}\text{+}{\rho }_{\text{ad}}^{\text{+}}){\mathrm{F}}^{m}\right]\text{-}{F}_{\text{ad}}^{\text{+}}\mathrm{\nabla }{C}_{\text{ad}}^{\text{+}}\) eq 4.2.8-5

In isotropic cases, \({\lambda }_{{\text{lq}}_{}}^{H}\text{=}{\lambda }_{{\text{lq}}_{}}^{H}\text{.}1\) and \({\lambda }_{{\text{gz}}_{}}^{H}\text{=}{\lambda }_{{\text{gz}}_{}}^{H}\text{.}1\)