7. Description of document versions

Version Aster	Author (s) Organization (s)	Description of changes
5	C.Chavant EDF -R&D/ AMA	Initial Release
7.4	C.Chavant, S.Granet, EDF -R&D/ AMA
9.2	S.Granet EDF -R&D/ AMA	Van Genuchten’s law
10.2	Meunier, EDF R&D/ AMA	Saturated Hydraulic Modeling
11.2	F.Voldoire EDF -R&D/ AMA	Equation formatting changes, some corrections here and there.
12.1	S.Granet EDF -R&D/ AMA	Introduction of transverse isotropy

Generalized constraints and internal variables

The constraints:

Number	Aster component name	Content
1	SIXX	\({\mathrm{\sigma }}_{\text{xx}}^{\text{'}}\)
2	SIYY	\({\sigma }_{\text{yy}}^{\text{'}}\)
3	SIZZ	\({\sigma }_{\text{zz}}^{\text{'}}\)
4	SIXY	\({\sigma }_{\text{xy}}^{\text{'}}\)
5	SIXZ	\({\sigma }_{\text{xz}}^{\text{'}}\)
6	SIYZ	\({\sigma }_{\text{yz}}^{\text{'}}\)
7	SIPXX	\({\sigma }_{{\mathrm{p}}_{\text{xx}}}\)
8	SIPYY	\({\sigma }_{{\mathrm{p}}_{\text{yy}}}\)
9	SIPZZ	\({\sigma }_{{\mathrm{p}}_{\text{zz}}}\)
10	SIPXY	\({\sigma }_{{\mathrm{p}}_{\text{xy}}}\)
11	SIPXZ	\({\sigma }_{{\mathrm{p}}_{\text{xz}}}\)
12	SIPYZ	\({\sigma }_{{\mathrm{p}}_{\text{yz}}}\)
13	M11	\({m}_{w}\)
14	FH11X	\({M}_{{w}_{x}}\)
15	FH11Y	\({M}_{{w}_{y}}\)
16	FH11Z	\({M}_{{w}_{z}}\)
17	ENT11	\({h}_{w}^{m}\)
18	M12	\({m}_{{}_{\text{vp}}}\)
19	FH12X	\({M}_{{\text{vp}}_{x}}\)
20	FH12Y	\({M}_{{\text{vp}}_{y}}\)
21	FH12Z	\({M}_{{\text{vp}}_{z}}\)
22	ENT12	\({h}_{\text{vp}}^{m}\)
23	M21	\({m}_{{}_{\text{as}}}\)
24	FH21X	\({M}_{{\text{as}}_{x}}\)
25	FH21Y	\({\mathrm{M}}_{{\text{as}}_{y}}\)
26	FH21Z	\({M}_{{\text{as}}_{z}}\)
27	ENT21	\({h}_{\text{as}}^{m}\)
28	M22	\({m}_{{}_{\text{ad}}}\)
29	FH22X	\({M}_{{\text{ad}}_{x}}\)
30	FH22Y	\({M}_{{\text{ad}}_{y}}\)
31	FH22Z	\({M}_{{\text{ad}}_{z}}\)
32	ENT22	\({h}_{\text{ad}}^{m}\)
33	QPRIM	\(Q\text{'}\)
34	FHTX	\({q}_{x}\)
35	FHTY	\({q}_{y}\)
36	FHTZ	\({q}_{z}\)

In the case without mechanics, and for the laws of behavior (LIQU_VAPE_GAZ, LIQU_VAPE,, LIQU_AD_GAZ_VAPE and LIQU_AD_GAZ) the internal variables are:

Number	Aster component name	Content
1	V1	\({\rho }_{\text{lq}}-{\rho }_{{}_{\text{lq}}}^{0}\)
2	V2	\(\varphi -{\varphi }^{0}\)
3	V3	\({p}_{\text{vp}}-{p}_{\text{vp}}^{0}\)
4	V4	\({S}_{\text{lq}}\)

In the case without mechanics, and for the laws of behavior (LIQU_GAZ, LIQU_GAZ_ATM,), the internal variables are:

Number	Aster component name	Content
1	V1	\({\rho }_{\text{lq}}-{\rho }_{{}_{\text{lq}}}^{0}\)
2	V2	\(\varphi -{\varphi }^{0}\)
3	V3	\({S}_{\text{lq}}\)

In the case without mechanics, and for the laws of behavior (LIQU_SATU,) the internal variables are:

Number	Aster component name	Content
1	V1	\({\rho }_{\text{lq}}-{\rho }_{{}_{\text{lq}}}^{0}\)
2	V2	\(\varphi -{\varphi }^{0}\)

In the case with mechanics the first numbers will be those corresponding to mechanics (V1 in the elastic case, V1 and following for plastic models). The number of the above internal variables must then be incremented by the same amount.

Material data

One gives here the correspondence between the vocabulary of Aster commands and the notations used in this note for the various characteristic quantities of materials.

A2.1 Key factor THM_LIQU

♦	RHO	\({\rho }_{\mathrm{lq}}^{0}\)
◊	UN_SUR_K	\(\frac{1}{{K}_{\mathrm{lq}}}\)
◊	ALPHA	\({\alpha }_{\mathrm{lq}}\)
◊	CP	\({C}_{\mathrm{lq}}^{p}\)
◊	VISC	\({\mu }_{\mathrm{lq}}(T)\)
◊	D_ VISC_TEMP	\(\frac{\partial {\mu }_{\mathrm{lq}}}{\partial T}(T)\)

A2.2 Key factor THM_GAZ

◊	MASS_MOL	\({M}_{\mathrm{as}}^{\mathrm{ol}}\)
◊	CP	\({C}_{\mathrm{lq}}^{P}\)
◊	VISC	\({\mu }_{\text{as}}(T)\)
◊	D_ VISC_TEMP	\(\frac{\partial {\mu }_{\text{as}}}{\partial T}(T)\)

A2.3 Key factor THM_VAPE_GAZ

◊	MASS_MOL	\({M}_{\text{VP}}^{\text{ol}}\)
◊	CP	\({C}_{\text{vp}}^{p}\)
◊	VISC	\({\mu }_{\text{vp}}(T)\)
◊	D_ VISC_TEMP	\(\frac{\partial {\mu }_{\text{vp}}}{\partial T}(T)\)

A2.4 Key factor THM_AIR_DISS

◊	CP	\({C}_{\text{ad}}^{p}\)
◊	COEF_HENRY	\({K}_{H}\)

A2.5 Key factor THM_INIT

♦	TEMP	\({}^{\mathrm{init}}T\)
♦	PRE1	\({P}_{1}\)
♦	PRE2	\({}^{\mathrm{init}}{P}_{2}\)
♦	PORO	\({\phi }^{0}\)
♦	PRES_VAPE	\({P}_{\mathrm{vp}}^{0}\)

It is recalled that, according to the modeling, the two pressures and represent:

	LIQU_SATU	LIQU_VAPE	LIQU_GAZ_ATM		GAZ	LIQU_VAPE_GAZ
\({P}_{1}\)	\({p}_{w}\)	\({p}_{w}\)		\({p}_{c}\text{=}\text{-}{p}_{w}\)	\({p}_{\text{gz}}\)	\({p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}\)
\({P}_{2}\)					\({p}_{\text{gz}}\)

	LIQU_GAZ	LIQU_AD_GAZ_VAPE	LIQU_AD_GAZ
\({P}_{1}\)	\({p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}\)	\({p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}\text{-}{p}_{\text{ad}}\)	\({p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}\text{-}{p}_{\text{ad}}\)
\({P}_{2}\)	\({p}_{\text{gz}}\)	\({p}_{\text{gz}}\)	\({p}_{\text{gz}}\)

A2.6 Key factor THM_DIFFU

♦	R_ GAZ	\(R\)
◊	RHO	\({r}_{0}\)
◊	CP	\({C}_{\sigma }^{S}\)
◊	BIOT_COEF	\(b\)
◊	BIOT_L	\({b}_{L}\)
◊	BIOT_N	\({b}_{N}\)
◊	BIOT_T	\({b}_{T}\) (2D case)
◊	SATU_PRES	\({S}_{\mathrm{lq}}({p}_{c})\)
◊	D_ SATU_PRES	\(\frac{\partial {S}_{\mathrm{lq}}}{\partial {p}_{c}}({p}_{c})\)
◊	PESA_X	\({F}_{x}^{m}\)
◊	PESA_Y	\({F}_{y}^{m}\)
◊	PESA_Z	\({F}_{z}^{m}\)
◊	PERM_IN	\({K}^{\text{int}}(\phi )\)
◊	PERMIN_L	\({K}_{L}^{\text{int}}(\varphi )\)
◊	PERMIN_N	\({K}_{N}^{\text{int}}(\varphi )\)
◊	PERMIN_T	\({K}_{T}^{\text{int}}(\varphi )\) (2D case)
◊	PERM_LIQU	\({K}_{\mathrm{lq}}^{\mathrm{rel}}({S}_{\mathrm{lq}})\)
◊	D_ PERM_LIQU_SATU	\(\frac{\partial {k}_{\mathrm{lq}}^{\mathrm{rel}}}{\partial {S}_{\mathrm{lq}}}({S}_{\mathrm{lq}})\)
◊	PERM_GAZ	\({k}_{\mathrm{gz}}^{\mathrm{rel}}({S}_{\mathrm{lq}},{p}_{\mathrm{gz}})\)
◊	D_ PERM_SATU_GAZ	\(\frac{\partial {k}_{\mathrm{gz}}^{\mathrm{rel}}}{\partial {S}_{\mathrm{lq}}}({S}_{\mathrm{lq}},{p}_{\mathrm{gz}})\)
◊	D_ PERM_PRES_GAZ	\(\frac{\partial {k}_{\mathrm{gz}}^{\mathrm{rel}}}{\partial {p}_{\mathrm{gz}}}({S}_{\mathrm{lq}},{p}_{\mathrm{gz}})\)
◊	FICKV_T	\({f}_{\text{vp}}^{T}(T)\)
◊	FICKV_S	\({f}_{\text{vp}}^{S}(S)\)
◊	FICKV_PG	\({f}_{\text{vp}}^{\text{gz}}({P}_{g})\)
◊	FICKV_PV	\({f}_{\text{vp}}^{\text{vp}}({P}_{\text{vp}})\)
◊	D_ FV_T	\(\frac{\partial {f}_{\text{vp}}^{T}}{\partial T}(T)\)
◊	D_ FV_PG	\(\frac{\partial {f}_{\text{vp}}^{\text{gz}}}{\partial {P}_{\text{gz}}}({P}_{\text{gz}})\)
◊	FICKA_T	\({f}_{\text{ad}}^{T}(T)\)
◊	FICKA_S	\({f}_{\text{ad}}^{S}(S)\)
◊	FICKA_PA	\({f}_{\text{ad}}^{\text{ad}}({P}_{\text{ad}})\)
◊	FICKA_PL	\({f}_{\text{ad}}^{\text{lq}}({P}_{\text{lq}})\)
◊	D_ FA_T	\(\frac{\partial {f}_{\text{vp}}^{T}}{\partial T}(T)\)
◊	LAMB_T	\({\lambda }_{T}^{T}(T)\)
◊	LAMB_T_L	\({\lambda }_{T}^{T}(T)\) according to L
◊	LAMB_T_N	\({\lambda }_{T}^{T}(T)\) according to N
◊	LAMB_T_T	\({\lambda }_{T}^{T}(T)\) according to T (2D)
◊	D_ LB_T	\(\frac{\mathrm{\partial }{\lambda }_{T}^{T}(T)}{\mathrm{\partial }T}\)
◊	D_ LB_T_L	\(\frac{\partial {\lambda }_{T}^{T}(T)}{\partial T}\) according to L
◊	D_ LB_T_N	\(\frac{\partial {\lambda }_{T}^{T}(T)}{\partial T}\) according to N
◊	D_ LB_T_T	\(\frac{\partial {\lambda }_{T}^{T}(T)}{\partial T}\) according to T
◊	LAMB_PHI	\({\lambda }_{\varphi }^{T}(\varphi )\)
◊	D_ LB_PHI	\(\frac{\partial {\lambda }_{\varphi }^{T}(\varphi )}{\partial \varphi }\)
◊	LAMB_S	\({\lambda }_{S}^{T}(S)\)
◊	DLAMBS	\(\frac{\partial {\lambda }_{S}^{T}(S)}{\partial S}\)
◊	LAMB_CT	\({\lambda }_{\text{CT}}^{T}\)
◊	LAMB_CL	\({\lambda }_{\text{CT}}^{T}\) according to L
◊	LAMB_CN	\({\lambda }_{\text{CT}}^{T}\) according to N
◊	LAMB_CT	\({\lambda }_{\text{CT}}^{T}\) according to T

Note:

For models involving thermal energy, and for the calculation of homogenized specific heat, the relationship is used: \({C}_{\sigma }^{0}\text{=}(1\text{-}\varphi ){\rho }_{s}{C}_{\sigma }^{s}\text{+}{\rho }_{\text{lq}}{S}_{\text{lq}}\varphi {C}_{\text{lq}}^{p}\text{+}(1\text{-}{S}_{\text{lq}})\varphi ({\rho }_{\text{vp}}{C}_{\text{vp}}^{p}\text{+}{\rho }_{\text{as}}{C}_{\text{as}}^{p})\). In this formula, we confuse \({\rho }_{s}\) with its initial value \({\rho }_{S}^{0}\) whose value is read under the keyword RHOdu keyword factor ELAS.

Pressure derivatives as a function of generalized deformations

Here we detail the calculation of pressure derivatives as a function of generalized deformations. Recall that the equation [éq 3.2.6.3] is \(\frac{{\text{dp}}_{\text{vp}}}{{\rho }_{\text{vp}}}\text{=}\frac{{\text{dp}}_{w}}{{\rho }_{w}}\text{+}L\frac{\text{dT}}{T}\) with \(L={h}_{\text{vp}}^{m}-{h}_{w}^{m}\). Also \({\text{dp}}_{\text{ad}}\text{=}{\text{dp}}_{\text{lq}}\text{-}{\text{dp}}_{w}\text{=}\frac{R}{{K}_{H}}{p}_{\text{as}}\text{dT}\text{+}\frac{\text{RT}}{{K}_{H}}{\text{dp}}_{\text{as}}\) and \({\text{dp}}_{\text{as}}={\text{dp}}_{\text{gz}}-{\text{dp}}_{\text{vp}}\). By combining these equations we obtain:

\(\{\begin{array}{c}{\text{dp}}_{\text{vp}}\left[\frac{\text{RT}}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\right]\text{=}(\text{-}L{\rho }_{w}\text{+}{p}_{\text{ad}})\frac{\text{dT}}{T}\text{+}(\frac{\text{RT}}{{K}_{H}}\text{-}1){\text{dp}}_{\text{gz}}\text{+}{\text{dp}}_{c}\\ {\text{dp}}_{w}\left[\frac{{\rho }_{\text{vp}}}{{\rho }_{w}}\frac{\text{RT}}{{K}_{H}}\text{-}1\right]\text{=}(\text{-}\text{LR}\frac{{\rho }_{\text{vp}}}{{K}_{H}}\text{+}\frac{{p}_{\text{ad}}}{T})\text{dT}\text{+}(\frac{\text{RT}}{{K}_{H}}\text{-}1){\text{dp}}_{\text{gz}}\text{+}{\text{dp}}_{c}\end{array}\)

We can therefore write the partial derivatives of water and steam as a function of generalized deformations:

\(\frac{\partial {p}_{w}}{\partial T}\text{=}\frac{\text{-}\text{LR}\frac{{\rho }_{\text{vp}}}{{K}_{H}}\text{+}\frac{{p}_{\text{ad}}}{T}}{\frac{{\rho }_{\text{vp}}}{{\rho }_{w}}\frac{\text{RT}}{{K}_{H}}\text{-}1};\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}}\text{=}\frac{\frac{\text{RT}}{{K}_{H}}\text{-}1}{\frac{{\rho }_{\text{vp}}}{{\rho }_{w}}\frac{\text{RT}}{{K}_{H}}\text{-}1};\frac{\partial {p}_{w}}{\partial {p}_{c}}\text{=}\frac{1}{\frac{{\rho }_{\text{vp}}}{{\rho }_{w}}\frac{\text{RT}}{{K}_{H}}\text{-}1}\)

\(\frac{\partial {p}_{\text{vp}}}{\partial T}\frac{(\text{-}L{\rho }_{w}\text{+}{p}_{\text{ad}})}{\frac{\text{RT}}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}}\text{.}\frac{1}{T};\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}\frac{\frac{\text{RT}}{{K}_{H}}\text{-}1}{\frac{\text{RT}}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}};\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}\frac{1}{\frac{\text{RT}}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}}\)

The relationships \({\text{dp}}_{\text{as}}={\text{dp}}_{\text{gz}}-{\text{dp}}_{\text{vp}}\) and \({\text{dp}}_{\text{ad}}\text{=}{\text{dp}}_{\text{gz}}\text{-}{\text{dp}}_{c}\text{-}{\text{dp}}_{w}\) make it possible to derive all the pressures, since we will have:

\(\frac{\partial {p}_{\text{as}}}{\partial T}\text{=}\text{-}\frac{\partial {p}_{\text{vp}}}{\partial T};\frac{\partial {p}_{\text{as}}}{\partial {p}_{\text{gz}}}\text{=}1\text{-}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}};\frac{\partial {p}_{\text{as}}}{\partial {p}_{c}}\text{=}\text{-}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}\)

and

\(\frac{\partial {p}_{\text{ad}}}{\partial T}\text{=}\text{-}\frac{\partial {p}_{w}}{\partial T};\frac{\partial {p}_{\text{ad}}}{\partial {p}_{\text{gz}}}\text{=}1\text{-}\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}};\frac{\partial {p}_{\text{ad}}}{\partial {p}_{c}}\text{=}\text{-}1\text{-}\frac{\partial {p}_{w}}{\partial {p}_{c}}\)

Second derivatives of vapor and dissolved air pressures as a function of generalized deformations

Here we calculate the second-order partial derivatives of the vapour pressure required for section [§5.5.2]. The following will be noted:

\(\mathrm{A1}\text{=}\frac{\text{RT}}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\) and \(\mathrm{A2}=\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}(\frac{\mathrm{RT}}{{K}_{H}}-1)\)

\(\mathrm{A3}=\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial T}-\frac{1}{T}-\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial T}-3{\alpha }_{w}\)

\(\mathrm{A4}=-\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial T}+\frac{1}{T}+\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial T}-3{\alpha }_{w}\)

Second derivatives of vapour pressure:

\(\frac{\partial }{\partial {p}_{c}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}\text{=}\frac{\mathrm{A2}}{{\mathrm{A1}}^{2}}(\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{c}}\text{-}\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}})\)

\(\frac{\partial }{\partial {p}_{\text{gz}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}\text{=}\frac{\mathrm{A2}}{{\mathrm{A1}}^{2}}(\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}}\text{-}\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}})\)

\(\frac{\partial }{\partial T}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}\text{=}\frac{R}{{K}_{H}\mathrm{A1}}\text{-}\frac{1}{{\mathrm{A1}}^{2}}(\frac{\text{RT}}{{K}_{H}}\text{-}1)(\frac{R}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\mathrm{\rho }}_{\text{vp}}}\mathrm{A4})\)

\(\frac{\partial }{\partial {p}_{c}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}\text{=}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}\text{-}\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{c}})\)

\(\frac{\partial }{\partial {p}_{\text{gz}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}\text{=}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}\text{-}\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}})\)

\(\frac{\partial }{\partial T}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}\text{=}\text{-}\frac{1}{{\mathrm{A1}}^{2}}(\frac{R}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\mathrm{\rho }}_{\text{vp}}}\mathrm{A4})\)

\(\frac{\partial }{\partial {p}_{c}}\frac{\partial {p}_{\text{vp}}}{\partial T}=-\frac{1}{T}\frac{1}{{\mathrm{A1}}^{2}}(\mathrm{A1}(1-\frac{\partial {p}_{w}}{\partial {p}_{c}}(1\text{+}L\frac{{\rho }_{w}}{{K}_{W}}))\text{+}({p}_{\text{ad}}\text{-}L{\rho }_{w})\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}(\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{c}}-\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}))\)

\(\frac{\partial }{\partial {p}_{\text{gz}}}\frac{\partial {p}_{\text{vp}}}{\partial T}=-\frac{1}{T}\frac{1}{{\mathrm{A1}}^{2}}(\mathrm{A1}(1-\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}}(1\text{+}L\frac{{\rho }_{w}}{{K}_{W}}))\text{+}({p}_{\text{ad}}-L{\rho }_{w})\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}(\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}}-\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}))\)

\(\frac{\partial }{\partial T}\frac{\partial {p}_{\text{vp}}}{\partial T}=\frac{1}{T\text{.}\mathrm{A1}}(\frac{\partial {p}_{\text{ad}}}{\partial T}-L(\frac{{\rho }_{w}}{{K}_{W}}\frac{\partial {p}_{w}}{\partial T}-3{\alpha }_{w}{\rho }_{w}))-\frac{1}{{T}^{2}\text{.}{\mathrm{A1}}^{2}}(\frac{\text{RT}}{{K}_{H}}-\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}+T(\frac{R}{{K}_{H}}-\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\mathrm{A4}))({p}_{\text{ad}}-L{\rho }_{w})\)

Second derivatives of dissolved air pressure:

\(\frac{\partial }{\partial {p}_{c}}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{\text{gz}}}=\frac{\text{RT}}{{K}_{H}}\frac{\mathrm{A2}}{{\mathrm{A1}}^{2}}(\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}-\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{c}})\)

\(\frac{\partial }{\partial {p}_{\text{gz}}}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{\text{gz}}}=\frac{\text{RT}}{{K}_{H}}\frac{\mathrm{A2}}{{\mathrm{A1}}^{2}}(\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}-\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}})\)

\(\frac{\partial }{\partial T}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{\text{gz}}}=-{R}_{{K}_{H}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{\mathrm{A1}}+{(\frac{{\rho }_{w}}{{\rho }_{\text{vp}}})}^{2}\frac{R}{{K}_{H}}\frac{\mathrm{A2}}{{\mathrm{A1}}^{2}}(1\text{+}\mathrm{A3}\text{.}T)\)

\(\frac{\partial }{\partial {p}_{c}}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{c}}=\frac{\text{RT}}{{K}_{H}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}-\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{c}})\)

\(\frac{\partial }{\partial {p}_{\text{gz}}}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{c}}=\frac{\text{RT}}{{K}_{H}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}-\frac{1}{{K}_{w}}\frac{\partial {p}_{w}}{\partial {p}_{\text{gz}}})\)

\(\frac{\partial }{\partial T}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{c}}=\frac{R}{{K}_{H}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(1+\mathrm{A3}\text{.}T)\)

\(\frac{\partial }{\partial {p}_{c}}\frac{\partial {p}_{\text{ad}}}{\partial T}=\frac{1}{\mathrm{A1}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}(\frac{\text{LR}}{{K}_{H}}\frac{{\rho }_{\text{vp}}}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{c}}-\frac{1}{T}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{c}})-\frac{\text{RT}}{{K}_{H}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(\frac{\text{LR}}{{K}_{H}}{\rho }_{\text{vp}}-\frac{{p}_{\text{ad}}}{T})\text{.}\mathrm{A3}\)

\(\frac{\partial }{\partial {p}_{\text{gz}}}\frac{\partial {p}_{\text{ad}}}{\partial T}=\frac{1}{\mathrm{A1}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}(\frac{\text{LR}}{{K}_{H}}\frac{{\rho }_{\text{vp}}}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial {p}_{\text{gz}}}-\frac{1}{T}\frac{\partial {p}_{\text{ad}}}{\partial {p}_{\text{gz}}})-\frac{\text{RT}}{{K}_{H}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(\frac{\text{LR}}{{K}_{H}}{\rho }_{\text{vp}}-\frac{{p}_{\text{ad}}}{T})\text{.}\mathrm{A3}\)

\(\frac{\partial }{\partial T}\frac{\partial {p}_{\text{ad}}}{\partial T}=\frac{1}{\mathrm{A1}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}(\frac{\text{LR}{\rho }_{\text{vp}}}{{K}_{H}}(\frac{1}{{p}_{\text{vp}}}\frac{\partial {p}_{\text{vp}}}{\partial T}-\frac{1}{T})+\frac{{p}_{\text{ad}}}{{T}^{2}}-\frac{1}{T}\frac{\partial {p}_{\text{ad}}}{\partial T})\text{-}\frac{\text{RT}}{{K}_{H}}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}\frac{1}{{\mathrm{A1}}^{2}}(\frac{\text{LR}}{{K}_{H}}{\rho }_{\text{vp}}-\frac{{p}_{\text{ad}}}{T})\text{.}(\mathrm{A3}+\frac{1}{T})\)

Equivalence with formulations ANDRA

In order to be able to be integrated into the ALLIANCE platform, it is necessary to be consistent with the formulations put forward by ANDRA. Here we propose an equivalence between the ratings that would be dissimilar. These differences only concern the writing of:

• The energy equation

• Henry’s law

• Diffusion in the liquid

• Diffusion in gas

Note about enthalpies:

It is necessary to have consistency between the two models that the Code_Aster user takes: \({h}_{\text{lq}}^{{m}_{0}}\text{=}0\) and \({h}_{\text{vp}}^{{m}_{0}}\text{=}{L}_{0}\) .

A5.1 Energy equation

The table above recalls the two formulations:

Notes Code_Aster	Ratings ANDRA
\({h}_{\text{lq}}^{m}\)	\(\text{=}\frac{{\varphi }_{w}}{{\rho }_{w}{S}_{w}n}\)
\({h}_{\text{vp}}^{m}\)	\(\text{=}\frac{{\varphi }_{v}}{{\rho }_{v}(1\text{-}{S}_{w})n}\)
\({h}_{\text{as}}^{m}\)	\(\text{=}\frac{{\varphi }_{\text{as}}}{{\rho }_{\text{as}}(1\text{-}{S}_{w})n}\)
\({M}_{\text{lq}}\)	\(\text{=}{\rho }_{w}{f}_{w}\)
\({M}_{\text{as}}\)	\(\text{=}{\rho }_{\text{as}}{f}_{\text{as}}\)
\({M}_{\text{vp}}\)	\(\text{=}{\rho }_{v}{f}_{v}\)

Rewriting the energy equation of*Code_Aster* with these notations, we find:

\(\begin{array}{}\frac{d\phi }{\text{dt}}+\text{Div}({\phi }_{w}\frac{{f}_{w}}{{S}_{w}n}+{\phi }_{\text{as}}\frac{{f}_{\text{as}}}{(1-{S}_{w})n}\text{+}{\phi }_{v}\frac{{f}_{v}}{(1-{S}_{w})n})+\text{Div}(q)-(T-{T}^{0})\frac{d}{\text{dt}}\left[(1-n){\rho }_{s}{C}_{s}\right]\\ +3{\alpha }_{0}{K}_{0}T\frac{d{\varepsilon }_{v}}{\text{dt}}\text{+}3{\alpha }_{\text{lq}}^{m}T\frac{{\text{dp}}_{c}}{\text{dt}}-3({\alpha }_{\text{lq}}^{m}+{\alpha }_{\text{gz}}^{m})T\frac{{\text{dp}}_{\text{gz}}}{\text{dt}}-9{\text{TK}}_{0}{\alpha }_{{0}^{2}}\frac{\text{dT}}{\text{dt}}\\ =({\rho }_{w}{f}_{w}\text{+}{\rho }_{v}{f}_{v}\text{+}{\rho }_{\text{as}}{f}_{\text{as}})g\text{+}\Theta \end{array}\)

The first line being that of ANDRA and the others being a prima facie negligible.

A5.2 Henry’s Law

In the formulation of ANDRA, Henry’s formulation is given by \({\omega }_{l}^{a}\text{=}\frac{{P}_{\text{as}}}{H}\frac{{M}_{\text{as}}^{\text{ol}}}{{M}_{w}}{\rho }_{w}\) with the concentration of air in water that can be reduced to a density such as \({\omega }_{l}^{a}\text{=}{\rho }_{\text{ad}}\). \(H\) is expressed in \(\mathit{Pa}\).

In the formulation of Code_Aster, we recall that Henry’s law is expressed in the form: \({C}_{\text{ad}}^{\text{ol}}\text{=}\frac{{\rho }_{\text{ad}}}{{M}_{\text{ad}}^{\text{ol}}}\) with \({C}_{\text{ad}}^{\text{ol}}\text{=}\frac{{p}_{\text{as}}}{{K}_{H}}\). \({K}_{H}\) is expressed in \(\mathit{Pa.}{m}^{3}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1}\).

So we have the equivalence:

\({K}_{H}\text{=}H\frac{{M}_{w}}{{\rho }_{w}}\)

A5.3 Diffusion of steam in the air

In formulation ANDRA the flow of water vapor in the air as a function of the concentration of water vapor in the air or relative humidity is noted:

\({f}_{{\text{Diff}}_{v}}=-{D}_{v}\text{.}\nabla {\omega }_{g}^{e}\)

with the concentration defined as the molar ratio in the gas: \({\omega }_{g}^{e}\text{=}\frac{{n}_{\nu }}{{n}_{g}}\).

In Code_Aster, this same flow is written: \({f}_{{\text{Diff}}_{v}}={F}_{\text{vp}}\nabla {C}_{\text{vp}}\) with the Fick vapor coefficient \({F}_{\text{vp}}=\frac{{D}_{\text{vp}}}{{C}_{\text{vp}}(1-{C}_{\text{vp}})}\) and \({D}_{\mathit{vp}}\) the Fick diffusion coefficient of the gas mixture. \({C}_{\mathit{vp}}\) is defined as the pressure ratio such as: \({C}_{\text{vp}}\text{=}\frac{{p}_{\text{vp}}}{{p}_{\text{gz}}}\).

The ideal gas law allows us to write that \({C}_{\text{vp}}\text{=}{\omega }_{g}^{e}\) therefore \(\nabla {\omega }_{g}^{e}\text{=}\nabla {C}_{\text{vp}}\) and \({f}_{{\text{Diff}}_{v}}\text{=}{D}_{v}\text{.}\nabla {C}_{\text{vp}}\).

So the Code_Aster/ANDRA equivalent is simply written:

\({F}_{\text{vp}}\text{=}{D}_{v}\).

A5.4 Diffusion of air dissolved in water

In formulation ANDRA the flow of air dissolved in water is expressed

\({f}_{{a}_{{\text{ds}}_{e}}}\text{=}{D}_{a}\text{.}\nabla {\omega }_{l}^{a}\)

with \({\omega }_{l}^{a}\text{=}\frac{{\rho }_{\text{ad}}}{{M}_{\text{ad}}^{\text{ol}}}\).

In Code_Aster, this same flow is written: \({f}_{{a}_{{\text{ds}}_{v}}}\text{=}{F}_{\text{ad}}\nabla {C}_{\text{ad}}\) with the air-dissolved Fick coefficient \({F}_{\text{ad}}=\frac{{D}_{\text{ad}}}{{C}_{\text{ad}}(1-{C}_{\text{ad}})}\) and \({D}_{\mathit{ad}}\) the Fick diffusion coefficient of the liquid mixture. \({C}_{\mathit{ad}}\) is defined as: \({C}_{\text{ad}}\text{=}{w}_{l}^{a}\). So:

\({F}_{\text{ad}}\text{=}{D}_{a}\).