Description of document versions ==================================== .. csv-table:: "**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", "S. 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: .. csv-table:: "**Number**", "**Aster component name**", "**Content**" "1"," SIXX "," :math:`{\mathrm{\sigma }}_{\text{xx}}^{\text{'}}`" "2"," SIYY "," :math:`{\sigma }_{\text{yy}}^{\text{'}}`" "3"," SIZZ "," :math:`{\sigma }_{\text{zz}}^{\text{'}}`" "4"," SIXY "," :math:`{\sigma }_{\text{xy}}^{\text{'}}`" "5"," SIXZ "," :math:`{\sigma }_{\text{xz}}^{\text{'}}`" "6"," SIYZ "," :math:`{\sigma }_{\text{yz}}^{\text{'}}`" "7"," SIPXX "," :math:`{\sigma }_{{\mathrm{p}}_{\text{xx}}}`" "8"," SIPYY "," :math:`{\sigma }_{{\mathrm{p}}_{\text{yy}}}`" "9"," SIPZZ "," :math:`{\sigma }_{{\mathrm{p}}_{\text{zz}}}`" "10"," SIPXY "," :math:`{\sigma }_{{\mathrm{p}}_{\text{xy}}}`" "11"," SIPXZ "," :math:`{\sigma }_{{\mathrm{p}}_{\text{xz}}}`" "12"," SIPYZ "," :math:`{\sigma }_{{\mathrm{p}}_{\text{yz}}}`" "13", "M11"," :math:`{m}_{w}`" "14"," FH11X "," :math:`{M}_{{w}_{x}}`" "15"," FH11Y "," :math:`{M}_{{w}_{y}}`" "16"," FH11Z "," :math:`{M}_{{w}_{z}}`" "17"," ENT11 "," :math:`{h}_{w}^{m}`" "18", "M12"," :math:`{m}_{{}_{\text{vp}}}`" "19"," FH12X "," :math:`{M}_{{\text{vp}}_{x}}`" "20"," FH12Y "," :math:`{M}_{{\text{vp}}_{y}}`" "21"," FH12Z "," :math:`{M}_{{\text{vp}}_{z}}`" "22"," ENT12 "," :math:`{h}_{\text{vp}}^{m}`" "23", "M21"," :math:`{m}_{{}_{\text{as}}}`" "24"," FH21X "," :math:`{M}_{{\text{as}}_{x}}`" "25"," FH21Y "," :math:`{\mathrm{M}}_{{\text{as}}_{y}}`" "26"," FH21Z "," :math:`{M}_{{\text{as}}_{z}}`" "27"," ENT21 "," :math:`{h}_{\text{as}}^{m}`" "28", "M22"," :math:`{m}_{{}_{\text{ad}}}`" "29"," FH22X "," :math:`{M}_{{\text{ad}}_{x}}`" "30"," FH22Y "," :math:`{M}_{{\text{ad}}_{y}}`" "31"," FH22Z "," :math:`{M}_{{\text{ad}}_{z}}`" "32"," ENT22 "," :math:`{h}_{\text{ad}}^{m}`" "33"," QPRIM "," :math:`Q\text{'}`" "34"," FHTX "," :math:`{q}_{x}`" "35"," FHTY "," :math:`{q}_{y}`" "36"," FHTZ "," :math:`{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: .. csv-table:: "**Number**", "**Aster component name**", "**Content**" "1", "V1"," :math:`{\rho }_{\text{lq}}-{\rho }_{{}_{\text{lq}}}^{0}`" "2", "V2"," :math:`\varphi -{\varphi }^{0}`" "3", "V3"," :math:`{p}_{\text{vp}}-{p}_{\text{vp}}^{0}`" "4", "V4"," :math:`{S}_{\text{lq}}`" In the case without mechanics, and for the laws of behavior (LIQU_GAZ, LIQU_GAZ_ATM,), the internal variables are: .. csv-table:: "**Number**", "**Aster component name**", "**Content**" "1", "V1"," :math:`{\rho }_{\text{lq}}-{\rho }_{{}_{\text{lq}}}^{0}`" "2", "V2"," :math:`\varphi -{\varphi }^{0}`" "3", "V3"," :math:`{S}_{\text{lq}}`" In the case without mechanics, and for the laws of behavior (LIQU_SATU,) the internal variables are: .. csv-table:: "**Number**", "**Aster component name**", "**Content**" "1", "V1"," :math:`{\rho }_{\text{lq}}-{\rho }_{{}_{\text{lq}}}^{0}`" "2", "V2"," :math:`\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 .. csv-table:: "♦ "," RHO "," :math:`{\rho }_{\mathrm{lq}}^{0}`" "◊", "UN_SUR_K "," :math:`\frac{1}{{K}_{\mathrm{lq}}}`" "◊", "ALPHA "," :math:`{\alpha }_{\mathrm{lq}}`" "◊", "CP", ":math:`{C}_{\mathrm{lq}}^{p}`" "◊", "VISC "," :math:`{\mu }_{\mathrm{lq}}(T)`" "◊", "D_ VISC_TEMP "," :math:`\frac{\partial {\mu }_{\mathrm{lq}}}{\partial T}(T)`" **A2.2** **Key factor** THM_GAZ .. csv-table:: "◊", "MASS_MOL "," :math:`{M}_{\mathrm{as}}^{\mathrm{ol}}`" "◊", "CP", ":math:`{C}_{\mathrm{lq}}^{P}`" "◊", "VISC "," :math:`{\mu }_{\text{as}}(T)`" "◊", "D_ VISC_TEMP "," :math:`\frac{\partial {\mu }_{\text{as}}}{\partial T}(T)`" **A2.3** **Key factor** THM_VAPE_GAZ .. csv-table:: "◊", "MASS_MOL "," :math:`{M}_{\text{VP}}^{\text{ol}}`" "◊", "CP", ":math:`{C}_{\text{vp}}^{p}`" "◊", "VISC "," :math:`{\mu }_{\text{vp}}(T)`" "◊", "D_ VISC_TEMP "," :math:`\frac{\partial {\mu }_{\text{vp}}}{\partial T}(T)`" **A2.4** **Key factor** THM_AIR_DISS .. csv-table:: "◊", "CP", ":math:`{C}_{\text{ad}}^{p}`" "◊", "COEF_HENRY "," :math:`{K}_{H}`" **A2.5** **Key factor** THM_INIT .. csv-table:: "♦ "," TEMP "," :math:`{}^{\mathrm{init}}T`" "♦ "," PRE1 "," :math:`{P}_{1}`" "♦ "," PRE2 "," :math:`{}^{\mathrm{init}}{P}_{2}`" "♦ "," PORO "," :math:`{\phi }^{0}`" "♦ "," PRES_VAPE "," :math:`{P}_{\mathrm{vp}}^{0}`" It is recalled that, according to the modeling, the two pressures and represent: .. csv-table:: "", "LIQU_SATU "," LIQU_VAPE "," LIQU_GAZ_ATM "," "," GAZ "," LIQU_VAPE_GAZ" ":math:`{P}_{1}` "," :math:`{p}_{w}` "," :math:`{p}_{w}` "," "," :math:`{p}_{c}\text{=}\text{-}{p}_{w}` "," :math:`{p}_{\text{gz}}` "," :math:`{p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}`" ":math:`{P}_{2}` ", "", "", "", "", ":math:`{p}_{\text{gz}}`" .. csv-table:: "", "LIQU_GAZ "," LIQU_AD_GAZ_VAPE "," LIQU_AD_GAZ" ":math:`{P}_{1}` "," :math:`{p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}` "," :math:`{p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}\text{-}{p}_{\text{ad}}` "," :math:`{p}_{c}\text{=}{p}_{\text{gz}}\text{-}{p}_{w}\text{-}{p}_{\text{ad}}`" ":math:`{P}_{2}` "," :math:`{p}_{\text{gz}}` "," :math:`{p}_{\text{gz}}` "," :math:`{p}_{\text{gz}}`" **A2.6** **Key factor** THM_DIFFU .. csv-table:: "♦ ", "R_ GAZ "," :math:`R`" "◊", "RHO "," :math:`{r}_{0}`" "◊", "CP", ":math:`{C}_{\sigma }^{S}`" "◊", "BIOT_COEF "," :math:`b`" "◊", "BIOT_L "," :math:`{b}_{L}`" "◊", "BIOT_N "," :math:`{b}_{N}`" "◊", "BIOT_T "," :math:`{b}_{T}` (2D case)" "◊", "SATU_PRES "," :math:`{S}_{\mathrm{lq}}({p}_{c})`" "◊", "D_ SATU_PRES "," :math:`\frac{\partial {S}_{\mathrm{lq}}}{\partial {p}_{c}}({p}_{c})`" "◊", "PESA_X "," :math:`{F}_{x}^{m}`" "◊", "PESA_Y "," :math:`{F}_{y}^{m}`" "◊", "PESA_Z "," :math:`{F}_{z}^{m}`" "◊", "PERM_IN "," :math:`{K}^{\text{int}}(\phi )`" "◊", "PERMIN_L "," :math:`{K}_{L}^{\text{int}}(\varphi )`" "◊", "PERMIN_N "," :math:`{K}_{N}^{\text{int}}(\varphi )`" "◊", "PERMIN_T "," :math:`{K}_{T}^{\text{int}}(\varphi )` (2D case)" "◊", "PERM_LIQU "," :math:`{K}_{\mathrm{lq}}^{\mathrm{rel}}({S}_{\mathrm{lq}})`" "◊", "D_ PERM_LIQU_SATU "," :math:`\frac{\partial {k}_{\mathrm{lq}}^{\mathrm{rel}}}{\partial {S}_{\mathrm{lq}}}({S}_{\mathrm{lq}})`" "◊", "PERM_GAZ "," :math:`{k}_{\mathrm{gz}}^{\mathrm{rel}}({S}_{\mathrm{lq}},{p}_{\mathrm{gz}})`" "◊", "D_ PERM_SATU_GAZ "," :math:`\frac{\partial {k}_{\mathrm{gz}}^{\mathrm{rel}}}{\partial {S}_{\mathrm{lq}}}({S}_{\mathrm{lq}},{p}_{\mathrm{gz}})`" "◊", "D_ PERM_PRES_GAZ "," :math:`\frac{\partial {k}_{\mathrm{gz}}^{\mathrm{rel}}}{\partial {p}_{\mathrm{gz}}}({S}_{\mathrm{lq}},{p}_{\mathrm{gz}})`" "◊", "FICKV_T "," :math:`{f}_{\text{vp}}^{T}(T)`" "◊", "FICKV_S "," :math:`{f}_{\text{vp}}^{S}(S)`" "◊", "FICKV_PG "," :math:`{f}_{\text{vp}}^{\text{gz}}({P}_{g})`" "◊", "FICKV_PV "," :math:`{f}_{\text{vp}}^{\text{vp}}({P}_{\text{vp}})`" "◊", "D_ FV_T "," :math:`\frac{\partial {f}_{\text{vp}}^{T}}{\partial T}(T)`" "◊", "D_ FV_PG "," :math:`\frac{\partial {f}_{\text{vp}}^{\text{gz}}}{\partial {P}_{\text{gz}}}({P}_{\text{gz}})`" "◊", "FICKA_T "," :math:`{f}_{\text{ad}}^{T}(T)`" "◊", "FICKA_S "," :math:`{f}_{\text{ad}}^{S}(S)`" "◊", "FICKA_PA "," :math:`{f}_{\text{ad}}^{\text{ad}}({P}_{\text{ad}})`" "◊", "FICKA_PL "," :math:`{f}_{\text{ad}}^{\text{lq}}({P}_{\text{lq}})`" "◊", "D_ FA_T "," :math:`\frac{\partial {f}_{\text{vp}}^{T}}{\partial T}(T)`" "◊", "LAMB_T "," :math:`{\lambda }_{T}^{T}(T)`" "◊", "LAMB_T_L "," :math:`{\lambda }_{T}^{T}(T)` according to L" "◊", "LAMB_T_N "," :math:`{\lambda }_{T}^{T}(T)` according to N" "◊", "LAMB_T_T "," :math:`{\lambda }_{T}^{T}(T)` according to T (2D)" "◊", "D_ LB_T "," :math:`\frac{\mathrm{\partial }{\lambda }_{T}^{T}(T)}{\mathrm{\partial }T}`" "◊", "D_ LB_T_L "," :math:`\frac{\partial {\lambda }_{T}^{T}(T)}{\partial T}` according to L" "◊", "D_ LB_T_N "," :math:`\frac{\partial {\lambda }_{T}^{T}(T)}{\partial T}` according to N" "◊", "D_ LB_T_T "," :math:`\frac{\partial {\lambda }_{T}^{T}(T)}{\partial T}` according to T" "◊", "LAMB_PHI "," :math:`{\lambda }_{\varphi }^{T}(\varphi )`" "◊", "D_ LB_PHI "," :math:`\frac{\partial {\lambda }_{\varphi }^{T}(\varphi )}{\partial \varphi }`" "◊", "LAMB_S "," :math:`{\lambda }_{S}^{T}(S)`" "◊", "DLAMBS "," :math:`\frac{\partial {\lambda }_{S}^{T}(S)}{\partial S}`" "◊", "LAMB_CT "," :math:`{\lambda }_{\text{CT}}^{T}`" "◊", "LAMB_CL "," :math:`{\lambda }_{\text{CT}}^{T}` according to L" "◊", "LAMB_CN "," :math:`{\lambda }_{\text{CT}}^{T}` according to N" "◊", "LAMB_CT "," :math:`{\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:* :math:`{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* :math:`{\rho }_{s}` *with its initial value* :math:`{\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 [:ref:`éq 3.2.6.3 <éq 3.2.6.3>`] is :math:`\frac{{\text{dp}}_{\text{vp}}}{{\rho }_{\text{vp}}}\text{=}\frac{{\text{dp}}_{w}}{{\rho }_{w}}\text{+}L\frac{\text{dT}}{T}` with :math:`L={h}_{\text{vp}}^{m}-{h}_{w}^{m}`. Also :math:`{\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 :math:`{\text{dp}}_{\text{as}}={\text{dp}}_{\text{gz}}-{\text{dp}}_{\text{vp}}`. By combining these equations we obtain: :math:`\{\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: :math:`\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}` :math:`\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 :math:`{\text{dp}}_{\text{as}}={\text{dp}}_{\text{gz}}-{\text{dp}}_{\text{vp}}` and :math:`{\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: :math:`\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 :math:`\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 [:ref:`§5.5.2 <§5.5.2>`]. The following will be noted: :math:`\mathrm{A1}\text{=}\frac{\text{RT}}{{K}_{H}}\text{-}\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}` and :math:`\mathrm{A2}=\frac{{\rho }_{w}}{{\rho }_{\text{vp}}}(\frac{\mathrm{RT}}{{K}_{H}}-1)` :math:`\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}` :math:`\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: :math:`\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}})` :math:`\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}}})` :math:`\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})` :math:`\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}})` :math:`\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}}})` :math:`\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})` :math:`\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}}))` :math:`\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}}}))` :math:`\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: :math:`\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}})` :math:`\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}}})` :math:`\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)` :math:`\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}})` :math:`\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}}})` :math:`\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)` :math:`\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}` :math:`\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}` :math:`\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:* :math:`{h}_{\text{lq}}^{{m}_{0}}\text{=}0` *and* :math:`{h}_{\text{vp}}^{{m}_{0}}\text{=}{L}_{0}` *.* **A5.1** **Energy equation** The table above recalls the two formulations: .. csv-table:: "**Notes** *Code_Aster*", "**Ratings ANDRA**" ":math:`{h}_{\text{lq}}^{m}` "," :math:`\text{=}\frac{{\varphi }_{w}}{{\rho }_{w}{S}_{w}n}`" ":math:`{h}_{\text{vp}}^{m}` "," :math:`\text{=}\frac{{\varphi }_{v}}{{\rho }_{v}(1\text{-}{S}_{w})n}`" ":math:`{h}_{\text{as}}^{m}` "," :math:`\text{=}\frac{{\varphi }_{\text{as}}}{{\rho }_{\text{as}}(1\text{-}{S}_{w})n}`" ":math:`{M}_{\text{lq}}` "," :math:`\text{=}{\rho }_{w}{f}_{w}`" ":math:`{M}_{\text{as}}` "," :math:`\text{=}{\rho }_{\text{as}}{f}_{\text{as}}`" ":math:`{M}_{\text{vp}}` "," :math:`\text{=}{\rho }_{v}{f}_{v}`" Rewriting the energy equation of*Code_Aster* with these notations, we find: :math:`\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 :math:`{\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 :math:`{\omega }_{l}^{a}\text{=}{\rho }_{\text{ad}}`. :math:`H` is expressed in :math:`\mathit{Pa}`. In the formulation of *Code_Aster*, we recall that Henry's law is expressed in the form: :math:`{C}_{\text{ad}}^{\text{ol}}\text{=}\frac{{\rho }_{\text{ad}}}{{M}_{\text{ad}}^{\text{ol}}}` with :math:`{C}_{\text{ad}}^{\text{ol}}\text{=}\frac{{p}_{\text{as}}}{{K}_{H}}`. :math:`{K}_{H}` is expressed in :math:`\mathit{Pa.}{m}^{3}\mathrm{.}{\mathit{mol}}^{\mathrm{-}1}`. So we have the equivalence: :math:`{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: :math:`{f}_{{\text{Diff}}_{v}}=-{D}_{v}\text{.}\nabla {\omega }_{g}^{e}` with the concentration defined as the molar ratio in the gas: :math:`{\omega }_{g}^{e}\text{=}\frac{{n}_{\nu }}{{n}_{g}}`. In *Code_Aster*, this same flow is written: :math:`{f}_{{\text{Diff}}_{v}}={F}_{\text{vp}}\nabla {C}_{\text{vp}}` with the Fick vapor coefficient :math:`{F}_{\text{vp}}=\frac{{D}_{\text{vp}}}{{C}_{\text{vp}}(1-{C}_{\text{vp}})}` and :math:`{D}_{\mathit{vp}}` the Fick diffusion coefficient of the gas mixture. :math:`{C}_{\mathit{vp}}` is defined as the pressure ratio such as: :math:`{C}_{\text{vp}}\text{=}\frac{{p}_{\text{vp}}}{{p}_{\text{gz}}}`. The ideal gas law allows us to write that :math:`{C}_{\text{vp}}\text{=}{\omega }_{g}^{e}` therefore :math:`\nabla {\omega }_{g}^{e}\text{=}\nabla {C}_{\text{vp}}` and :math:`{f}_{{\text{Diff}}_{v}}\text{=}{D}_{v}\text{.}\nabla {C}_{\text{vp}}`. So the *Code_Aster*/ANDRA equivalent is simply written: :math:`{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 :math:`{f}_{{a}_{{\text{ds}}_{e}}}\text{=}{D}_{a}\text{.}\nabla {\omega }_{l}^{a}` with :math:`{\omega }_{l}^{a}\text{=}\frac{{\rho }_{\text{ad}}}{{M}_{\text{ad}}^{\text{ol}}}`. In Code_Aster, this same flow is written: :math:`{f}_{{a}_{{\text{ds}}_{v}}}\text{=}{F}_{\text{ad}}\nabla {C}_{\text{ad}}` with the air-dissolved Fick coefficient :math:`{F}_{\text{ad}}=\frac{{D}_{\text{ad}}}{{C}_{\text{ad}}(1-{C}_{\text{ad}})}` and :math:`{D}_{\mathit{ad}}` the Fick diffusion coefficient of the liquid mixture. :math:`{C}_{\mathit{ad}}` is defined as: :math:`{C}_{\text{ad}}\text{=}{w}_{l}^{a}`. So: :math:`{F}_{\text{ad}}\text{=}{D}_{a}`.