Lagamine

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====== WAVAT/WAVAT3 ======
===== Description =====
Water - air seepage - thermal coupled – vapour diffusion 2D/3D constitutive law for solid elements
==== The model ====
This law is only  used for water seepage - air seepage-thermal coupled and vapour diffusion for non linear analysis in 2D/3D porous media.

=== Conservation de la masse d’eau (Liquide + vapeur) ===
\[
\underbrace{\frac{\partial}{\partial t} (\rho_s . n . S_{r,w}) + div(\rho_w \vec{q_l})}_{\text{Liquide}} + \underbrace{\frac{\partial}{\partial t} (\rho_v . n . S_{r,g}) + div(\rho_v \vec{q_g})}_{Vapeur} = 0
\]

=== Ecoulement du liquide et de la vapeur ===
En partant de l’équation de Darcy, la vitesse du liquide (Volume de fluide par unité de surface de sol) est donnée par :
\[
\vec{q_l} = - \frac{k_w}{\mu_w}\left[ \vec{grad}(p_w) + g \rho_w \vec{grad}(y)\right]\ \text{où}\ k_w = K_w \frac{\mu_w}{\rho_w g}\left[ m^2\right]
\]

La vapeur se déplace uniquement dans les pores occupés par la phase gazeuse et le trajet qu'elle parcourt dépend de la tortuosité :
\[
\vec{i}_v = - n S_{r,g} \tau D \rho_s \vec{grad} \omega_v
\]
Où $\omega_v = \rho_v/\rho_g$ est la teneur massique d’air sec dans le mélange gazeux.

=== Equations d’état du liquide ===

  - Viscosité dynamique $\mu_w$: \[\mu_w (T) = \mu_{w,0} - \alpha_w^T \mu_{w0}(T-T_0)\] 
  - Masse volumique $\rho_w$: \[\rho_w (T, p_w) = \rho_{wo}\left[ 1+\frac{p_w-p_{w0}}{\chi_w} - \beta_w^T (T-T_0) \right]\]
  - Perméabilité intrinsèque $k_w$: \\ Dépendance avec le degré de saturation $S_w$ : $k_{r,w} = f(S_w)$ avec $k_{w,eff} = k_f k_{r,w}$
  - Degré de saturation $S_w$: \\ Dépendance avec la succion $s = p_a - p_w : S_w = f(s)$

=== Conservation de la masse de l’air sec ===
\[\frac{\partial}{\partial t} (\rho_a . n . S_{r,g}) + div(\rho_a \vec{q_g}) + div(\vec{i_a}) = 0\]

=== Ecoulement de la phase gazeuse et l’air sec ===
En partant de l’équation de Darcy , la vitesse de la phase gazeuse est donnée par :
\[
\vec{q_g} = - \frac{k_g}{\mu_g}\left[ \vec{grad}(p_g) + g \rho_g \vec{grad}(y)\right]\ \text{où}\ k_g = K_g \frac{\mu_g}{\rho_g g}\left[ m^2\right]
\]
La vitesse de diffusion de l'air sec est liée à un gradient de la masse volumique de l'air. En utilisant la théorie de la diffusion adaptée au cas des milieux poreux, nous pouvons écrire : 
\[
\vec{i}_a = - n S_{r,g} \tau D \rho_g \vec{grad} \omega_a = -\vec{I}_v
\]
Où $\omega_a = \rho_a/\rho_g$ est la teneur massique d’air sec dans le mélange gazeux.

=== Equation d’état de l’air sec ===

  - Viscosité dynamique $\mu_a$ : dépendance avec la température \[\mu_a (T) = \mu_{a,0} - \alpha_a^T \mu_{a0}(T-T_0)\]
  - Masse volumique $\rho_a$ :\\ //Hypothèse// : L’air est considéré comme un gaz parfait. \[\rho_a (T, p_a) = \rho_{a,0}\frac{p_a}{p_{a,0}}\frac{T_0}{T} \]
  - Perméabilité intrinsèque $k_g$: \\ Dépendance avec le degré de saturation $S_g$ : $k_{r,g} = f(S_g)$ avec $k_{g,effectif} = k_{g, intrinsic}k_{a,w}$
  - Degré de saturation en liquide $S_g$: \\ Dépendance avec la succion $s = p_g - p_w$ \\ $S_g = 1-S_w$ d’où voir partie liquide

=== Conservation en volume de la chaleur ===
\[\dot{S_T} + \dot{E}_{H_2O}^{w\rightarrow v}L + div(\vec{V}_T) - Q_T = 0\]
Où $S_T$ représente la quantité de chaleur emmagasinée, $L$ la chaleur latente de vaporisation, $\vec{V}_T$ le flux de chaleur et $Q_T$ une source de chaleur en volume.\\

Cette dernière équation peut être transformée en utilisant l'équation de bilan de la vapeur d'eau:
\[
\dot{S_T} + \dot{S}_vL + div(\vec{V}_T) + div(\vec{V}_v) L - Q_T = 0
\]

=== Quantité de chaleur emmagasinée par unité de volume ===
La quantité d'enthalpie du système est définie comme la somme des contributions de chaque espèce du système: 
\[
S_T = H_m = \sum_i H_i = \sum_i \theta_i \rho_i c_{p,i} (T-T_0)
\]
Les contributions de chaque composante à l’enthalpie du système s'exprime selon :
\[
\begin{array}{l}
	H_w = n.S_{r,w} \rho_w c_{p,w} (T-T_0) \\
	H_a = n(1-S_{r,g})\rho_ac_{p,a} (T-T_0)\\
	H_s = (1-n)\rho_s c_{p,s} (T-T_0)\\
	H_v = n(1-S_{r,g})\rho_v c_{p,v} (T-T_0)
\end{array}
\]
Un dernier terme, lié à la vaporisation de l’eau, contribue également à l'emmagasinement de chaleur et dépend de la quantité de vapeur et la chaleur latente de vaporisation.
\[
H_{Lat} = nS_{r,g} \rho_{v} L
\]

=== Transfert de la chaleur par unité de volume ===
\[
\vec{V_T} + \vec{V_v}L = \underbrace{- \Gamma_m \vec{\nabla}T}_{conduction} + \underbrace{\left[c_{p,w}\rho_w \vec{q}_l + c_{p,a}(\vec{i}_a + \rho_a \vec{q}_g) + c_{pv}(\vec{i}_v+\rho_v\vec{q}_g)\right](T-T_0)}_{convection} + \underbrace{\left[\vec{i}_v + \rho_v \vec{q}_g\right] L}_{Latente}
\]

=== Equations d’état ===

  - $\rho_w, \vec{f_w}, S_w$: Voir partie eau
  - $\rho_a, \vec{f_a}, S_a$ : Voir partie air
  - Les conductivités thermiques $\Gamma_w,\Gamma_a$ et $\Gamma_s$ : \[\begin{array}{l}\Gamma_w(T) = \Gamma_{w,0} + \Gamma_{w,0} \gamma_w^T (T-T_0)\\ \Gamma_a(T) = \Gamma_{a,0} + \Gamma_{a,0} \gamma_a^T (T-T_0)\\ \Gamma_s(T) = \Gamma_{s,0} + \Gamma_{s,0} \gamma_s^T (T-T_0) \end{array}\]
  - Les chaleurs spécifiques $c_{p,w},c_{p,a}$ et $c_{p,s}$ :\[\begin{array}{l} c_{p,w}(T) = c_{p,w0} + c_{p,w0} H_w^T (T-T_0) \\ c_{p,a}(T) = c_{p,a0} + c_{p,a0} H_a^T (T-T_0)\\ c_{p,s}(T) = c_{p,s0} + c_{p,s0} H_s^T (T-T_0)\end{array}\]

==== Files ====
Prepro: LWAVAT.F \\
===== Availability =====
|Plane stress state| NO |
|Plane strain state| YES |
|Axisymmetric state| YES |
|3D state| YES |
|Generalized plane state| NO |
===== Input file =====
==== Parameters defining the type of constitutive law ====
^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 171 (= 174 in LOI2 for 3D state) |
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|
==== Integer parameters ====
^ Line 1 (18I5) ^^
|IANI|= 0, isotropic permeability case|
|:::| ≠ 0, anisotropic permeability case|
|IKW|formulation index for $k_w$|
|IKA|formulation index for $k_a$|
|ISRW|formulation index for $S_W$|
|ITHERM|formulation index for $\Gamma_T$|
|IVAP|= 0 if no vapour diffusion in the problem|
|:::|= 1 else|
|IFORM|= 1  tangent formulation \[ \left\{\begin{array}{l}f_{we} = \dot{M}_w = (\dot{\varepsilon}_v S_w + n S_w\frac{\dot{\rho}_w}{\chi_w} + n \dot{S}_w) \rho_w \\ f_{ae} = \dot{M}_a = (\dot{\varepsilon}_V S_a + n S_a\frac{\dot{p}_a}{p_a} + n \dot{S}_a) \rho_a \end{array}\right. \]|
|:::|= 0  secant formulation \[ \left\{\begin{array}{l}	f_{we} = \dot{M}_w = (n^BS_w^B\rho_w^B - n^AS_w^A\rho_w^A)/\Delta t \\ f_{ae} = \dot{M}_a = (n^BS_a^B\rho_a^B - n^AS_a^A\rho_a^A)/\Delta t		\end{array}\right. \]|
|ICONV|= 0 if no convectif term in the heat transport problem|
|:::|= 1 else|
|ITEMOIN|= 0 if analytic matrix (can be used only if IVAP = 0  if no vapour diffusion in the problem)|
|:::|= 1 if semi-analytic matrix (can be used in all the problems)|
|IKRN|= 1 if Kozeni-Karmann formulation|
|:::|= 2, GDR Momas relation	$K=f(n)$|
|:::|= 3 Coupling permeability-deformation $K=f(\varepsilon_n)$ (only in 2D)|
|IGAS|= 0 if gas is air|
|:::|= 1 if gas is $H_2$|
|:::|= 2 if gas is $N_2$|
|:::|= 3 if gas is Ar|
|:::|= 4 if gas is He|
|:::|= 5 if gas is $CO_2$|
|IENTH|= 0 if we define $\rho$ and $C_p$ for each constituent \[\left\{\begin{array}{l}H_w = N.S_{r,w} \rho_w c_{p,w} (T-T_0) \\ H_v = n(1-S_{r,w})\rho_v c_{p,v} (T-T_0)\\ H_a = n(1-S_{r,w})\rho_ac_{p,a} (T-T_0)\\ H_{a-d} = n S_{r,w} H\rho_ac_{p,a} (T-T_0)\\ H_s = (1-n)\rho_s c_{p,s} (T-T_0) \end{array}\right.\]|
|:::|= 1 if we define $\rho C_p$ equivalent for the medium and constant	\[H_m = \rho C_p (T-T_0)\]|
|:::|= 2  if we define $\rho C_p$ equivalent for the medium depending on temperature: \[\left\{\begin{array}{l}			\rho C_p = RHOC1\ \text{if }\ T>T_u \\ \rho C_p = RHOC2\ \text{if }\ T<T_f \\ \rho C_p = \frac{RHOC1-RHOC2}{T_u-T_f}(T-T_u) + RHOC1\ \text{if }\ T_f\leq T \leq T_u\\ \end{array} \right.\] \[T_u = CLT3\] \[T_f = CLT4\]|
|IANITH|= 0, isotropic conductivity case|
|:::|= 1, anisotropic conductivity case|
|IVISCW|= 0, $\mu_w = \mu_{w,0}\left( 1-ALPHW0(T-T_0)) \right)$|
|:::|= 1, $\mu_w = 0.6612(T-229)^{-1.562}$|
|IXHIW|	= 0, constant water compressibility|
|:::|= 1, $\chi_w = \chi_w + \frac{H}{p_a}$ ( $p_a$ is partial pressure of air and H is Henry coefficient)|
|IDIFF|= 0, with diffusion of dissolved air|
|:::|≠ 0, divisor (integer becomes real) of diffusion coefficient of dissolved air|
|ISTRUCT|= 0, constant permeability|
|:::|= 1, permeability depends on microstructure evolution|
|ICOAL|= 0, solid conductivity: $\Gamma_s = \Gamma_{s,0}(1+GAMS0(T-T_0))$|
|:::|= 1, solid conductivity: $\Gamma_s = \Gamma_{s,0}GAMS0(T-T_0)^3$|

==== Real parameters: permeabilities definition ====
The permeability $k_f$ is an \underline{intrinsic} permeability ($\left[L^2\right]$) $\boxed{ \begin{array}{l} k_{f, intrinsic} = K_f \frac{\mu_f}{\rho_f g}\\ \left[ L^2 \right] = \left[ LT^{-1} \right] \frac{ \left[ ML^{-1}T^{-1}\right]}{\left[ML^{-3}\right]\left[LT^{-2}\right]} \end{array}}$\\
^If IANI ≠ 0 (4G10.0) - Repeated IANI times^^
|PERME(I)|soil anisotropic int. permeability ($k_f$) in the direction I|
|COSX(I)|director cosinus of the direction I \\ (in 3d state)|  	
|COSY(I)|:::|
|COSZ(I)|:::|
Permeabilities in different directions can be input ( $I_{max}= 10$ ).The effect of these permeabilities will be summed.
^If IANI = 0 (1G10.0)^^
|PERME|soil isotropic intrinsic permeability ($k_f$)|

==== Real parameters ====
^ Line 1 (5G10.0) ^^
|POROS|soil porosity $(= n)$|
|TORTU|soil tortuosity $(=\tau )$|
|T0|definition temperature  $(=T_0)\ \left[K\right]$|
|PW0|definition liquid pression $(=p_{w,0})\ \left[Pa\right]$|
|PA0|definition gaz pression $(=p_{a,0})\ \left[Pa\right]$|
^ Line 2 (7G10.0) ^^
|VISCW0|liquid dynamic viscosity $(=\mu_{w,0}\ \left[ Pa.s \right]$|	
|ALPHW0|liquid  dynamic viscosity thermal coefficient $(=\alpha_{w}^T)\ \left[K^{-1}\right]$|
|RHOW0|liquid density $(=\rho_{w,0})\ \left[ kg.m^{-3}\right]$
|UXHIW0|liquid compressibility coefficient $(=1/ \chi_{w})\ \left[ Pa^{-1}\right]$|
|BETAW0|liquid thermal expansion coefficient $(=\beta{w}^T\ \left[K^{-1}\right]$|
|CONW0|liquid thermal conductivity $(=\Gamma_{w,0})\ \left[W.m^{-1}.K^{-1}\right]$|
|GAMW0|liquid thermal conductivity coefficient $(=\gamma_{w}^T\ \left[K^{-1}\right]$|
^ Line 3 (3G10.0) ^^
|CPW0|liquid specific heat $(=c_{p,w0})\ \left[J.kg^{-1}.K^{-1}\right]$|
|HEATW0|liquid specific heat coefficient $(=H_{w}^T)\ \left[K^{-1}\right]$|
|EMMAG|storage coefficient $(=E_s)\ \left[ P_a^{-1}\right]$|
^ Line 4 (7G10.0) ^^
|VISCA0|gaz dynamic viscosity $(=\mu_{a,0})\ \left[Pa.s \right]$\
|ALPHW0|gaz  dynamic viscosity thermal coefficient$(=\alpha_{a}^T)\ \left[K^{-1}\right]$|
|RHOA0|gaz density$(=\rho_{a,0})\ \left[kg.m^{-3}\right]$|
|CONA0|gaz thermal conductivity $(=\Gamma_{a,0})\ \left[W.m^{-1}.K^{-1}\right]$|
|GAMA0|gaz thermal conductivity coefficient $(=\gamma_{a}^T)\ \left[K^{-1}\right]$|
|CPA0|gaz specific heat $(=c_{p,a0})\ \left[J.kg^{-1}.K^{-1}\right]$|
|HEATA0|gaz specific heat coefficient $(=H_{a}^T)\ \left[K^{-1}\right]$|
^ Line 5 (5G10.0) ^^
|BETAS0|solid thermal expansion coefficient $(=\beta_{s}^T)\ \left[K^{-1}\right]$|
|CONS0|solid thermal conduction$(=\Gamma_{s,0})\ \left[W.m^{-1}.K^{-1}\right]$|
|GAMS0|solid conduction coefficient $(=\gamma_{s}^T)\ \left[K^{-1}\right]$|
|CPS0|solid specific heat $(=c_{p,s0})\ \left[J.kg^{-1}. K^{-1}\right]$|
|HEATS0|solid specific heat coefficient $(=H_{s}^T)\ \left[ K^{-1}\right]$|
^ Line 6 (3G10.0) ^^
|CKW1|1st  coefficient of the function $k_{rw}$|
|CKW2|2nd coefficient of the function $k_{rw}$|
|CKW3|3rd coefficient of the function $k_{rw}$|
^ Line 7 (3G10.0) ^^
|CKA1|1st  coefficient of the function $k_{ra}$|
|CKA2|2nd coefficient of the function $k_{ra}$|
|CSR5|5th coefficient of the function $S_w$|
^ Line 8 (7G10.0/) ^^
|CSR1|1st  coefficient of the function $S_w$|
|CSR2|2nd coefficient of the function $S_w$|
|CSR3|3rd coefficient of the function $S_w$|
|CSR4|4th coefficient of the function $S_w$|
|SRES|residual saturation degree $(=S_{res})$|
|SRFIELD|field saturation degree $(=S_{r, field})$|
|AIREV|air entry value $\left[Pa\right]$|
^ Line 9 (7G10.0) ^^
|CLT1|1st  coefficient of the function $\Gamma_T$|
|CLT2|2nd coefficient of the function $\Gamma_T$|
|CLT3|3rd coefficient of the function $\Gamma_T$|
|CLT4|4th coefficient of the function $\Gamma_T$|
|RHOC|coefficient for enthalpie $\rho C_p$ (if ienth = 1)|
|RHOC1|1st coefficient for enthalpie $\rho C_p$ (if ienth = 2)|
|RHOC2|2nd coefficient for enthalpie $\rho C_p$ (if ienth = 2)|
^ Line 10 (4G10.0) ^^
|KRMIN|Minimum value of $kr$|
|HENRY|Henry coefficient|
|EXPM|for IKRN=1: $m$ Exponent of Kozeni-Karmann formulation|
|:::|for IKRN=3 : A parameter|
|EXPN|n Exponent of Kozeni-Karmann formulation|
^Line 11 - Only if IANITH ≠ 0 (4G10.0) - Repeat IANITH times^^
|CONDUC(I)|soil anisotropic conductivity in the direction I|
|COSX(I)|director cosinus of the direction I \\ (in 3d state)|
|COSY(I)|:::|
|COSZ(I)|:::|
|Thermal conductivities in different directions can be input (Imax = 10). The effect of these conductivities will be summed. In that case of anisotropic conductivity, conductivities remain constants for each direction: coefficients ITHERM, CLT1, CLT2, CLT3, CT4, CONS0, CONW0, and CONA0 are so not used in that case||
^If IANITH = 0: nothing^^
|Isotropic thermal conductivity is already defined by preceding coefficients ITHERM, CLT1, CLT2, CLT3, CT4, CONS0, CONW0, CONA0 …||

Following empirical formulations for describing the evolution of the relative permeability, the thermal conductivity and saturation with the suction are possible: see [[appendices:a8|Appendix 8]].
For any suction lower than air entry value (AIREV), the saturation is equal to SRFIELD value. \\ 
__Kozeny Karman formulation:__
\[K = C_0 \frac{n^{EXPN}}{(1-n)^{EXPM}}\]
$C_0$ is computed automatically from $C_0 = K_0 \frac{(1-n_0)^{EXPM}}{(n_0)^{EXPn}}$ \\
__GDR Momas formulation:__
\[
\frac{k}{k_0} = 1+EXPM\left[ \phi - \phi_0\right]^{EXPN}\ \text{où}\ EXPM = 2.10^{12}\ \text{et}\ EXPN = 3
\]

__Coupling permeability-deformation formulation:__ (only in 2D)
\[
K_{ij} = \sum_n K_n^0 (1+A\varepsilon_n^T)^3\beta_{ij} (\alpha_n)
\]
$\varepsilon_n^T$ tensile deformation \\
$\alpha_n$ crack orientation with horizontal \\
$A$ parameter of the crack
===== Stresses =====
==== Number of stresses ====
24
==== Meaning ====
__In 2D state :__
|SIG(1)|liquid velocity in the X direction $(=f_{wx})$||
|SIG(2)|liquid velocity in the Y direction $(=f_{wy})$||
|SIG(3)|liquid velocity stored  $(=f_{we})$||
|SIG(4)|none||
|SIG(5)|gas total velocity in the X direction $(=f_{ax})$|gas advection + \\ gas diffusion + \\ dissolved gas advection + \\ dissolved gas diffusion|
|SIG(6)|gas total velocity in the Y direction $(=f_{ay})$|:::|
|SIG(7)|gas total velocity stored $(=f_{ae})$|:::|
|SIG(8)|none|:::|
|SIG(9)|conductive heat flow in the X direction $(=f_{tx})$||
|SIG(10)|conductive heat flow in the Y direction $(=f_{ty})$||
|SIG(11)|energy accumulated by heat capacity $(=f_{te})$||
|SIG(12)|none||
|SIG(13)|Water vapour velocity in the X direction $(=f_{vx})$||
|SIG(14)|Water vapour velocity in the Y direction $(=f_{vy})$||
|SIG(15)|Water vapour stored $(=f_{ve})$||
|SIG(16)|none||
|SIG(17)|dissolved gas advection and diffusion velocity in the X direction||
|SIG(18)|dissolved gas advection and diffusion velocity in the Y direction||
|SIG(19)|dissolved gas advection and diffusion velocity stored||
|SIG(20)|none||
|SIG(21)|dissolved gas diffusion velocity in the X direction||
|SIG(22)|dissolved gas diffusion velocity in the Y direction||
|SIG(23)|dissolved gas and diffusion velocity stored||
|SIG(24)| none||
__In 3D state :__
|SIG(1)|liquid velocity in the X direction $(=f_{wx})$||
|SIG(2)|liquid velocity in the Y direction $(=f_{wy})$||
|SIG(3)|liquid velocity in the Z direction $(=f_{wz})$||
|SIG(4)|liquid velocity stored $(=f_{we})$||
|SIG(5)|gas total velocity in the X direction $(=f_{ax})$|gas advection + \\ gas diffusion + \\ dissolved gas advection + \\ dissolved gas diffusion|
|SIG(6)|gas total velocity in the Y direction $(=f_{ay})$|:::|
|SIG(7)|gas total velocity in the Z direction $(=f_{az})$|:::|
|SIG(8)|gas total velocity stored $(=f_{az})$|:::|
|SIG(9)|conductive heat flow in the X direction $(=f_{tx})$||
|SIG(10)|conductive heat flow in the Y direction $(=f_{ty})$||
|SIG(11)|conductive heat flow in the Z direction $(=f_{tz})$||
|SIG(12)|energy accumulated by heat capacity $(=f_{te})$||
|SIG(13)|Water vapour velocity in the X direction $(=f_{yx})$||
|SIG(14)|Water vapour velocity in the Y direction $(=f_{yy})$||
|SIG(15)|Water vapour velocity in the Z direction $(=f_{yz})$||
|SIG(16)|Water vapour stored $(=f_{ye})$||
|SIG(17)|dissolved gas advection and diffusion velocity in the X direction ||
|SIG(18)|dissolved gas advection and diffusion velocity in the Y direction ||
|SIG(19)|dissolved gas advection and diffusion velocity in the Z direction ||
|SIG(20)|dissolved gas advection and diffusion velocity stored ||
|SIG(21)|dissolved gas diffusion velocity in the X direction ||
|SIG(22)|dissolved gas diffusion velocity in the Y direction ||
|SIG(23)|dissolved gas diffusion velocity in the Z direction ||
|SIG(24)|dissolved gas and diffusion velocity stored||

===== State variables =====
==== Number of state variables ====
= 26 in 2D cases \\
= 16 in 3D cases
==== List of state variables ====
|Q(1)|water relative permeability $(=k_{rw})$ |
|Q(2)|air relative permeability $(=k_{ra})$ |
|Q(3)|Soil porosity (= n) |
|Q(4)|Soil saturation degree $(=S_w)$ |
|Q(5)|Suction $(=p_c = p_a-p_w)$ |
|Q(6)|water specific mass $(=\rho_w)$ |
|Q(7)|air specific mass $(=\rho_a)$ |
|Q(8)|"Pe number"	= convective effect / conductive effect \[= \frac{\rho_f . c_f . T . \vec{q}}{\Gamma_{av} . \vec{grad} (T)}\]|
|Q(9)|Water content (=w) |
|Q(10)|Vapour specific mass $(=\rho_v)$ |
|Q(11)|Vapour pressure $(=p_v)$ |
|Q(12)|Relative humidity $(=H_r)$ |
|Q(13)|Liquid water mass per unit soil volume |
|Q(14)|Dry air mass per unit soil volume |
|Q(15)|Vapour mass per unit soil volume |
|Q(16)|Intrinsic permeability |
|Q(17)|Gas soil saturation degree $(=S_g)$ |
|Q(18)|$\alpha (H_2, N_2, …)$ partial pressure $(=p_a^g = p^g - p_{H_2O}^g = \text{gas pressure-vapour pressure})$ |
|Q(19)|Area associated to one integration point |
|Q(20)|Dissolved air concentration $=\frac{\rho_{a-d}}{\rho_w + \rho_{a-d}} = \frac{H_a \rho_a}{\rho_w + H_a \rho_a}$|
|Q(21)|$K_{xx}$ (or zero if IANI = 0) |
|Q(22)|$K_{yy}$ (or zero if IANI = 0) |
|Q(23)|$K_{xy}$ (or zero if IANI = 0)  |
|Q(24)|$\varepsilon_1$ |
|Q(25)|$\varepsilon_2$ |
|Q(26)|$\alpha$ (= angle between principal stress and horizontal) |
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