====== WAPET / WAPET3 ======

===== Description =====
Water-oil seepage-thermal coupled 2D-3D constitutive law for solid elements. \\
\\
Implemented by: F. Collin, J.P. Radu, R. Charlier, 2000

==== The model ====

This law is used for water seepage - oil seepage - thermal coupled for non-linear analysis in 2D/3D porous media.

==== Files ====

Prepro: LWAPET.F \\
Lagamine: WAPET.F, WAPET3.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| 173|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (7I5) ^^
|IANI| = 0 : isotropic permeability case |
|:::|≠ 0 : anisotropic permeability case |
|IKW| Formulation index for $k_w$ |
|IKP| Formulation index for $k_p$ |
|ISRW| Formulation index for $S_w$ |
|ITHERM| Formulation index for $\Gamma_T$ | 
|IFORM| = 1 : tangent formulation \\ \[\left\{\begin{array}{l} f_{we} = \dot{M}_w=\left(\dot{\varepsilon}_v.S_w+n.S_w.\frac{\dot{\rho}_w}{\xi_w}+n.\dot{S}_w\right)\rho_w \\ f_{pe} = \dot{M}_p=\left(\dot{\varepsilon}_v.S_p+n.S_p.\frac{\dot{\rho}_p}{\xi_p}+n.\dot{S}_p\right)\rho_p \end{array}\right.\] |
|:::| = 0 : secant formulation \\ \[\left\{\begin{array}{l} f_{we} = \dot{M}_w = \frac{(n^BS_w^B\rho_w^B-n^AS_w^A\rho_w^A)}{\Delta t} \\ f_{pe} = \dot{M}_p = \frac{(n^BS_p^B\rho_p^B-n^AS_p^A\rho_p^A)}{\Delta t}\end{array}\right.\] |
|ICONV| = 0 if no convectif term in the heat transport problem |
|:::| = 1 else |
|ITEMOIN| = 0 if analytic matrix |
|:::| = 1 if semi-analytic matrix |
|IKRN| = 1 if Kozeni-Karmann formulation |

==== Real parameters : Permeabilities definition ====

The permeability $k_f$ is an __intrinsic__ permeability ([L$^2$]) : \[k_{f, intrinsic} = K_f.\frac{\mu_f}{\rho_f.g}\]\[[L^2]=[LT]^{-1}\frac{[ML^{-1}T^{-1}]}{[ML^{-3}][LT^{-2}]}\]

__If IANI ≠ 0__
^ Line 1 (4G10.0) - Repeat IANI times (I=1,IANI)^^
|PERME(I)| Soil anisotropic intrinsic permeability ($k_f$) in the direction I |
|COSX(I)| Director cosinus of the direction I (in 3D state) |
|COSY(I)| Director cosinus of the direction I (in 3D state) |
|COSZ(I)| Director cosinus of the direction I (in 3D state) |

Permeabilities in different directions can be input ($I_{max} = 10$). The effect of these permeabilities will be summed.

__Else if IANI = 0__
^ Line 1 (1G10.0) ^^
|PERME| Soil isotropic intrinsic permeability ($k_f$) |

==== Real parameters ====

^ Line 1 (4G10.0) ^^^
|POROS| Soil porosity (=$n$) ||
|T0| Definition temperature (=$T_0$) | [°K] |
|PW0| Definition liquid pression (=$p_{w,0}$) | [Pa] |
|PP0| Definition oil pression (=$p_{p,0}$) | [Pa] |
^ Line 2 (7G10.0) ^^^
|VISCW0| Liquid dynamic viscosity (=$\mu_{w,0}$) | [Pa.s] |	
|ALPHW0| Liquid  dynamic viscosity thermal coefficient (= $\alpha_w^T$) | [°K$^{-1}$] |
|RHOW0|Liquid density (=$\rho_{w,0}$) |[$kg.m^{-3}$] |
|UXHIW0| Liquid compressibility coefficient (= 1/$\xi_w$) | [Pa$^{-1}$] |
|BETAW0| Liquid thermal expansion coefficient (=$\beta_w^T$) | [°K$^{-1}$] |
|CONW0|Liquid thermal conductivity (=$\Gamma_{w,0}$) | [$W.m^{-1}$.°K$^{-1}$] |
|GAMW0| Liquid thermal conductivity coefficient (=$\gamma_w^T$) | [ °K$^{-1}$] |
^ Line 3 (2G10.0) ^^^
|CPW0| Liquid specific heat (=$c_{p,wo}$) | [J.kg$^{-1}$.°K$^{-1}$] |
|HEATW0| Liquid specific heat coefficient (=$H_w^T$) | [°K$^{-1}$] |
^ Line 4 (7G10.0) ^^^
|VISCP0| Oil dynamic viscosity (=$\mu_{w,0}$) | [Pa.s] |	
|ALPHA0| Oil dynamic viscosity thermal coefficient (= $\alpha_w^T$) | [°K$^{-1}$] |
|RHOP0| Oil density (=$\rho_{w,0}$) | [kg.m$^{-3}$] |
|UXHIP0| Oil compressibility coefficient (= 1/$\xi_w$) | [Pa$^{-1}$] |
|BETAP0| Oil thermal expansion coefficient (=$\beta_w^T$) | [°K$^{-1}$] |
|CONP0| Oil thermal conductivity (=$\Gamma_{w,0}$) | [W.m$^{-1}$.°K$^{-1}$] |
|GAMP0|Oil thermal conductivity coefficient (=$\gamma_w^T$) | [°K$^{-1}$] |
^ Line 5 (2G10.0) ^^^
|CPP0| Oil specific heat (=$c_{p,wo}$) | [J.kg$^{-1}$.°K$^{-1}$] |
|HEATP0| Oil specific heat coefficient (=$H_w^T$) | [°K$^{-1}$] |
^ Line 6 (5G10.0) ^^^
|BETAS0| Solid thermal expansion coefficient (=$\beta^T_s$) | [°K$^{-1}$] |
|CONS0| Solid thermal conduction (=$\Gamma_{s,0}$) | [W.m$^{-1}$.°K$^{-1}$] |
|GAMS0|Solid conduction coefficient (=$\gamma^T_s$) | [°K$^{-1}$] |
|CPS0| Solid specific heat (=$c_{p,so}$) | [J.kg$^{-1}$.°K$^{-1}$] |
|HEATS0| Solid specific heat coefficient (=$H_s^T$) | [°K$^{-1}$] |
^ Line 7 (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 8 (2G10.0) ^^^
|CKP1| 1st  coefficient of the function $k_{rp}$ ||
|CKP2| 2nd coefficient of the function $k_{rp}$ ||
^ Line 9 (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 [Pa] ||
^ Line 10 (4G10.0) ^^^
|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 $\Gamma_T$ ||
^ Line 11 (3G10.0) ^^^
|KRMIN| Minimum value of $k_r$ ||
|EXPM| m Exponent of Kozeni-Karmann formulation ||
|EXPN|n Exponent of Kozeni-Karmann formulation ||

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]]

===== Stresses =====
==== Number of stresses ====

24 for both 2D and 3D state \\

==== 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)| Oil velocity in the X direction (=$f_{ax}$) |
|SIG(6)| Oil velocity in the Y direction (=$f_{ay}$) |
|SIG(7)| Oil 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 to 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)| Oil velocity in the X direction (=$f_{ax}$) |
|SIG(6)| Oil velocity in the Y direction (=$f_{ay}$) |
|SIG(7)| Oil velocity in the Z direction (=$f_{az}$) |
|SIG(8)| Oil velocity stored (=$f_{ae}$) |
|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 to 24)| None |

===== State variables =====
==== Number of state variables ====
16
==== List of state variables ====
|Q(1)| Water relative permeability (=$k_{rw}$) |
|Q(2)| Oil relative permeability (=$k_{rp}$) |
|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)| Oil specific mass (=$\rho_p$) |
|Q(8)| "Pe number" = convective effect/conductive effect \[=\frac{\rho_f.c_f.T.\underline{q}}{\Gamma_{av}.\underline{grad}(T)}\] |
|Q(9)| Water content (=$w$) |
|Q(10)| Volume related to the Gauss point |
|Q(11)| Porous volume related to the Gauss point |
|Q(12)| Not used | 
|Q(13)| Water mass |
|Q(14)| Oil mass |
|Q(15)| PERMINT |