====== WPRO ======
===== Description =====

Constitutive law for well limit condition for element [[elements:wepro|WEPRO]] \\
\\
Implemented by: F. Collin, 2001

==== The model ====

This constitutive law is used for non-linear flow analysis of solids and allows to model fluids exchanges between a well and the reservoir.\\

The retention curve allows to compute the relative permeabilities and to see which fluids is mobile.


==== Files ====

Prepro: LWPRO.F \\
Lagamine: WPRO.F

===== Availability =====


|Plane stress state| NO |
|Plane strain state| YES|
|Axisymmetric state| YES|
|3D state| NO |
|Generalized plane state| NO |

===== Input file ===== 

==== Parameters defining the type of constitutive law ====

^ Line 1 (2I5, 60A1)^^
|IL|Law number|
|ITYPE| 197|
|COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing|

==== Integer parameters ====

^ Line 1 (4I5) ^^
|ISR| Formulation index for $S_w$ (see [[appendices:a8|Appendix 8]]) |
|IKW| Formulation index for $k_{r,w}$ (see [[appendices:a8|Appendix 8]]) |
|IKP| Formulation index for $k_{r,p}$ (see [[appendices:a8|Appendix 8]]) |
|IINJ| Index to specify the type of well |
|:::| = 0 : production well |
|:::| = 1 : injection well |

==== Real parameters ====

^ Line 1 (1G10.0) ^^
|TFACT| Soil geometric factor |
^ Line 2 (4G10.0) ^^
|VISCW| Water dynamic viscosity |
|RHOW| Water density |
|VISCP| Oil dynamic viscosity |
|RHOP| Oil density |
^ Line 3 (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 4 (3G10.0) ^^
|CKW1| 1st  coefficient of the function $k_{r,w}$ |
|CKW2| 2nd coefficient of the function $k_{r,w}$ |
|CKW3| 3rd coefficient of the function $k_{r,w}$ |
^ Line 5 (2G10.0) ^^
|CKP1| 1st  coefficient of the function $k_{r,p}$ |
|CKP2| 2nd coefficient of the function $k_{r,p}$ |
^ Line 6 (1G10.0) ^^
|KRMIN| Minimum value of $k_r$ |

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

2

==== Meaning ====

|SIG(1)| Current value of water flow |
|SIG(2)| Current value of oil flow |

===== State variables =====

==== Number of state variables ====

5

==== List of state variables ====

|Q(1)| none |
|Q(2)| Water saturation degree |
|Q(3)| Element jacobian |
|Q(4)| Well pressure at the integration point |
|Q(5)| Generalised reservoir pressure at the integration point |

===== Comments =====

For IINJ = 0 (production well), the fluid flow has the following expression : \[SIG(1) = f_w=T^{well}.\frac{k_{rw}}{\mu_w}(p_w-p_{well})\]\[SIG(2) = f_o=T^{well}.\frac{k_{ro}}{\mu_o}(p_o-p_{well})\]

For IINJ = 1 (injection well), the water injection flow has the following expression : \[SIG(1) = f_w = T^{well}.\left(\frac{k_{rw}}{\mu_w}+\frac{k_{ro}}{\mu_o}\right).(p_w-p_{well})\]\[SIG(2)=0\]