Water-oil seepage-thermal coupled 2D-3D constitutive law for solid elements.
Implemented by: F. Collin, J.P. Radu, R. Charlier, 2000
This law is used for water seepage - oil seepage - thermal coupled for non-linear analysis in 2D/3D porous media.
Prepro: LWAPET.F
Lagamine: WAPET.F, WAPET3.F
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | NO |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 173 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| 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 |
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 ($k_f$) |
| 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 Appendix 8
24 for both 2D and 3D state
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 |
16
| 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 |