Constitutive law of longitudinal and transversal flows (water, gas and thermal) in porous media for an interface element 2D (FAIF2B) or 3D (FAIF3B).
Implemented by: J.P. Radu, 2007-2008
This law is used for non-linear analysis of longitudinal seepage (water, gas and thermal) in porous media interface element.
The case of free surface seepage is also treated.
Transversal fluid (water, gas and thermal) transfer between the bodies depends upon the contact state:
Prepro: LINTFL.F
Lagamine: INTFL2.F, INTFL3.F
| Plane stress state | YES |
| Plane strain state | YES |
| Axisymmetric state | YES |
| 3D state | YES |
| Generalized plane state | YES |
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 119 (Rem: = 120 in LOI2 for 3D state) |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
| Line 1 (15I5) | |
|---|---|
| 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 \[ \left\{\begin{array}f_{we}=\dot{M}_w=\left(\dot{\varepsilon}_v.S_w+n.S_w.\frac{\dot{\rho}_w}{\xi_w}+n.\dot{S}_w\right)\rho_w \\ f_{ae}=\dot{M}_a=\left(\dot{\varepsilon}_v.S_a+n.S_a.\frac{\dot{\rho}_a}{p_a}+n.\dot{S}_a\right)\rho_a\end{array}\right. \] |
| = 0 : secant formulation \[ \left\{\begin{array}f_{we}=\dot{M}_w=\frac{n^BS_w^B\rho_w^B-n^AS_w^A\rho_w^A}{\Delta t} \\ f_{ae}=\dot{M}_a=\frac{n^BS_a^B\rho_a^B-n^AS_a^A\rho_a^A}{\Delta t} \end{array}\right.\] |
|
| ICONV | = 0 if no convective term in the heat transport problem |
| = 1 else | |
| ITEMOIN | = 0 if analytic matrix (can be used only if IVAP = 0 $\rightarrow$ 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 |
| IGAS | = 0 if gas is air |
| = 1 if gas is hydrogen | |
| = 2 if gas is nitrogen | |
| IENTH | = 0 if we define $\rho$ and $C_p$ for each constituent \[\left\{\begin{array}{l} H_w = n.S_{r,w}.\rho_w.c_{p,w}.(T-T_0) \\ H_v = n.(1-S_{r,w}).\rho_v.c_{p,v}.(T-T_0) \\ H_a = n.(1-S_{r,w}).\rho_a.c_{p,a}.(T-T_0) \\ H_{a-d}=n.S_{r,w}.H.\rho_a.c_{p,a}.(T-T_0) \\ H_s = (1-n).\rho_s.c_{pas}.(T-T_0) \end{array}\right.\] |
| = 1 if we define $\rho.C_p$ equivalent for the medium \[H_m=\rho.C_p.(T-T_0) \] | |
| INDIC | = 0, 1, 2 to define the outside pressures/temperature used in case of no contact (see The model) |
| IKE | = index of the longitudinal intrinsic permeability formulation |
| = 0 : $k_l=k_{l0}$ | |
| = 1 : $k_l=f(d)=k_{l0}.\dfrac{(d_0+d)^{exp}}{12}$ | |
| ITW | Formulation index for relative transversal transmissivity $t_w$ |
| ITA | Formulation index for relative transversal transmissivity $t_a$ |
| Line 1 (1G10.0) | ||
|---|---|---|
| PERMEA | Fault longitudinal intrinsic permeability (=$k_{l0}$) | |
| Line 2 (5G10.0) | ||
| POROS | Soil porosity (=$n$) | |
| TORTU | Soil tortuosity (=$\tau$) | |
| T0 | Definition temperature (=$T_0$) | [°K] |
| PW0 | Definition liquid pression (=$p_{w,0}$) | [Pa] |
| PA0 | Definition gaz pression (=$p_{a,0}$) | [Pa] |
| Line 3 (7G10.0) | ||
| VISCW0 | Liquid dynamic viscosity (=$\mu_{w,0}$) | [Pa.s] |
| ALPHW0 | Liquid dynamic viscosity thermal coefficient (=$\alpha^T_w$) | [°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 (=$\beta^T_w$) | [°K$^{-1}$] |
| CONW0 | Liquid thermal conductivity (=$\Gamma_{w,0}$) | [W.m$^{-1}$.°K$^{-1}$] |
| GAMW0 | Liquid thermal conductivity coefficient (=$\gamma^T_w$) | [°K$^{-1}$] |
| Line 4 (3G10.0) | ||
| CPW0 | Liquid specific heat (=$c_{p,wo}$) | [J.kg$^{-1}$.°K$^{-1}$] |
| HEATW0 | Liquid specific heat coefficient (=$H_w^T$) | [°K$^{-1}$] |
| EMMAG | Storage coefficient (=$E_s$) | [Pa$^{-1}$] |
| Line 5 (7G10.0) | ||
| VISCA0 | Gaz dynamic viscosity (=$\mu_{a,0}$) | [Pa.s] |
| ALPHW0 | Gaz dynamic viscosity thermal coefficient (=$\alpha^T_a$) | [°K$^{-1}$] |
| RHOA0 | Gaz density (=$\rho_{a,0}$) | [kg.m$^{-3}$] |
| CONA0 | Gaz thermal conductivity (=$\Gamma_{a,0}$) | [W.m$^{-1}$.°K$^{-1}$] |
| GAMA0 | Gaz thermal conductivity coefficient (=$\gamma^T_a$) | [°K$^{-1}$] |
| CPA0 | Gaz specific heat (=$c_{p,a0}$) | [J.kg$^{-1}$.°K$^{-1}$] |
| HEATA0 | Gaz specific heat coefficient (=$H_a^T$) | [°K$^{-1}$] |
| Line 6 (5G10.0) | ||
| BETAS0 | Solid thermal expansion coefficient (=$\beta_s^T$) | [°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) | ||
| CKA1 | 1st coefficient of the function $k_{ra}$ | |
| CKA2 | 2nd coefficient of the function $k_{ra}$ | |
| 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 (5G10.0) | ||
| 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 enthalpy $\rho.C_p$ (if IENTH = 1) | |
| Line 11 (4G10.0) | ||
| KRMIN | Minimum value of $k_r$ | |
| HENRY | Henry coefficient | |
| EXPM | m Exponent of Kozeni-Karmann formulation | |
| EXPN | n Exponent of Kozeni-Karmann formulation | |
| Line 12 (2G10.0) | ||
| D0 | Maximal fault closure in absolute value (correspond to D0 from INTME2 mechanical law) for formulation (IKE=1) | |
| EXP | Exponent (=$exp$) = 2 for cubic law | |
| EPAIS | Fault thickness (useful only if no Goodman's formulation in mechanical law) | |
| Line 13 (3G10.0) | ||
| THCONW | Fault water intrinsic transverse transmissivity ($T_{t,c}$) when contact occurs | |
| CONVECW | Fault water intrinsic transverse transmissivity ($T_{t,nc}$) when contact does not occur | |
| PWAMB | Atmosphere water pressure | |
| Line 14 (3G10.0) | ||
| THCONG | Fault gas intrinsic transverse transmissivity ($T_{t,c}$) when contact occurs | |
| CONVECG | Fault gas intrinsic transverse transmissivity ($T_{t,nc}$) when contact does not occur | |
| PGAMB | Atmosphere gas pressure | |
| Line 15 (3G10.0) | ||
| THCONT | Fault thermal transverse transmissivity ($T_{t,c}$) when contact occurs | |
| CONVECT | Fault thermal transverse transmissivitty ($T_{t,nc}$) when contact does not occur | |
| TAMB | Atmosphere temperature | |
| Line 16 (3G10.0) | ||
| CTW1 | 1st coefficient of the relative transversal transmissivity function $t_{rw}$ | |
| CTW2 | 2nd coefficient of the relative transversal transmissivity function $t_{rw}$ | |
| CTW3 | 3rd coefficient of the relative transversal transmissivity function $t_{rw}$ | |
| Line 17 (2G10.0) | ||
| CTA1 | 1st coefficient of the relative transversal transmissivity function $t_{ra}$ | |
| CTA2 | 2nd coefficient of the relative transversal transmissivity function $t_{ra}$ | |
The longitudinal permeability $k$ is an intrinsic permeability ([$L^2$]) where $K_l$ is the permeability coefficient [$LT^{-1}$]. \[k_{l,intrinsic}=K_l\frac{\mu_f}{\rho_f g}\]\[[L^2]=[LT^{-1}]\frac{[ML^{-1}T^{-1}]}{[ML^{-3}][LT^{-2}]}\]
Following empirical formulations for describing the evolution of the relative permeability, the thermal conductivity and saturation with the suction are possible : see Appendix 8.
For any suction lower than air entry value (AIREV), the saturation is equal to SRFIELD value.
The longitudinal permeability of the fault is computed according to IKE value :
20 for 3D state
16 for the other cases
In 2D state :
| SIG(1) | Longitudinal water flow in the interface element |
| SIG(2) | Stored water flow |
| SIG(3) | 1st transversal water flow in the interface element |
| SIG(4) | 2nd transversal water flow in the interface element |
| SIG(5) | Longitudinal gas flow in the interface element |
| SIG(6) | Stored gas flow |
| SIG(7) | 1st transversal gas flow in the interface element |
| SIG(8) | 2nd transversal gas flow in the interface element |
| SIG(9) | Longitudinal thermal flow in the interface element |
| SIG(10) | Stored thermal flow |
| SIG(11) | 1st transversal thermal flow in the interface element |
| SIG(12) | 2nd transversal thermal flow in the interface element |
| SIG(13) | Longitudinal vapour flow in the interface element |
| SIG(14) | Stored vapour flow |
| SIG(15) | 1st transversal vapour flow in the interface element |
| SIG(16) | 2nd transversal vapour flow in the interface element |
In 3D state:
| SIG(1) | Mass water flow in the x local direction of the interface element |
| SIG2) | Mass water flow in the y local direction of the interface element |
| SIG(3) | Stored water flow |
| SIG(4) | 1st transversal water flow in the interface element |
| SIG(5) | 2nd transversal water flow in the interface element |
| SIG(6) | Mass gas flow in the x local direction of the interface element |
| SIG(7) | Mass gas flow in the y local direction of the interface element |
| SIG(8) | Stored gas flow |
| SIG(9) | 1st transversal gas flow in the interface element |
| SIG(10) | 2nd transversal gas flow in the interface element |
| SIG(11) | Thermal flow in the x local direction of the interface element |
| SIG(12) | Thermal flow in the y local direction of the interface element |
| SIG(13) | Stored thermal flow |
| SIG(14) | 1st transversal thermal flow in the interface element |
| SIG(15) | 2nd transversal thermal flow in the interface element |
| SIG(16) | Mass vapour flow in the x local direction of the interface element |
| SIG(17) | Mass vapour flow in the y local direction of the interface element |
| SIG(18) | Stored vapour flow |
| SIG(19) | 1st transversal vapour flow in the interface element |
| SIG(20) | 2nd transversal vapour flow in the interface element |
25
| 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 \[=\frac{\rho_f.c_f.T.\underline{q}}{\Gamma_{av}.\underline{grad}(T)}\] |
| 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) | Longitudinal intrinsic permeability (=$k_{long}$) |
| Q(17) | Gas soil saturation degree (=$S_g$) |
| Q(18) | none |
| Q(19) | Water pressure inside the fault |
| Q(20) | Gas pressure (if IVAP=0) or $\alpha$ (H2, N2 …) partial pressure ($p_{\alpha}^g=p^g-p_{H_2O}^g$ = gas pressure–vapour pressure (if IVAP=1)) inside the fault |
| Q(21) | Temperature inside the fault |
| Q(22) | Water transverse transmissivity ($T_{t,c}$ or $T_{t,nc}$), multiplied by $t_{rw}$ |
| Q(23) | Gas transverse transmissivity ($T_{t,c}$ or $T_{t,nc}$), multiplied by $t_{ra}$ |
| Q(24) | Thermal transverse transmissivity ($T_{t,c}$ or $T_{t,nc}$) |
| Q(25) | Thermal longitudinal conductivity |