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laws:hypofe2
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HYPOFE2 **(WIP)**
Description
Multiscale law for water-air seepage, pollutant diffusion and advection. Inspired from WAVAT and ADVEC.
Can be parallelized with ELEMB (macroscale) or at the perturbation loop (microscale).
Takes into account the hysteresis in the water retention law when used with FKRSAT.
The model
This law is only used for water seepage - air seepage- pollutant diffusion and advection (coupled) for non linear analysis in 2D porous media.
Mass conservation of liquid water
\[ \underbrace{\frac{\partial}{\partial t} (\rho_s . n . S_{r,w}) + div(\rho_w \vec{q_l})}_{\text{Liquide}} = 0 \]
Liquid flow
Starting from Darcy's law, the liquid water velocity is: \[ \vec{q_l} = - \frac{k_w}{\mu_w}\left[ \vec{grad}(p_w) \right]\ \text{where}\ k_w = K_w \frac{\mu_w}{\rho_w g}\left[ m^2\right] \]
Liquid State Equations
- Density: $\rho_w$: \[\rho_w (p_w) = \rho_{wo}\left[ 1+\frac{p_w-p_{w0}}{\chi_w}\right]\]
- Intrinsic Permeability $k_w$:
Depending on the water saturation degree $S_w$ : $k_{r,w} = f(S_w)$ with $k_{w,eff} = k_f k_{r,w}$ - Saturation degree $S_w$:
Depending on succion $s = p_a - p_w : S_w = f(s)$
Mass conservation of dry air
\[\frac{\partial}{\partial t} (\rho_a . n . S_{r,g}) + div(\rho_a \vec{q_g}) = 0\]
Gas flows
Starting from Darcy's law, the gas velocity is: \[ \vec{q_g} = - \frac{k_g}{\mu_g}\left[ \vec{grad}(p_g) + \right]\ \text{où}\ k_g = K_g \frac{\mu_g}{\rho_g g}\left[ m^2\right] \]
Gas State Equation
- Density $\rho_a$ :
Hypothesis : The air is supposed to be a perfect gas. \[\rho_a (p_a) = \rho_{a,0}\frac{p_a}{p_{a,0}} \] - Intrinsic Permeability $k_g$:
Depending on the saturation degree $S_g$ : $k_{r,g} = f(S_g)$ with $k_{g,effectif} = k_{g, intrinsic}k_{a,w}$
Balance Equation of Pollutant
\[\frac{\partial}{\partial x_i} (v_i^p) = 0\]
Pollutant flows
\[ v_i^p = v_i^{advection} + v_i^{diffusion+dispersion} = C_M v_i^w - D \frac{\partial C_m}{\partial x_i} \]
With C_M and C_m [-] the concentration in pollutant at the macroscale and subscale, respectively. $v_i^w$ is the water velocity obtained from Darcy's law and $D$ [m$^2$/s] is the diffusion and dispersion coefficient.
Files
Prepro: LHYPOFE2.F & EHYPOFE2A.F
Lagamine: HYPOFE2.F & EHYPOFE2B.F
Availability
| Plane stress state | NO |
| Plane strain state | YES |
| Axisymmetric state | NO |
| 3D state | YES |
| Generalized plane state | NO |
Input file
Parameters defining the type of constitutive law
| Line 1 (2I5, 60A1) | |
|---|---|
| IL | Law number |
| ITYPE | 629 |
| COMMENT | Any comment (up to 60 characters) that will be reproduced on the output listing |
Integer parameters
| Line 1 (3I10,2G10.0) | |
|---|---|
| NLAWFEM2 | Number of constitutive laws at the subscale |
| KFLU | Number of DOF: 1=Pw, 2=Pw+C, 3=Pw+Pg, 4=Pw+C+Pg with C the concentration in pollutant |
| MITER | Maximum number of iterations at the subscale |
| CNORM | Norm for the solver of the subscale |
| FACONV | Units of conversion of the RVE (it has a size of 1[-]) |
Real parameters
| Line 1 (3E10.2,2G10.0) | |
|---|---|
| VISCW0 | Liquid dynamic viscosity $(=\mu_{w,0})\ \left[ Pa.s \right]$ |
| RHOW0 | Liquid density $(=\rho_{w,0})\ \left[ kg.m^{-3}\right]$ |
| UXHIW | Liquid compressibility coefficient $(=1/ \chi_{w})\ \left[ Pa^{-1}\right]$ |
| PW0 | Initial water pressure $\left[ Pa\right]$ |
| T0 | Initial temperature $\left[ K\right]$ |
| Line 2 (1G10.0) | |
| CPINI | Initial pollutant concentration $\left[ -\right]$ |
| Line 3 (3E10.2,2G10.0) | |
| VISCA0 | Gas dynamic viscosity $(=\mu_{a,0})\ \left[Pa.s \right]$ |
| RHOA0 | Gaz density $(=\rho_{a,0})\ \left[kg.m^{-3}\right]$ |
| PMGAS | Gas molar mass $[g/mol]$ |
| PA0 | Initial gas pressure $\left[ Pa\right]$ |
| PHENRY | Henry coefficient |
| Line 4 (1I10) | |
| IVAP | = 1 for vapour, = 0 if liquid water only (VAPOUR NOT IMPLEMENTED YET) |
| Line 5 (3I10) | |
| ISR | Retention curve (=53 for Van Genuchten with hysteresis) |
| IKW | Water relative permeability curve (=7 for Van Genuchten) |
| IKA | Gas relative permeability curve (=6 for Van Genuchten) |
| Line 6 (3G10.0) | |
| CKW1 | First parameter of IKW |
| CKW2 | Second paremeter of IKW |
| CKW3 | Third parameter of IKW |
| Line 7 (2G10.0) | |
| CKA1 | First parameter of IKA |
| CKA2 | Second parameter of IKA |
| Line 8 (5G10.0) | |
| CSR1 | First parameter of ISR |
| CSR2 | Second parameter of ISR |
| CSR3 | Third parameter of ISR |
| CSR4 | Fourth parameter of ISR |
| CSR5 | Fifth parameter of ISR |
| Line 9 (5G10.0) | |
| SRES | Residual saturation degree $(=S_{res})$ |
| SRFIELD | Field saturation degree $(=S_{r, field})$ |
| AIREV | Air entry pressure $\left[Pa\right]$ |
| AKRMIN | Minimum value of relative permeabikity |
| SRINI | Initial saturation degree |
Subscale parameters
To be repeated as many time as NLAWFEM2.
| Line 1 (2I5) | |
|---|---|
| ILAW2 | Number of the subscale constitutive law (=1:NLAWFEM2) |
| ITYPE2 | Type of subscale law (=1 for Hydraulic pollutant microscale law) |
| Line 2 (4G10.0) | |
| POROS | Material porosity ($=n$) |
| PERMINT | Material intrinsic permeability ($=k_{int}$) $[m^2]$ |
| DIFFC | Material diffusion coefficient of the pollutant ($D_{app}$) $[m^2/s]$ |
| TORTU | Material tortuosity ($=\tau$) |
Stresses
Number of stresses
17
Meaning
In 2D state :
| SIG(1) | $Sigma_x$ (unused) |
| SIG(2) | $Sigma_y$ (unused) |
| SIG(3) | $Sigma_{xy}$ (unused) |
| SIG(4) | $Sigma_z$ (unused) |
| SIG(5) | Homogenised liquid flow along $x$ $(=f_{wx})$ |
| SIG(6) | Homogenised liquid flow along $y$ $(=f_{wy})$ |
| SIG(7) | Homogenised liquid flow stored $(=f_{we})$ |
| SIG(8) | Homogenised mean flow of the pollutant along $x$ $(=(f_{px,a}+f_{px,b})/2)$ |
| SIG(9) | Homogenised mean flow of the pollutant along $y$ $(=(f_{py,a}+f_{py,b})/2)$ |
| SIG(10) | Homogenised pollutant flow stored (takes advection into account) $(=f_{pe})$ |
| SIG(11) | Homogenised diffusive flow of the pollutant along $x$ for the current step $(=f_{px,b}) |
| SIG(12) | Homogenised diffusive flow of the pollutant along $y$ for the current step $(=f_{py,b}) |
| SIG(13) | Homogenised gas flow along $x$ $(=f_{gx})$ |
| SIG(14) | Homogenised gas flow along $y$ $(=f_{gy})$ |
| SIG(15) | Homogenised gas flow stored $(=f_{ge})$ |
| SIG(16) | Advective flow of dissolved gas (unused) |
| SIG(17) | Advective flow of dissolved gas (unused) |
| SIG(5) | liquid velocity in the Y direction $(=f_{wy})$ | |
| SIG(6) | liquid velocity stored $(=f_{we})$ | |
| SIG(7) | 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 | |
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) |
laws/hypofe2.1700817002.txt.gz · Last modified: 2023/11/24 10:10 by arthur
