laws:hypofe2
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laws:hypofe2 [2025/09/10 14:15] arthur [The model] |
laws:hypofe2 [2025/09/10 14:39] (current) arthur [Number of state variables] |
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| This law is only used for water seepage - air seepage- pollutant diffusion and advection (coupled with water or gas flows) for non linear analysis in 2D porous media. | This law is only used for water seepage - air seepage- pollutant diffusion and advection (coupled with water or gas flows) for non linear analysis in 2D porous media. | ||
| - | === Mass conservation of liquid water === | + | === Mass conservation of water (liquid and vapour) === |
| \[ | \[ | ||
| - | \underbrace{\frac{\partial}{\partial t} (\rho_s . n . S_{r,w}) + div(\rho_w \vec{q_l})}_{\text{Liquide}} = 0 | + | \underbrace{\frac{\partial}{\partial t} (\rho_s . n . S_{r,w}) + div(\rho_w . \vec{q_l})}_{\text{Liquide}} + \underbrace{\frac{\partial}{\partial t} (\rho_v . n . S_{r,g}) + div(\rho_v . \vec{q_g})}_{\text{Vapeur}} = 0 |
| \] | \] | ||
| - | === Liquid flow === | + | === Liquid and vapour flows === |
| Starting from Darcy's law, the liquid water velocity is: | Starting from Darcy's law, the liquid water velocity is: | ||
| \[ | \[ | ||
| - | \vec{q_l} = - \frac{k_w}{\mu_w}\left[ \vec{grad}(p_w) \right]\ \text{where}\ k_w = K_w \frac{\mu_w}{\rho_w g}\left[ m^2\right] | + | \vec{q_l} = - \frac{k_w}{\mu_w}\left[ \vec{grad}(p_w) + g \; \rho_w \; \vec{grad}(y) \right]\ \text{where}\ k_w = K_w\; \frac{\mu_w}{\rho_w\; g}\left[ m^2\right] |
| \] | \] | ||
| + | |||
| + | The water vapour only flows in unsaturated pores and depends on the tortuosity of the path: | ||
| + | \[ | ||
| + | \vec{i}_v = - n \; S_{r,g} \; \tau D\; \rho_s \; \vec{grad} \omega_v | ||
| + | \] | ||
| + | Where $\omega_v = \rho_v/\rho_g$ is the dry air mass content in the gaseous mix. | ||
| + | |||
| === Liquid State Equations === | === Liquid State Equations === | ||
| - | - Density: $\rho_w$: \[\rho_w (p_w) = \rho_{wo}\left[ 1+\frac{p_w-p_{w0}}{\chi_w}\right]\] | + | - 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 $k_{w,eff} = k_f k_{r,w}$ | - 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 $s = p_a - p_w : S_w = f(s)$ | + | - Saturation degree $S_w$: \\ Depending on suction $s = p_g - p_w : S_w = f(s)$ |
| === Saturation degree equation (with FKRSAT) === | === Saturation degree equation (with FKRSAT) === | ||
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| The ISR=53 parameters are: CSRW1=$a_d$, CSRW2=$n_d$, CSRW3=$a_w$, CSRW4=$n_w$ and CSRW5=$b$ | The ISR=53 parameters are: CSRW1=$a_d$, CSRW2=$n_d$, CSRW3=$a_w$, CSRW4=$n_w$ and CSRW5=$b$ | ||
| + | |||
| + | === Osmotic suction model === | ||
| + | |||
| + | TO BE COMPLETED. | ||
| === Mass conservation of dry air === | === Mass conservation of dry air === | ||
| - | \[\frac{\partial}{\partial t} (\rho_a . n . S_{r,g}) + div(\rho_a \vec{q_g}) = 0\] | + | \[\frac{\partial}{\partial t} (\rho_a . n . S_{r,g}) + div(\rho_a \vec{q_g}) + div(\vec{i_a}) = 0\] |
| - | === Gas flows === | + | === Dry air and dissolved gas flows === |
| Starting from Darcy's law, the gas velocity is: | Starting from Darcy's law, the gas velocity is: | ||
| \[ | \[ | ||
| - | \vec{q_g} = - \frac{k_g}{\mu_g}\left[ \vec{grad}(p_g) + \right]\ \text{où}\ k_g = K_g \frac{\mu_g}{\rho_g g}\left[ m^2\right] | + | \vec{q_g} = - \frac{k_g}{\mu_g}\left[ \vec{grad}(p_g) + g \rho_g \vec{grad}(y) \right]\ \text{où}\ k_g = K_g \frac{\mu_g}{\rho_g g}\left[ m^2\right] |
| + | \] | ||
| + | |||
| + | The diffusion velocity of dry air is proportional to a density gradient. Using the diffusion theory adapted to porous medium, one writes: | ||
| + | \[ | ||
| + | \vec{i}_a = - n S_{r,g} \tau D \rho_g \vec{grad} \omega_a = -\vec{I}_v | ||
| \] | \] | ||
| + | Where $\omega_a = \rho_a/\rho_g$ is the dry air mass content inside the gas mix. | ||
| === Gas State Equation === | === Gas State Equation === | ||
| Line 56: | Line 73: | ||
| - Density $\rho_a$ :\\ //Hypothesis// : The air is supposed to be a perfect gas. \[\rho_a (p_a) = \rho_{a,0}\frac{p_a}{p_{a,0}} \] | - 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 $k_{g,effectif} = k_{g, intrinsic}k_{a,w}$ | - Intrinsic Permeability $k_g$: \\ Depending on the saturation degree $S_g$ : $k_{r,g} = f(S_g)$ with $k_{g,effectif} = k_{g, intrinsic}k_{a,w}$ | ||
| + | - Saturation degree $S_g$: \\ Depending on suction $s = p_g - p_w : S_g = f(s) = 1 - S_w$ | ||
| === Balance Equation of Pollutant === | === Balance Equation of Pollutant === | ||
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| === Pollutant flows === | === Pollutant flows === | ||
| - | \[ v_i^p = v_i^{advection} + v_i^{diffusion+dispersion} = C_M v_i^w - D \frac{\partial C_m}{\partial x_i} \]\\ | + | \[ v_i^p = v_i^{advection} + v_i^{diffusion+dispersion} = C_M v_i^{w/g} - D \frac{\partial C_m}{\partial x_i} \]\\ |
| - | 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. | + | With $C_M$ and $C_m$ [-] the concentration in pollutant at the macroscale and subscale, respectively. $v_i^{w/g}$ is the water or gas velocity obtained from Darcy's law and $D$ [m$^2$/s] is the diffusion and dispersion coefficient. |
| ==== Files ==== | ==== Files ==== | ||
| Prepro: LHYPOFE2.F \\ | Prepro: LHYPOFE2.F \\ | ||
| Line 81: | Line 99: | ||
| ^ Line 1 (3I10,2G10.0) ^^ | ^ Line 1 (3I10,2G10.0) ^^ | ||
| |NLAWFEM2|Number of constitutive laws at the subscale| | |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| | + | |KFLU|Number of DOF at the microscale: 1 = $P_w$, 2 = $P_w+C$, 3 = $P_w+P_g$, 4 = $P_w+C+P_g$ with $C$ the concentration in pollutant| |
| |MITER|Maximum number of iterations at the subscale| | |MITER|Maximum number of iterations at the subscale| | ||
| |CNORM|Norm for the solver of the subscale| | |CNORM|Norm for the solver of the subscale| | ||
| - | |FACONV|Units of conversion of the RVE (it has a size of 1[-])| | + | |FACONV|Units of conversion of the RVE (it has a size of 1*FACONV[-])| |
| Line 101: | Line 119: | ||
| |RHOA0|Gaz density $(=\rho_{a,0})\ \left[kg.m^{-3}\right]$| | |RHOA0|Gaz density $(=\rho_{a,0})\ \left[kg.m^{-3}\right]$| | ||
| |PMGAS|Gas molar mass $[g/mol]$| | |PMGAS|Gas molar mass $[g/mol]$| | ||
| - | |PA0|Initial gas pressure $\left[ Pa\right]$| | + | |PG0|Initial gas pressure $\left[ Pa\right]$| |
| |PHENRY|Henry coefficient| | |PHENRY|Henry coefficient| | ||
| - | ^ Line 4 (1I10) ^^ | + | ^ Line 4 (4I10) ^^ |
| - | |IVAP|= 1 for vapour, = 0 if liquid water only (VAPOUR NOT IMPLEMENTED YET)| | + | |IVAP|= 1 for vapour, = 0 if liquid water only| |
| - | ^ Line 5 (3I10) ^^ | + | |IGAS|= 0 for air, =1 for $H_2$, =2 for $N_2$, = 3 for $Ar$, = 4 for $He$, = 5 for $CO_2$, = 6 for $CH_4$| |
| + | |IOSMOTIC|= 0 to neglect osmotic suction, = 1 for osmotic suction with Van't Hoff model, = 2 for osmotic suction with Kelvin (water activity) and Pitzer model| | ||
| + | |IDIFF|= 0 for the pollutant to diffuse through water, = 1 through gas| | ||
| + | ^ Line 5 (4I10) ^^ | ||
| |ISR|Retention curve (=53 for Van Genuchten with hysteresis)| | |ISR|Retention curve (=53 for Van Genuchten with hysteresis)| | ||
| |IKW|Water relative permeability curve (=7 for Van Genuchten)| | |IKW|Water relative permeability curve (=7 for Van Genuchten)| | ||
| |IKA|Gas relative permeability curve (=6 for Van Genuchten)| | |IKA|Gas relative permeability curve (=6 for Van Genuchten)| | ||
| + | |N_SUBINCR|Number of additional multiplicator for the number of subincrement in the hysteresis model| | ||
| ^ Line 6 (3G10.0)^^ | ^ Line 6 (3G10.0)^^ | ||
| |CKW1|First parameter of IKW| | |CKW1|First parameter of IKW| | ||
| Line 162: | Line 184: | ||
| |SIG(16)|Advective flow of dissolved gas along $x$ (unused)| | |SIG(16)|Advective flow of dissolved gas along $x$ (unused)| | ||
| |SIG(17)|Advective flow of dissolved gas along $y$ (unused)| | |SIG(17)|Advective flow of dissolved gas along $y$ (unused)| | ||
| - | |SIG(18)|Unused| | + | |SIG(18)|Vapour flow along $x$ $(=f_{vx})$| |
| - | |SIG(19)|Unused| | + | |SIG(19)|Vapour flow along $y$ $(=f_{vy})$| |
| - | |SIG(20)|Unused| | + | |SIG(20)|Vapour flow stored $(=f_{ve})$| |
| |SIG(21)|Unused| | |SIG(21)|Unused| | ||
| |SIG(22)|Unused| | |SIG(22)|Unused| | ||
| Line 176: | Line 198: | ||
| ===== State variables ===== | ===== State variables ===== | ||
| ==== Number of state variables ==== | ==== Number of state variables ==== | ||
| - | 10 + 5*(Number of Subscale Nodes)\\ | + | 11 + 5*(Number of Subscale Nodes)\\ |
| - | /!\ The state variables vector also contains the following information for each subscale node: X,Y,Pw,C,Pg | + | /!\ The state variables vector also contains the following information for each subscale node: $X$, $Y$, $P_w$, $C$, $P_g$ |
| ==== List of state variables ==== | ==== List of state variables ==== | ||
| |Q(1)|Liquid water mass at the RVE| | |Q(1)|Liquid water mass at the RVE| | ||
| Line 187: | Line 209: | ||
| |Q(7)|Homogenised gas relative permeability| | |Q(7)|Homogenised gas relative permeability| | ||
| |Q(8)|Homogenised macroscale tortuosity| | |Q(8)|Homogenised macroscale tortuosity| | ||
| - | |Q(9)|Vapour mass at the RVE (unused)| | + | |Q(9)|Vapour mass at the RVE| |
| - | |Q(10)|Homogenised succion| | + | |Q(10)|Homogenised total suction $(= p_g - p_w + osmotic)$| |
| - | |Q(11 + (i-1)*5)|$X_i$| | + | |Q(11)|Homogenised osmotic suction $(= osmotic)$| |
| - | |Q(11 + (i-1)*5 +1)|$Y_i$| | + | |Q(12 + (i-1)*5)|$X_i$| |
| - | |Q(11 + (i-1)*5 +2)|$P_{w,i}$| | + | |Q(12 + (i-1)*5 +1)|$Y_i$| |
| - | |Q(11 + (i-1)*5 +3)|$C_i$| | + | |Q(12 + (i-1)*5 +2)|$P_{w,i}$| |
| - | |Q(11 + (i-1)*5 +4)|$P_{g,i}$| | + | |Q(12 + (i-1)*5 +3)|$C_i$| |
| + | |Q(12 + (i-1)*5 +4)|$P_{g,i}$| | ||
laws/hypofe2.1757506504.txt.gz · Last modified: 2025/09/10 14:15 by arthur
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