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--- appendices:a8 [2023/12/07 11:27]
arthur [Thermal conductivity]
+++ appendices:a8 [2024/06/17 11:21] (current)
arthur
@@ Line 86: / Line 86: @@
 \[SATUR  = (ONE+(PC/CSW1)**CSW2)**(-COEFM)*(ONE-PC/CSW3)**CSW4\]
-^ISR = 22 - Romera et al., 2011^^
+^ISR = 22 - Romero et al., 2011^^
 ^ISR = 23 - Unconstant parameters, function of the porosity^^
@@ Line 162: / Line 162: @@
 ===== Water relative permeability =====
-  * **__IKW = 0__** \[k_{rw} = 1\]
+^IKW = 0^^
-  * **__IKW = 1__** \[k_{rw} = CKW3 - CKW1 (1-S_{r,w})^{CKW2}\] __Example__: CKW1 = 2.207; CKW2 = 0.953; CKW3 = 1
+\[k_{rw} = 1\]
-  * **__IKW = 2__** \[k_e^{rel} (S_e) = \left(1+\left(S_{r,w}^{CKW1} - 1\right)^{CKW2}\right)^{-1}\] __Example__: Momas: CKW1 = -2.429; CKW2 = 1.176
-  * **__IKW = 3__** \[k_{r,w} = \begin{cases} \exp(CKW1*S_w+CKW2*S_w^2) & \quad \text{if } S_w \geq S_{res} \\ k_{r,min} & \quad \text{if } S_w<S_{res} \end{cases} \]
+^IKW = 1^^
-  * **__IKW = 4__** \[k_{r,w} = \begin{cases} \frac{(S_w-S_{res})^{CKW1}}{(S_{r,field}-S_{res})^{CKW2}} & \quad \text{if } S_w \geq S_{res} \\ k_{r,min} & \quad \text{if } S_w<S_{res} \end{cases} \] __Example__: $CKW1 = 4$; $CKW2 = 4$; $S_{r,field} = 1$; $S_{res}=0.1$
+\[k_{rw} = CKW3 - CKW1 (1-S_{r,w})^{CKW2}\]
-  * **__IKW = 7__** \[k_{rw}=\sqrt{S_{rw}} \left(1-\left(1-S_{rw}^{\frac{1}{CKW1}}\right)^{CKW1}\right)^2\]
+Example: CKW1 = 2.207; CKW2 = 0.953; CKW3 = 1
-  * **__IKW = 8__** \[k_{rw} = S_{rw}^3\]
-  * **__IKW = 9__** \[k_{rw}=\sqrt{S_{we}} \left(1-\left(1-S_{we}^{\frac{1}{CKW1}}\right)^{CKW1}\right)^2\] \[S_e=\frac{S_{rw}-S_{rw,res}}{1-S_{rw,res}-S_{rg,res}}\] \[S_{rw,res}=CKW2\] \[S_{rg,res}=CKW3\]
+^IKW = 2^^
+\[k_e^{rel} (S_e) = \left(1+\left(S_{r,w}^{CKW1} - 1\right)^{CKW2}\right)^{-1}\]
+Example: Momas: CKW1 = -2.429; CKW2 = 1.176
+^IKW = 3^^
+\[k_{r,w} = \begin{cases} \exp(CKW1*S_w+CKW2*S_w^2) & \quad \text{if } S_w \geq S_{res} \\ k_{r,min} & \quad \text{if } S_w<S_{res} \end{cases} \]
+^IKW = 4^^
+\[k_{r,w} = \begin{cases} \frac{(S_w-S_{res})^{CKW1}}{(S_{r,field}-S_{res})^{CKW2}} & \quad \text{if } S_w \geq S_{res} \\ k_{r,min} & \quad \text{if } S_w<S_{res} \end{cases} \] __Example__: $CKW1 = 4$; $CKW2 = 4$; $S_{r,field} = 1$; $S_{res}=0.1$
+^IKW = 7^^
+\[k_{rw}=\sqrt{S_{rw}} \left(1-\left(1-S_{rw}^{\frac{1}{CKW1}}\right)^{CKW1}\right)^2\]
+^IKW = 8^^
+ \[k_{rw} = S_{rw}^3\]
+^IKW = 9^^
+\[k_{rw}=\sqrt{S_{we}} \left(1-\left(1-S_{we}^{\frac{1}{CKW1}}\right)^{CKW1}\right)^2\]
+\[S_e=\frac{S_{rw}-S_{rw,res}}{1-S_{rw,res}-S_{rg,res}}\] \[S_{rw,res}=CKW2\] \[S_{rg,res}=CKW3\]
 ^IKW = 55, Relative permeability function for water considering gas entry pressure (EURAD-Gas Task4.2).^^
@@ Line 189: / Line 208: @@
 ===== Air relative permeability =====
-  * **__IKA = 0__** \[k_{ra}=1\]
+^IKA = 0^^
-  * **__IKA = 1__** \[k_{ra} = (1-S_e)^{CKA1}(1-S_e^{CKA2})\] \[S_e=\frac{S_{rw}-S_{rw,u}}{1-S_{rw,u}}\] __Example__: CKA1 = 2; CKA2 = 5/3
+\[k_{ra}=1\]
-  * **__IKA = 2__** \[k_{r,a}=CKA1\]
-  * **__IKA = 3__** \[S_e = \frac{S_{r,w}-S_{r,u}}{1-S_{rw,u}} \\ \begin{cases} \text{If } S_e<0 => S_e = 0 \\ \text{If } 0<S_e<0.55 => k_{ra}=(0.55-S_e)^{CKA1}(1-S_e^{CKA2}) \\ \text{If } S_e>0.55 => k_{ra}=k_{rmin} \end{cases} \] __Remark__:  \[k_{w,effectif}=k_{f,intrinsic}k_{rw} \\ k_{a,effectif}=k_{f,intrinsic}k_{aw}\]
+^IKA = 1^^
-  * **__IKA = 6__** \[k_{ra}=\sqrt{1-S_{r,w}}\left(1-S_{r,w}^{\frac{1}{CKA1}}\right)^{2CKA1}\]
+\[k_{ra} = (1-S_e)^{CKA1}(1-S_e^{CKA2})\] \[S_e=\frac{S_{rw}-S_{rw,u}}{1-S_{rw,u}}\]
-  * **__IKA = 7__** \[k_{ra}=\sqrt{1-S_{we}}\left(1-S_{we}^{\frac{1}{CKA1}}\right)^{2CKA1}\] \[S_e=\frac{S_{rw}-S_{rw,res}}{1-S_{rw,res}-S_{rg,res}}\] \[S_{rw,res}=CKA2\] \[S_{rg,res}=CKA3\]
+Example: CKA1 = 2; CKA2 = 5/3
-  * **__IKA = 8__** \[k_{ra}=CKA2(1-S_{we})^{CKA1}\] \[S_e=\frac{S_{rw}-S_{rw,res}}{1-S_{rw,res}-S_{rg,res}}\]
+^IKA = 2^^
+\[k_{r,a}=CKA1\]
+^IKA = 3^^
+\[S_e = \frac{S_{r,w}-S_{r,u}}{1-S_{rw,u}} \\ \begin{cases} \text{If } S_e<0 => S_e = 0 \\ \text{If } 0<S_e<0.55 => k_{ra}=(0.55-S_e)^{CKA1}(1-S_e^{CKA2}) \\ \text{If } S_e>0.55 => k_{ra}=k_{rmin} \end{cases} \]
+Remark:  \[k_{w,effectif}=k_{f,intrinsic}k_{rw} \\ k_{a,effectif}=k_{f,intrinsic}k_{aw}\]
+^IKA = 6^^
+\[k_{ra}=\sqrt{1-S_{r,w}}\left(1-S_{r,w}^{\frac{1}{CKA1}}\right)^{2CKA1}\]
+^IKA = 7^^
+\[k_{ra}=\sqrt{1-S_{we}}\left(1-S_{we}^{\frac{1}{CKA1}}\right)^{2CKA1}\]
+\[S_e=\frac{S_{rw}-S_{rw,res}}{1-S_{rw,res}-S_{rg,res}}\] \[S_{rw,res}=CKA2\]
+\[S_{rg,res}=CKA3\]
+^IKA = 8^^
+\[k_{ra}=CKA2(1-S_{we})^{CKA1}\]
+\[S_e=\frac{S_{rw}-S_{rw,res}}{1-S_{rw,res}-S_{rg,res}}\]
+^IKA = 55, Relative permeability function for gas considering gas entry pressure (EURAD-Gas Task4.2). ^^
+Similar to **ISR = 55**, $p_c$ is the capillary pressure ($p_c=p_{air}-p_{water}$), $α$ is the inverse of air-entry pressure i.e., $P_r$, $S_e$ is the effective degree of water saturation, $S_l$ is the degree of water saturation, $S_r$  is residual degree of water saturation, $S_e^*$ is the effective degree of saturation considering the explicit gas entry pressure i.e., $P_e$, $m$ and $n$  are fitting parameters. Additionally, $f_{g}$ is the ratio of intrinsic permeability values for Gas ($K_{Gas}$) to Water ($K_{Water}$).
+\[ k_{rg}= \begin{cases} f_{g}\sqrt{1-S_{e}}\left [ \frac{\left ( 1-\left (S_{e}^{*}\right )^{1/m} \right )^{m}-\left ( 1-\left (S_{e}^{*}S_{e} \right )^{1/m} \right )^{m}}{\left ( 1-\left (S_{e}^{*}\right )^{1/m} \right )^{m}-1} \right ]^2, \quad \text{if} \; S_{e}\leq 1
+\\
+, \quad \text{if} \; S_{e} = 1
+\end{cases}\].
+\[ f_{g}=\frac{K_{Gas}}{K_{Water}}\]
+\[ S_{e}=\frac{S_{l}-S_{r}}{1-S_{r}}\]
+\[ S_{e}^{*}=\left ( 1+\left (\alpha P_{e}  \right )^{n}  \right )^{-m}\]
+\[ m=\left ( 1-\frac{1}{n} \right )\]
+\[ \alpha =\frac{1}{P_{r}}\]
+/**NOTE**\ Similar to IKW=55, the above formulation is implemented in conjunction with either **ISR=5** or **ISR=55**. In case of ISR=55, it will automatically adopt the required parameters from the definition of soil water retention curve, EXCEPT the parameter $f_{g}$.
+So, CKA1 = $f_{g}$ i.e. $\frac{K_{Gas}}{K_{Water}}$
+Whereas, in case of ISR=5 (Classical Van Genuchten formulation) CSR3 will represent the gas entry pressure i.e. $P_{e}$. The definition of remaining parameters will be same except the parameter CKA1 which will represent $f_{g}$ i.e. $\frac{K_{Gas}}{K_{Water}}$.
 ===== Thermal conductivity =====
 ^  ITHERM  ^  $\Gamma_T$  ^
 |  \[1\]  |  \[nS_w\Gamma_w+nS_a\Gamma_a+(1-n)\Gamma_s\]|
 |  \[2\]  |  \[CLT1*S_w +CLT2\]|
 |  \[3\]  |  \[CLT1 -\frac{CLT2}{1 + exp\left(\frac{S_w -CLT3}{CLT4}\right)}\]|

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