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--- appendices:a8 [2023/11/30 11:59]
arthur [Water Retention Curves]
+++ 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 92: / Line 92: @@
 ^ISR = 25 - Van Genuchten with "true" air-entry pressure (aept).^^
-\[COEFM  = ONE-ONE/CSW2\]
+\[COEFM  = 1-\frac{1}{CSW2}\]
-\[SATUR  = SRES+(SRFIELD-SRES)*(ONE+((PC-AEPT)/CSW1)**CSW2)**(-COEFM)\]
+\[S_w  = S_{res}+(S_{r,field}-S_{res})*\left(1+\left(\frac{s-AEPT}{CSW1}\right)^{CSW2}\right)^{-COEFM}\]
 ^ISR = 26 - Double structure mechanism adsorption^^
@@ Line 121: / Line 121: @@
 The ISR=53 parameters are: CSRW1=$a_d$, CSRW2=$n_d$, CSRW3=$a_w$, CSRW4=$n_w$ and CSRW5=$b$.
-/!\ Two parameters must be passed through the DUM argument of FKRST: $DUM(1) = s^{t-1}$ and $DUM(2) = S_w^{t-1}$.
+/!\ Two parameters must be passed through the DUM argument of FKRST: DUM(1)=$s^{t-1}$ and DUM(2)=$S_w^{t-1}$.
+^ISR = 55, Soil water retention curve considering gas entry pressure (EURAD-Gas Task4.2).^^
+In the below formulations, $p_c$ and 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., $P_e$, $ε$ is a numerical parameter (0.01 or 0.001), $m$ and $n$  are fitting parameters.
+\[ p_c= \begin{cases} -\frac{1}{\alpha}\left ( \left ( S_{e}^{*}S_{e}\right )^{-\frac{1}{m}}-1\right)^{\frac{1}{n}}, \quad \text{if} \; S_{e}\leq 1-\varepsilon
+\\
+-\frac{1}{\alpha}\left ( \left ( S_{e}^{*}S_{e}\right )^{-\frac{1}{m}}-1\right)^{\frac{1}{n}}.\left ( \frac{1-S_{e}}{\varepsilon } \right ), \quad \text{if} \; \left ( 1-\varepsilon \right ) < S_{e}< 1
+\\
+, \quad \text{if} \; S_{e}=1
+\end{cases}\].
+\[ S_{e}=\frac{S_{l}-S_{r}}{1-S_{r}}\]
+\[ S_{e}^{*}=\left ( 1+\left (\alpha P_{e}  \right )^{n}  \right )^{-m}\]
+\[ m=\left ( 1-\frac{1}{n} \right )\]
+\[ \alpha =\frac{1}{P_{r}}\]
+So as per the above formulations, the required parameters are as follows:
+**CSR1** = Air-entry pressure i.e., $P_r$
+**CSR2** = $n$
+**CSR3** = Gas entry pressure i.e., $P_e$
+**CSR4** = $\varepsilon$
+**CSR5** = Residual degree of water saturation i.e., $S_r$
+**CSR6** = Max. degree of water saturation i.e., 1
+**CSR7** = NIL
+/**NOTE**\ The above water retention curve is implemented in conjunction with the water and air relative permeability functions which also consider the effect of gas entry pressure. It is advised to go through these formulations i.e., **IKW=55** for relative permeability function for water and **IKA=55** for air.
 ===== 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).^^
+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.
+\[ k_{rw}= \begin{cases} \sqrt{S_{e}}\left [ \frac{1-\left ( 1-\left (S_{e}^{*}S_{e} \right )^{1/m} \right )^{m}}{1-\left ( 1-\left (S_{e}^{*}\right )^{1/m} \right )^{m}} \right ]^{2}, \quad \text{if} \; S_{e}\leq 1
+\\
+, \quad \text{if} \; S_{e} = 1
+\end{cases}\].
+\[ S_{e}=\frac{S_{l}-S_{r}}{1-S_{r}}\]
+\[ S_{e}^{*}=\left ( 1+\left (\alpha P_{e}  \right )^{n}  \right )^{-m}\]
+\[ m=\left ( 1-\frac{1}{n} \right )\]
+\[ \alpha =\frac{1}{P_{r}}\]
+/**NOTE**\ 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. 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.
 ===== 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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