# Crop Water Relations {#Crop-Water-Relations}



## Crop water demand {#Crop-water-demand}


Following \citet{sinclair1986waterand}, transpiration demand is modeled
as a function of the current day's potential crop growth rate, estimated
by the potential biomass accumulation associated with intercepted
radiation ($\Delta Q_{r}$, see Equation \@ref(eq:BiomassProduction)),
divided by the transpiration efficiency.

\begin{equation}

W_{d}=\frac{\Delta Q_{r}-R}{TE} (\#eq:soilWaterDemand)

\end{equation}

where $R$ is respiration rate and equal to zero in the current version
of APSIM-Wheat, $TE$ is transpiration efficiency. $TE$ is related
to the daylight averaged vapour pressure deficit ($VPD$, Equation \@ref(eq:VPD))
and a multiple of CO\textsubscript{2} factor \citep{reyenga1999modelling}.

\begin{equation}

TE=f_{c,\,TE}\frac{f_{TE}}{VPD} (\#eq:TranspirationEfficiency)

\end{equation}

where $f_{c,\,TE}$ is the CO\textsubscript{2} factor for transpiration
efficiency, which is a function of carbon dioxide concentration and
is defined by parameters `x_co2_te_modifier` and `y_co2_te_modifier`
in wheat.xml and linearly interpolated by APSIM (Fig. \@ref(fig:wdCO2TE)).
$f_{c,\,TE}$ linearly increases from 1 to 1.37 when CO\textsubscript{2}
concentration increases from 350 ppm to 700 ppm \citep{reyenga1999modelling}.
$f_{TE}$ is the coefficient of transpiration efficiency, which values
are defined in wheat.xml by parameters `transp_eff_cf` in
wheat.xml for the different growth stages and are linearly interpolated
by APSIM (Fig. \@ref(fig:wdCoefficientOfTE)). 


```{r wdCO2TE,fig.cap='Relationship between factor of carbon dioxide for transpiration efficiency ($f_{c,\\, TE}$) and CO2 concentration.' }  

p <- wdVisXY(wheat_xml, 
		"x_co2_te_modifier", "y_co2_te_modifier",
		xlab = "Carbon dioxide concentration",
		ylab = "Transpiration efficiency factor")
print(p)

```



```{r wdCoefficientOfTE,fig.cap='Change in the coefficient of transpiration efficiency with growth stages.' }  

p <- wdVisXY(wheat_xml, 
		"stage_code", "transp_eff_cf",
		xlab = "Stage",
		ylab = "Coefficient of \ntranspiration efficiency")
print(p)

```


$VPD$ is the vapour pressure deficit, which is estimated using the
method proposed by \citet{tanner1983efficient} and only requires
daily maximum and minimum temperatures.


\begin{equation}

VPD=f_{v}[6.1078\times\exp(\frac{17.269\times T_{max}}{237.3+T_{max}})-6.1078\times\exp(\frac{17.269\times T_{min}}{237.3+T_{min}})] (\#eq:VPD)

\end{equation}

In this method, it is assumed that the air is saturated at the minimum
temperature. The saturated vapour pressure is calculated at both the
maximum and minimum temperatures, and the default vapour pressure
deficit for the day is taken as 75\% ($f_{v}$, defined by `svp_fract`
in wheat.xml) of the difference between these two vapour pressures. 

Crop water demand is capped to below a given multiple of potential
ET (taken as Priestly-Taylor Eo from the water balance module) as
specified by \hyperlink{eo_crop_factor_default}{eo_crop_factor_default}  
in the wheat.xml file (default value 1.5). This limits water use to
reasonable values on days with high VPD or in more arid environments.


## Potential and actual extractable soil water


Potential and actual extractable soil water is the sum of root water
contents available to the crop from each profile layer occupied by
roots. If roots are only partially through a layer available soil
water is scaled to the portion that contains roots. Potential extractable
soil water ($\text{ESW}_{p}$) is the difference between drained
upper limit soil water content (DUL) and lower limit of plant-extractable
soil water (LL) for each soil layer. The actual extractable soil water
($esw_{a}$) is the difference between the soil water content (SW)
and lower limit of plant-extractable soil water (LL) for each soil
layer.

\begin{equation}

\begin{array}{c}
\text{ESW}_{p}(i)=\text{DUL}(i)-\text{LL}(i)\\
\text{ESW}a(i)=\text{SW}(i)-\text{LL}(i)\\
\text{ESW}_{p}=\sum_{i=1}^{I}[\text{DUL}(i)-\text{LL}(i)]\\
\text{ESW}_{a}=\sum_{i=1}^{I}[\text{SW}(i)-\text{LL}(i)]
\end{array} (\#eq:SoilWaterESW)

\end{equation}

where $i$ indicates soil layers (where roots are present), and $I$
indicates the deepest soil water of root presented. Similar variables
are calculated for the entire soil profile (i.e. roots may not occupy
all the layers). 

\begin{equation}

\begin{array}{c}
\text{PAWC}=\sum_{i}^{N_{s}}[\text{DUL}(i)-\text{LL}(i)]\\
\text{ESW}=\sum_{i}^{N_{s}}[\text{SW}(i)-\text{LL}(i)]
\end{array}

\end{equation}

where $i$ indicates soil layers, $N_{s}$ indicates the number of
soil layers, and PAWC is the plant available water capacity. 


## Crop water supply, i.e. potential soil water uptake


The APSIM-Wheat module can be coupled to either the SWIM2 module (see
module documentation) or the SOILWAT2 module (default). When the APSIM-Wheat
module is coupled to APSIM-SOILWAT2, potential soil water uptake (or
water supply, $W_{s}$) is calculated using the approach first advocated
by Monteith (1986). Crop water supply is considered as the sum of
potential root water uptake from each profile layer occupied by root.
If roots are only partially through a layer available soil water is
scaled to the portion that contains roots. The potential rate of extraction
in a layer is calculated using a rate constant (KL) as actual extractable
soil water. The KL defines the fraction of available water able to
be extracted per day. The KL factor is empirically derived, incorporating
both plant and soil factors which limit rate of water uptake. Root
water extraction values (KL) must be defined for each combination
of crop species and soil type.

\begin{equation}

\begin{array}{c}
\begin{array}{cc}
W_{s}(i) & =\text{KL}(i)[\text{SW}(i)-\text{LL}(i)]\qquad\qquad if\,i\leq I-1\\
 & =\frac{D_{r}(i)}{D_{s}(i)}\text{KL}(i)[\text{SW}(i)-\text{LL}(i)]\qquad\qquad if\,i=I
\end{array}\\
W_{s}=\sum_{i=1}^{I}W_{s}(i)
\end{array} (\#eq:WaterSupply)

\end{equation}

where $i$ is the soil layer, $I$ is the deepest soil layer where
roots are present, $W_{s}(i)$ is the water supply available from
layer $i$, $W_{s}$ is the crop water supply, $\text{SW}(i)$ is
the soil water content in layer $i$, $\text{LL}(i)$ is the lower
limit of plant-extractable soil water in layer $i$, $\text{KL}(i)$
is the root water extraction values in layer $i$, $D_{r}(i)$ is
the root depth within the soil layer ($i$) where roots are present,
and $D_{s}(i)$ is the thickness of this layer $i$. 


## Actual soil water uptake


The actual rate of water uptake is the lesser of the potential soil
water supply ($W_{s}$, Equation \@ref(eq:WaterSupply)) and the soil water
demand ($W_{d}$, Equation \@ref(eq:soilWaterDemand)), which is determining
whether biomass production is limited by radiation or water uptake
(Equation \@ref(eq:actualBiomassProduction))


\begin{equation}

W_{u}=\min(W_{d},\,W_{s}) (\#eq:WaterUpdate)

\end{equation}

If the potential soil water supply (accessible by the roots) exceeds
the crop water demand, then the actual soil water uptake ($W_{u}$)
is removed from the occupied layers in proportion to the values of
potential root water uptake in each layer. If the computed soil water
supply from the profile is less than the demand then, and the actual
root water uptake from a layer is equal to the computed potential
uptake. If there are not soil water supply and demand, soil water
update equals to zero.

\begin{equation}

\begin{array}{c}
\Delta W_{s}(i)=-W_{s}(i)\times\frac{W_{d}}{W_{s}}\qquad if\;W_{s}<W_{d}\\
\Delta W_{s}(i)=-W_{s}(i)\qquad if\;W_{s}>W_{d}\\
\Delta W_{s}(i)=0\qquad if\;W_{s}=W_{d}=0
\end{array}

\end{equation}

where $\Delta W_{s}(i)$ is the daily change in soil water content
at layer $i$ (where roots are present), and $W_{s}(i)$ is the water
supply available from layer $i$ (Equation \@ref(eq:WaterSupply)) .


## Soil water stresses affecting plant growth


Soil water deficit factors are calculated to simulate the effects
of water stress on different plant growth-and-development processes.
Three water deficit factors are calculated which correspond to four
plant processes, each having different sensitivity to water stress
i.e. photosynthesis, leaf expansion, and phenology. 

Each of these factors is capped between 0 and 1, where the value of
0 corresponds to a complete stress, while 1 corresponds to no stress.

Leaf expansion is considered more sensitive to stress than photosynthesis,
while soil water has no impact on crop phenology in the current APSIM-Wheat
version. 


### Phenology


Soil water stress of phenology is determined by the soil water deficiency. 


\begin{equation}

f_{W,\,pheno}=h_{w,\,pheno}(\frac{esw_{a}}{esw_{p}}) (\#eq:SoilWaterStress)

\end{equation}

where $esw_{a}$ is the actual extractable soil water in root layers,
$esw_{p}$ is the potential extractable soil water in root layers.
$h_{w,\,pheno}$ is a function of soil water available ratio and soil
water stress, which is defined by parameters `x_sw_avail_ratio`
and `y_swdef_pheno` (default value 1) in wheat.xml and linearly
interpolated by APSIM. In the current version of APSIM-Wheat module,
no soil water stress for phenology is applied (Fig. \@ref(fig:wdSoilWaterStressPhenology)).
The soil water stress of phenology for flowering (`x_sw_avail_ratio_flowering`
and `y_swdef_pheno_flowering`) and grain filling (`x_sw_avail_ratio_start_grain_fill`
and `y_swdef_pheno_start_grain_fill`) phases are calculated
in the source code, but don't have influence on the phenology of wheat
in the current APSIM-Wheat version (default value of 1).


```{r wdSoilWaterStressPhenology,fig.cap='Relationship between soil water stress factor affecting phenology ($f_{W,\\, pheno}$) and the ratio of available soil water ($\\frac{esw_{a}}{esw_{p}}$).' }  

p <- wdVisXY(wheat_xml, 
		"x_sw_avail_ratio", "y_swdef_pheno ",
		xlab = "Ratio of available soil water",
		ylab = "Soil water stress \nof phenology")
print(p)

```



### Photosynthesis


Soil water stress of biomass accumulation ($f_{w,\,photo}$) is calculated
as follows.

\begin{equation}

f_{w,\,photo}=\frac{W_{u}}{W_{d}} (\#eq:swstressphoto)

\end{equation}

where $W_{u}$ is the total daily water uptake from root system (Equation \@ref(eq:WaterUpdate)),
$W_{d}$ is the soil water demand of `Leaf` and `Head`
parts (Equation \@ref(eq:soilWaterDemand)). 

Finally, the potential biomass production (radiation-limited$\Delta Q$)
can limit by water uptake ($f_{w,\,photo}<1$, i.e. when $W_{u}<W_{d}$),
or not (when $f_{w,\,photo}=1$, i.e. when $W_{u}=W_{d}$)


\begin{equation}

\Delta Q_{w}=\Delta Q_{r}f_{w,\,photo}=\Delta Q_{r}\frac{W_{u}}{W_{d}} (\#eq:WaterStressBiomassProduction)

\end{equation}


$f_{w,photo}$ also affect the senescence of the leaves.


### Leaf expansion


Soil water stress of leaf expansion is determined by the deficit of
soil water. 


\begin{equation}

f_{W,\,expan}=h_{w,\,expan}(\frac{W_{u}}{W_{d}}) (\#eq:WaterStressLeafExpansion)

\end{equation}

where $W_{u}$ is the crop water uptake (Equation \@ref(eq:WaterUpdate)),
$W_{d}$ is the crop water demand (Equation \@ref(eq:soilWaterDemand)).
$h_{w,\,expan}$ is a function of soil water content and stress, and
is defined by parameters `x_sw_demand_ratio` and `y_swdef_leaf`
in the wheat.xml, which is linearly interpolated by APSIM (Fig. \@ref(fig:wdWaterStress4LeafExpansion)). 


```{r wdWaterStress4LeafExpansion,fig.cap='Relationship between the soil water stress factor affecting expansion ($f_{W,\\, expan}$) and supply:demand ratio ($\\frac{W_{e}}{W_{d}}$).' }  

p <- wdVisXY(wheat_xml, 
		"x_sw_demand_ratio", "y_swdef_leaf",
		xlab = "Soil water supply:demand ratio",
		ylab = "Soil water stress of expansion")
print(p)

```



## KL factor


APSIM 7.5 introduces a modifying factor on KL (rate of maximum daily
water uptake per day) where there is an excess of chloride concentration
(Cl), exchangeable sodium percentage (ESP), or electrical conductivity
(EC) properties in the soil \citep{hochman2007simulating}. The KL
modifier is optional and triggered by setting the ModifyKL parameter
to `yes`.

When the KL modifier is activated, KL values are modified for each
layer, by factors (concerning Cl, ESP, EC; Fig. \@ref(fig:wdKLFactoring))
applied to default KL values. The modifiers are calculated using one
of the limiting factors in order of preference (Cl, ESP, EC), i.e.
KL is modified only if there are no soil parameters for Cl. The parameters
in the wheat.xml that control this mechanism are ClA, CLB, ESPA, ESPB,
ECA, ECB (slope and intercept of linear relationship for Cl, ESP and
EC).


```{r wdKLFactoring,fig.height=6,fig.cap='The KL factor in response to chloride concentration (Cl mg kg$^{-1}$, Exchangeable sodium percentage (ESP, \\%) and soil electrical conductivity (EC, dS m$^{-1}$.' }  

p <- wdKLFactoring(wheat_xml)
print(p)

```



