# Biochar Amended Soils and Water Systems: Investigation of Physical and Structural Properties
## 2 Materials and Methods
### Soil-Water Retention, Plant Available Water, and Aggregate Stability  
Three replicated air-dried soil samples for each soil and for the chosen BC rate, prepared as described above, were placed in stainless steel rings (5 cm diameter, 4.5 cm height, 88.4 cm\({}^{3}\) volume) and saturated, placed into the pressure plate apparatus [41], and then subjected to pneumatic potentials equal to 10.2, 102, 337, 1020, 3059 and 15,296 cm.  
The weight of the soil samples was determined at each equilibrium pressure and the sample height was also measured to check if shrinkage occurred at increasing pressure head, \(\left\lvert{h\,l}\right\rvert\), during the experiments in the pressure-plate apparatus [42].  
\(U\) (g/g) and \(\left\lvert{h\,l}\right\rvert\), corresponding respectively to the gravimetric water contents and pressure heads at equilibrium with the pneumatic potentials, were determined as:  
\[U=\frac{W_{w}-W_{d}}{W_{d}} \tag{3}\]  
Volumetric water content, \(\theta\) (cm\({}^{3}\)/cm\({}^{3}\)), was determined considering the wet soil sample, \(W_{w}\) (g), and the weight of the dry soil sample, \(W_{d}\) (g):  
\[\theta=\frac{W_{w}-W_{d}}{W_{d}}\frac{\rho_{b}}{\rho_{w}}=U\frac{\rho_{b}}{ \rho_{w}} \tag{4}\]  
where \(\rho_{b}\) (g/cm\({}^{3}\)) is the soil dry bulk density and \(\rho_{w}\) (g/cm\({}^{3}\)) is density of water. Plant available water, PAW (cm\({}^{3}\)/cm\({}^{3}\)), was calculated as:  
\[\text{PAW}=\left(\theta_{fc}-\theta_{wp}\right) \tag{5}\]  
PAW was also calculated as the difference between the gravimetric field capacity and wilting point, and indicated as PAW\({}_{\text{U}}\) (g/g), where \(\theta_{fc}\) is field capacity (cm\({}^{3}\)/cm\({}^{3}\)), i.e., the \(\theta\) value measured at \(\left\lvert{h\,l}\right\rvert\) = 336 cm, and \(\theta_{wp}\) (cm\({}^{3}\)/cm\({}^{3}\)) is the wilting point, i.e., the \(\theta\) value measured at \(\left\lvert{h\,l}\right\rvert\) = 15,296 cm. The two-tailed \(t\)-test was used to test the significance of differences between treatment means, and for the separation of the means a least significant difference test (significance level \(p\) = 0.05) was used.  
The van Genuchten (VG) retention function [43]:  
\[U=U_{r}+\frac{U_{s}-U_{r}}{\left(1+\left(\alpha\left\lvert{h}\right\rvert \right)^{n}\right)^{m}} \tag{6}\]  
the (\(U\), \(\left\lvert{h\,l}\right\rvert\)) experimental pairs was fitted by using in Equation (6) in which the saturated water content (g/g) is \(U_{s}\), the residual water content (g/g) is \(U_{r}\), the pressure head (cm) is \(\left\lvert{h\,l}\right\rvert\), and the empirical parameters are expressed by \(\alpha\) (cm\({}^{-1}\)), \(n\), and \(m=1-1/n\)[36]. The RETC code was used to estimate the VG parameters _Us_, _Ur_, \(\alpha\) and \(n\)[44].  
The pore diameter, \(D\) (\(\mu\)m), corresponding to (\(\left\lvert{h\,l}\right\rvert\)) (kPa), was estimated according to the Young-Laplace equation [37], which restrictively agrees with the assumption that pores are perfectly cylindrical, uniform and equally drained, as \(D=300/\left\lvert{h\,l}\right\rvert\).  
The differential water capacity curve was provided by the van Genuchten first derivative retention function that whose maximum is centred on the pressure head value associated to the modal suction, \(\mid\tau_{d}\mid\), and to the most common pore size diameter, \(D(\tau_{d})\):  
\[\frac{dU}{dt}=-\alpha(U_{s}-U_{r})(1-n)(\alpha|h|)^{n-1}\big{(}1+(\alpha|h|)^{n }\big{)}^{-(1+m)} \tag{7}\]  
Moreover, \(\mid\tau_{d}\mid\) (cm), represented as modal suction, was determined per [PERSON] et al. [36]:  
\[\mid\tau_{d}\mid=\frac{1}{\alpha}\left(\frac{n-1}{1+m\,n}\right)^{1/n} \tag{8}\]  
The soil structural index (SI, g/g cm\({}^{-1}\)) was calculated per [PERSON] and [PERSON] [45]:  
\[\text{SI}=\frac{\Delta U_{g}}{\mid\tau_{d}\mid} \tag{9}\]  
where \(\Delta U_{g}\) (g/g) is the volume of trainable pores, calculated as \(U_{s}-U_{r}\).