# Real-Time Behaviour of Dredged Slurry Treated by Air-Booster Vacuum Consolidation
## 8 Discussion
### Crack Initiation Pressure  
Both vertical and oblique cracks were observed in the PIV model test. The vertical cracks were parallel to the booster plate, and their positions basically overlapped the projection of the booster plate, as indicated in Figures 15-17. In comparison, oblique cracks emerged only in soils above the head of the booster plate. With these crack shapes in mind, it would be straightforward to attribute the vertical and oblique cracks to tensile and shear strength failures of the slurry soil, respectively. [PERSON] et al. [22] proposed two equations for evaluating crack-initiation pressures for the tensile and shear failures of soils when high-pressure air is injected into a borehole:  
\[P_{\mathrm{f}}\ =\ 2\sigma_{3}+\sigma_{t}, \tag{1}\]  
\[P_{\mathrm{f}}\ =\ (1+\sin\varphi)\sigma_{3}+c\cos\varphi \tag{2}\]  
where \(\sigma_{3}\) is the minimum principal stress in soil before air pressurisation; \(c\) and \(\varphi\) are the cohesion and friction angles of the soil, respectively; and \(\sigma_{t}\) is the tensile strength of soil. Generally, \(\sigma_{t}\) is small for soft soil, and a range of \(\sigma_{t}\ =\ (\frac{1}{6}\mathrm{to}\frac{1}{5})\ q_{\mathrm{u}}\) can be adopted [23], where \(q_{\mathrm{u}}\) denotes the unconfined uniaxial compression strength of soil. For the undrained behaviour of saturated cohesive soil, \(q_{\mathrm{u}}\ =\ 2c\). [PERSON] and [PERSON] [23] proposed a unified equation for determining the hydrofracturing pressure of cohesive soil, regardless of whether the fracture surface is vertical, horizontal or oblique, i.e.,  
\[P_{\mathrm{f}}\ =\ \sigma_{3}+q_{\mathrm{u}}, \tag{3}\]  
As indicated by the authors, the horizontal and oblique cracks are initialised by the shear failure of the soil near the borehole. Before fracture, soil displacements may already be observed during the pressurisation process. Under the small strain assumption of the soil, [PERSON] et al. [24] derived an explicit solution for the expansion displacement of cylindrical cavities in cohesive soil subjected to an internal pressure, \(p\):  
\[\frac{\delta}{a}\ =\ \frac{c}{2G}\exp{(\frac{p-p_{0}}{c}-1)}, \tag{4}\]  
where \(\delta\) is the expansion displacement of the cavity along the radial direction; \(a\) is the cavity radius; \(p_{0}\) is the hydrostatic stress of the soil before pressurisation; \(G\) is the shear modulus of the soil, which is related to [PERSON]'s modulus \(E\) by \(G\ =\ E/2(1+\vartheta)\); and \(\vartheta\)  
Figure 17: Schematic view of the connected cracks at the end of the air pressurisation. is the Poisson's ratio. When the expansion displacement is sufficiently large, the internal expansion pressure will reach its limit \(p_{L}\):  
\[p_{L}\ =\ c\left[1+\ln\left(\frac{G}{c}\right)\right]+p_{0}, \tag{5}\]  
It is noted that soil fracturing may occur before large radial displacement occurs. Thus, \(p_{L}\) can serve as the upper limit of the crack initiation pressure of soil. Equations (4) and (5) are applicable to the undrained behaviour of saturated soil. Since fracturing happens within a short time, the soil surrounding the booster plate in our model test should be undrained. At the same time, the slurry soil is consolidated by vacuum pressure before air pressurisation, thus guaranteeing saturation of the soil. For the undrained behaviour of the saturated cohesive soil, \(\varphi\ =\ 0\) and \(\vartheta\ =\ 0.5\) can be assumed. The cohesion, \(c\), of the test soil is best determined using an in situ test, e.g., cross-vane shear. However, the vane would puncture the membrane and cause air leakage because the vacuum consolidation is restarted directly after air boosting. Instead, we use the empirical formula of [PERSON] et al. (2015) to estimate the cohesion of the test soil:  
\[c\ =\ \frac{1}{\left(I_{L}-0.21\right)^{2}}, \tag{6}\]  
where \(I_{L}\) is the liquidity index.