## 2 Methodology
### Calculation of Jet-Grouted Bottom Sealing Barriers Considering Instability Failure  
For the instability failure mechanism (shown in Figure 2a), the limit equilibrium equation for the assessment of the instability of jet-grounded bottom sealing barriers can be expressed as:  
\[V=G+R \tag{1}\]  
where \(V\) is uplift action of the groundwater pressure acting at the bottom of jet-grounded bottom sealing barriers, \(G\) is stabilizing action that is provided by the self-weight of both untreated soils below the excavation bottom and jet-grounded bottom sealing barriers, and \(R\) is stabilizing action because of the shear force acting at the interface between the jet-grounded bottom sealing barriers and the underground enclosures. It is noted that stabilizing action of the shear force acting at the interface between untreated soils below the excavation bottom and underground enclosures is herein conservatively neglected. The water pressure acting on the bottom of jet-grounded bottom sealing barriers can be written as:  
\[V=\gamma_{w}h_{u}A_{p} \tag{2}\]  
where \(\gamma_{w}\) is unit weight of water, \(h_{u}\) is head height of water acting at the bottom of jet-grounded bottom sealing barriers, \(A_{p}\) is the area of jet-grounded bottom sealing barriers, for the long rectangle excavation, which can be computed as \(A_{p}=L\cdot B\), and \(L\) and \(B\) are the length and width of the long rectangle excavation, respectively. The structures' self-weight includes the self-weight of both untreated soils below the excavation bottom and jet-grounded bottom sealing barriers, the stabilizing action being provided by the structures' self-weight \(G\) can be computed as:  
\[G=\big{(}\gamma_{s}h_{s}+\gamma_{g}h_{g}\big{)}A_{p} \tag{3}\]  
where \(\gamma_{s}\) is unit weight of natural soil, \(\gamma_{g}\) is unit weight of jet-grounded bottom sealing barriers, \(h_{g}\) is the thickness of jet-grounded bottom-sealing barriers, and \(h_{s}\) is the depth of untreated soils with regard to the excavation bottom. The total shear force acting at the interface between the jet-grounded bottom sealing barriers and the underground enclosures \(R\) can be expressed as:  
\[R=p\int_{h_{exc}+h_{s}}^{h_{exc}+h_{s}}\tau dz=2(B+L)\int_{h_{exc}+h_{s}}^{h_ {exc}+h_{s}+h_{g}}\tau dz \tag{4}\]  
where \(p\) is perimeter of long rectangle excavation, which can be written as \(p\)= 2\(\cdot\)(\(L+B\)), \(h_{exc}\) is the excavation depth, and \(\tau\) is the shear stresses acting at the interface between the jet-grounded bottom sealing barriers and the underground enclosures. Herein, neglecting the frictional effect between jet-grounded bottom sealing barriers and the underground enclosures, the shear strength of the jet-grounded bottom sealing barriers is considered via the Tresca yield criterion. The shear strength \(\tau\) acting at the interface between the jet-grounded bottom sealing barriers and the underground enclosures can be assumed to coincide with the cohesion \(c_{g}\) of the jet-grounded bottom sealing barriers. Besides, the cohesion \(c_{g}\) can be expressed as a function of the unconfined compression strength \(q_{u}\) of the jet-grounded bottom sealing barriers, namely, the shear strength \(\tau\) acting at the interface between the jet-grounded bottom sealing barriers and the underground enclosures can be written as:  
\[\tau=c_{g}=\delta q_{u} \tag{5}\]  
where \(\delta\) is a constant, which typically ranges between 0.2 and 0.3 [27].