## 2 Methodology
### Calculation of Jet-Grouted Bottom sealing Barriers Considering Seepage Failure
The hydraulic head at the excavation bottom is herein assumed zero after dewatering (\(h_{0}=0\)), Equation (14) becomes as  
\[Q=k_{s}LBh_{1}/h_{s} \tag{15}\]  
By combining Equations (12), (13) and (15), the hydraulic head of groundwater at the bottom (\(h_{2}\)) and top (\(h_{1}\)) of the jet-grouted bottom sealing barriers as well as the flow rate \(Q\) can be, respectively, written as  
\[h_{2}=\frac{k_{u}k_{s}h_{g}+k_{u}k_{g}h_{s}}{k_{g}k_{u}+k_{u}k_{s}h_{g}+k_{u}k _{g}h_{s}}h_{w} \tag{16}\]  
\[h_{1}=\frac{k_{u}k_{g}h_{s}}{k_{g}k_{u}+k_{u}k_{s}h_{g}+k_{u}k_{g}h_{s}}h_{w} \tag{17}\]\[Q=\frac{k_{u}k_{g}k_{s}}{k_{s}k_{g}h_{u}+k_{u}k_{s}h_{g}+k_{u}k_{g}h_{s}}Lbh_{w} \tag{18}\]  
Ideally, the expected hydraulic conductivity (\(k_{g}\)) of the jet-grouted bottom sealing barriers is quite low, but the hydraulic conductivity (\(k_{g}\)) of the jet-grouted bottom sealing barriers in the field is obviously different from the expected value [20; 28]. In addition, it is impossible to accurately estimate the hydraulic conductivity of the jet-grouted bottom sealing barriers in advance during the design stage of the jet-grouted bottom sealing barriers, because it is influenced by geological conditions, construction technology, operation methods, and other factors. For this reason, the maximum acceptable seepage volume per unit area of jet-grouted bottom sealing barriers is taken as the performance control index during the design. When the actual seepage of jet-grouted bottom sealing barriers is less than the designed index, it can be considered that the jet-grouted bottom sealing barriers meets the design requirements. Thus, the following mathematical equation can be obtained based on the criterion of the allowable seepage of jet-grouted bottom sealing barriers:  
\[\frac{Q}{LB}=\frac{k_{u}k_{g}k_{s}}{k_{s}k_{g}h_{u}+k_{u}k_{s}h_{g}+k_{u}k_{g} h_{s}}h_{w}\leq[q] \tag{19}\]  
where \([q]\) is the designed maximum allowable seepage of jet-grouted bottom sealing barriers per unit area. By substituting the quantitative relation among \(h_{u}\), \(h_{w}\), \(h_{s}\) and \(h_{g}\) (i.e., \(h_{u}=h_{w}+h_{s}+h_{g}\)) into Equation (19), the limit equation for the allowable seepage of jet-grouted bottom sealing barriers can be expressed, as follows:  
\[Q=kLB\frac{dH}{dz} \tag{11}\]0  
When grouping all terms and dividing all terms by \(B/h_{w}\) in Equation (20), the dimensionless equation between the thickness (\(h_{g}\)) of jet-grouted bottom sealing barriers and the thickness (\(h_{s}\)) of untreated soils below the excavation bottom can be expressed, as follows:  
\[Q=kLB\frac{dH}{dz} \tag{11}\]1  
Safety factor \(F_{s}\) can be expressed as the ratio of the total effective gravity to the sum of seepage force (including jet-grouted barriers (\(h_{g}\)) and untreated soils (\(h_{s}\))), i.e.,  
\[Q=kLB\frac{dH}{dz} \tag{11}\]2  
where \(\gamma_{g}^{\prime}\) is effective unit weight of jet-grouted barriers, \(\gamma_{s}^{\prime}\) is effective unit weight of untreated soils, in the case of multi-layered soils, the value can be computed with a weighted value, and \(J_{g}\) is average hydraulic gradient of jet-grouted bottom sealing barriers, which is expressed as \(J_{g}=\left(h_{2}-h_{1}\right)/h_{g},J_{s}\) is the average hydraulic gradient of untreated soils, which is expressed as \(J_{s}=h_{1}/h_{s}\). By combining Equations (13), (15) and (22), safety factor \(F_{s}\) can be expressed as  
\[Q=kLB\frac{dH}{dz} \tag{11}\]3  
For this seepage failure mechanism, the designed index (\([q]\)) of the maximum allowable seepage of jet-grouted bottom sealing barriers per unit area is a given variable.