## 2 Mathematical Model
### Longitudinal Grout Filling Pressure Dissipation  
The force analysis of the longitudinal grout flow in an arbitrary longitudinal profile of the shield tail void is shown in Figure 5. The longitudinal flow of the grout is regarded as a one-dimensional motion along the \(z\)-axis.  
The force equilibrium relationship of the grout element in the longitudinal direction is expressed as follows:  
\[P_{l}dy-(P_{l}+dP_{l})dy+\tau dl-(\tau+d\tau)dl=0 \tag{35}\]  
where \(P_{l}\) is the longitudinal grout pressure (Pa), \(dl\) is the length of the grout element along the longitudinal direction (m), and \(dy\) is the grout element height perpendicular to the longitudinal flow direction (m).  
Equation (35) is transformed into:  
\[\frac{d\tau}{dy}=-\frac{dP_{l}}{dl} \tag{36}\]  
Defining \(\frac{dP_{l}}{dl}\) as \(B\) and combining (\(y=0\), \(\tau=0\)), Equation (36) is integrated as follows:  
\[\tau=-By \tag{37}\]  
Combining the boundary conditions (\(y=y_{p}\), \(\tau=\tau_{0}\)), the height of the longitudinal flow core \(y_{p}\) is expressed as follows:  
\[y_{p}=-\frac{\tau_{0}}{B} \tag{38}\]  
Based on the derivation of Equations (12)-(16), the average longitudinal flow velocity of grout can be expressed as:  
\[\overline{v}_{l}=\frac{1}{b\eta_{0}e^{\delta t}}\left[B\left(-\frac{2}{3}\frac {\tau_{0}}{B^{3}}-\frac{b^{3}}{12}\right)+\tau_{0}\left(\frac{\tau_{0}}{B^{2} }-\frac{b^{2}}{4}\right)\right] \tag{39}\]  
Figure 5: Force analysis of the longitudinal grout flow.  
When the initiation pressure gradient \(\lambda\) is equal to the ratio of \(2\tau_{0}\) to \(b\), Equation (39) is expressed as follows:  
\[\overline{v}_{l}=\frac{b^{2}}{12\mu_{0}e^{\xi t}}\bigg{(}-B-\frac{3}{2}\lambda+ \frac{\lambda^{3}}{2B^{2}}\bigg{)} \tag{40}\]  
According to the research of [PERSON] et al. (2022), \(\frac{dB}{dl}\) is much larger than \(\lambda\)[39]. \(\frac{\lambda^{3}}{2B^{2}}\) in Equation (40) can be neglected. So, Equation (40) can be simplified as follows:  
\[\overline{v}_{l}=\frac{b^{2}}{12\mu_{0}e^{\xi t}}\bigg{(}-B-\frac{3\tau_{0}}{b }\bigg{)} \tag{41}\]  
According to the conservation of mass, the total quantity of the injected grout is expressed as follows:  
\[qt_{1}=\overline{v}_{l}\ \pi\bigg{(}{R_{0}}^{2}-{R_{1}}^{2}\bigg{)}t_{1}+\varphi \pi\bigg{[}(R_{1}+mb)^{2}-R_{0}^{2}\bigg{]}\overline{v}_{d}t_{1} \tag{42}\]  
where \(t_{1}\) is the longitudinal filling time (s). Substituting Equation (41) into Equation (42), the average longitudinal flow velocity of grout is expressed as follows:  
\[\overline{v}_{l}=\frac{b^{2}}{12\mu_{0}e^{\xi t}}\bigg{(}-B-\frac{3\tau_{0}}{ b}\bigg{)}=\frac{q-\varphi\pi\bigg{[}(R_{1}+mb)^{2}-R_{0}^{2}\bigg{]}v_{d}}{\pi({R_{0}} ^{2}-{R_{1}}^{2})} \tag{43}\]  
Combined with the boundary conditions (\(P_{l}=P_{0}\) and \(l=0\)), the longitudinal grout filling pressure dissipation can be expressed as:  
\[\Delta P=P_{0}-P_{l}=\frac{12\mu_{0}e^{\xi t}\Big{\{}q-\varphi\pi\Big{[}(R_{1 }+mb)^{2}-R_{0}^{2}\Big{]}v_{d}\Big{\}}}{\pi b^{2}({R_{0}}^{2}-{R_{1}}^{2})}l+ \frac{3\tau_{0}}{b}l \tag{44}\]  
where \(P_{0}\) is the grout pressure formed at the completion of the circumferential filling (Pa), \(l\) is the distance of longitudinal tunneling of the shield (m), \(P_{l}\) is the grout pressure at longitudinal tunneling distance \(l\) (Pa).  
When \(l\) is equal to \(v_{d}\) multiplied by \(t_{1}\), Equation (44) can be expressed as follows:  
\[\Delta P=P_{0}-P_{l}=\frac{12\mu_{0}e^{\xi t}\Big{\{}q-\varphi\pi\Big{[}(R_{1 }+mb)^{2}-R_{0}^{2}\Big{]}v_{d}\Big{\}}}{\pi b^{2}({R_{0}}^{2}-{R_{1}}^{2})}v _{d}t_{1}+\frac{3\tau_{0}}{b}v_{d}t_{1} \tag{45}\]  
Based on Equation (45), the longitudinal grout filling pressure dissipation without considering the temporal variation in grout viscosity is described as follows:  
\[\Delta P=P_{0}-P_{l}=\frac{12\mu_{0}\Big{\{}q-\varphi\pi\Big{[}(R_{1}+mb)^{2}- R_{0}^{2}\Big{]}v_{d}\Big{\}}}{\pi b^{2}({R_{0}}^{2}-{R_{1}}^{2})}v_{d}t_{1}+ \frac{3\tau_{0}}{b}v_{d}t_{1} \tag{46}\]