## 2 Mathematical Model
### Radial Grout Pressure Dissipation  
The diffusion range of the grout presents an irregular distribution owing to the gravity and the grout pressure distribution formed during the circumferential filling phase. The grout diffusion quantity and range are difficult to obtain during synchronous grouting in shield tunnels. A typical empirical equation of the diffusion distance is proposed by [PERSON] et al. (2022) [35]:  
\[D=cP_{0}^{0.622}K_{w}^{0.533}\beta_{r}{}^{-0.534} \tag{26}\]  
where \(D\) is the grout diffusion distance (cm), \(c\) is the fitting coefficient, \(P_{0}\) is the grout pressure (kPa), \(K_{w}\) is the permeability coefficient of water (\(10^{-4}\) m/s). Existing studies have shown a nearly negative linear relationship between the porosity and stratum depth, and the permeability coefficient increases linearly with porosity \(\varphi\)[37]. \(\beta_{r}\) is the viscosity ratio of the average grout viscosity (\(\mu_{(t)}\)) to water viscosity (\(\mu_{w}\)).  
According to [PERSON] et al. (2020), the average diffusion distance \(D\) can be calculated by multiplying \(m\) and \(b\) (\(\theta=90^{\circ}\)) [38]. The grout diffusion radius \(R_{s}\) can be expressed as follows:  
\[R_{s}=R_{1}+D \tag{27}\]  
The fitting coefficient \(c\) can be obtained according to Equation (26). Therefore, the grout diffusion distance \(D\) and diffusion radius \(R_{s}\) at other positions can be calculated according to Equations (26) and (27).  
According to [PERSON]'s law:  
\[Q_{r}=-K_{g}\frac{dh_{r}}{dR_{s}}A_{r}t_{1} \tag{28}\]  
where \(Q_{r}\) is the grout diffusion quantity (m\({}^{3}\)), \(h_{r}\) is the hydraulic head (m), \(R_{s}\) is the diffusion radius of grout (m), \(K_{g}\) is the permeability coefficient of the grout in soil (m/s), which is the ratio of \(K_{w}\) to \(\beta_{r}\), \(A_{r}\) is the diffusion area (m\({}^{2}\)), \(t_{1}\) is the grout diffusion time (s).  
\(\beta_{r}\) in Equation (26) can be obtained by:  
\[\beta_{r}=\frac{\int_{0}^{t_{1}}\mu_{(t)}dt_{1}}{\mu_{w}t_{1}}=\frac{\mu_{0} \big{(}e^{\xi t_{1}}-1\big{)}}{\mu_{w}\xi t_{1}} \tag{29}\]  
The shield tunneling distance in time \(t_{1}\) is \(dz\). Equation (29) is substituted into Equation (28):  
\[Q_{r}=-\frac{2\pi K_{w}R_{s}\mu_{w}\xi dzt_{1}}{\mu_{0}\big{(}e^{\xi t_{1}}-1 \big{)}}\frac{dl_{r}}{dR_{s}} \tag{30}\]  
\(dh_{r}\) is equal to the ratio of the radial grout diffusion pressure gradient \(dP_{r}\) (Pa/m) to \(\rho g\). Equation (30) can be transformed into:  
\[Q_{r}=-\frac{2\pi K_{w}R_{s}\mu_{w}\xi dzt_{1}}{\mu_{0}\big{(}e^{\xi t_{1}}-1 \big{)}\rho g}\frac{dP_{r}}{dR_{s}} \tag{31}\]  
Combining the porosity \(\varphi\), Equation (31) can be transformed into:  
\[\frac{dP_{r}}{dR_{s}}=\frac{\mu_{0}\varphi\big{(}e^{\xi t_{1}}-1\big{)}\rho g }{2K_{w}\mu_{w}\xi t_{1}}\bigg{(}\frac{R_{0}^{2}}{R_{s}}-R_{s}\bigg{)} \tag{32}\]  
Combining Equation (32) with the boundary conditions (\(R_{s}=R_{0}\) and \(P_{r}=P_{r0}\)), the radial grout pressure dissipation can be expressed as follows:  
\[\Delta P_{r}=P_{r0}-P_{r}=\frac{\mu_{0}\varphi\big{(}e^{\xi t_{1}}-1\big{)} \rho g}{2K_{w}\mu_{w}\xi t_{1}}\bigg{[}\frac{R_{s}{}^{2}-R_{0}^{2}}{2}-R_{0}^{ 2}\ln\bigg{(}\frac{R_{s}}{R_{0}}\bigg{)}\bigg{]} \tag{33}\]where \(P_{r0}\) is the radial grout diffusion initial pressure (Pa), and \(P_{r}\) is the grout pressure at \(R_{s}\) (Pa).  
Based on Equation (33), the radial grout pressure dissipation without considering the temporal variation in grout viscosity is described as follows:  
\[\Delta P_{r}=P_{r0}-P_{r}=\frac{\mu_{0}\varphi\rho g}{2K_{w}\mu_{w}t_{1}}\left[ \frac{{{R_{s}}^{2}-R_{0}^{2}}}{2}-R_{0}^{2}\ln\left(\frac{R_{s}}{R_{0}}\right)\right] \tag{34}\]