## 2 Porosity Analysis of Injected Porous Media  
Cement-based materials are usually treated as Bingham fluids, and their flow rate through a single pore channel is  
\[q(\lambda)=-\frac{\pi\lambda^{3}+D_{T}}{128\mu L^{D_{t}-1}D_{T}}\frac{dp}{dL}- \frac{\tau_{0}\pi\lambda^{3}}{32\mu} \tag{1}\]  
where \(\tau_{0}\) is yield stress, \(\mu\) is the viscosity, \(D_{f}\) is the fractal dimension of the channel, and \(D_{t}\) is the tortuous fractal dimension of the channel. Then, the flow rate per unit cross-section can be expressed as,  
\[ \begin{split} Q(\lambda)=-\int_{\lambda_{\min}}^{\lambda_{\max} }q(\lambda)dN(\lambda)\\ =\int_{\lambda_{\min}}^{\lambda_{\max}}\left(-\frac{\pi\lambda^{3 +D_{T}}}{128\mu L^{D_{T}-1}D_{T}}\frac{dp}{dL}-\frac{\tau_{0}\pi\lambda^{3}}{ 32\mu}\right)\\ \times D_{f}\lambda_{\max}^{D_{f}}\lambda^{-(D_{f}+1)}d\lambda\\ =-\frac{\pi\lambda_{\max}^{3+D_{T}}D_{f}d\Phi}{128\mu L^{D_{T}-1 }D_{T}\left(3+D_{T}-D_{f}\right)dL}\left[1-\left(\frac{\lambda_{\min}}{ \lambda_{\max}}\right)^{3+D_{T}-D_{f}}\right]\\ =\frac{\pi\tau_{0}D_{f}}{32\mu\left(3-D_{f}\right)}\lambda_{\max }^{3}\left[1-\left(\frac{\lambda_{\min}}{\lambda_{\max}}\right)^{3-D_{f}} \right]\end{split}  \tag{2}\]  
where the diameters of this series of pore channels are (\(\lambda_{\min}\), \(\lambda_{\max}\)). The number of tortuous channels in a porous medium can be expressed as in [43; 44]:  
\[-dN(\lambda)=D_{f}\lambda_{\max}^{D_{f}}\lambda^{-D_{f}+1}d\lambda, \tag{3}\]  
The actual length \(L_{t}\) of the curved channel can be expressed as  
\[dL_{t}=\lambda^{1-D_{T}}L^{D_{T}-1}D_{T}dL, \tag{4}\]  
where \(\lambda\) is the diameter of the curved channel, and \(L_{t}\) and \(L_{0}\) are the actual and representative lengths of the channel, respectively. In an element, the pore area \(A_{P}\) of all pore channels on a certain cross-section can be expressed as  
\[ \begin{split} A_{P}=-\int_{\lambda_{\min}}^{\lambda_{\max}} \frac{1}{4}\pi\lambda^{2}dN(\lambda)\\ =\int_{\lambda_{\min}}^{\lambda_{\max}}\frac{1}{4}\pi\lambda^{2}D _{f}\lambda_{\max}^{D_{f}}\lambda^{-D_{f}+1}d\lambda\\ =\frac{\pi D_{f}}{4(2-D_{f})}\lambda_{\max}^{2}\left[1-\left( \frac{\lambda_{\min}}{\lambda_{\max}}\right)^{2-D_{f}}\right]\end{split}  \tag{5}\]Assuming that the dynamic porosity of the injected medium is \(\varphi\), then the total area \(A\) of a certain cross-section in the unit can be expressed as  
\[A=\frac{A_{P}}{\varphi} \tag{6}\]  
The average flow rate of slurry in porous medium is,  
\[ \begin{array}{c}v=\frac{Q(\lambda)}{A}\\ =-\frac{\lambda_{\max}^{1+D_{T}}\varphi(2-D_{f})dp}{32\mu L^{D_{f}-1}D_{T}(3+D_ {T}-D_{f})dL}\\ \times\left[1-\left(\frac{\lambda_{\min}}{\lambda_{\max}}\right)^{3+D_{T}-D_{ f}}\right]/\left[1-\left(\frac{\lambda_{\min}}{\lambda_{\max}}\right)^{2-D_{f}} \right]\\ -\frac{\lambda_{\max}\exp(2-D_{f})}{8\mu\left(3-D_{f}\right)}\left[1-\left( \frac{\lambda_{\min}}{\lambda_{\max}}\right)^{3-D_{f}}\right]/\left[1-\left( \frac{\lambda_{\min}}{\lambda_{\max}}\right)^{2-D_{f}}\right]\end{array}  \tag{7}\]  
From Equation (6), the initial porosity of injected media can be expressed as,  
\[\varphi_{0}=\frac{A_{P0}}{A}=\frac{\pi D_{f}}{4\left(2-D_{f}\right)}\lambda_{ \max}^{2}\left[1-\left(\frac{\lambda_{\min}}{\lambda_{\max}}\right)^{2-D_{f}} \right]/A \tag{8}\]  
Considering the percolation effect of cement slurry, the dynamic porosity of the injected medium can be expressed as,  
\[\varphi=\frac{A_{pt}}{A}=\frac{\pi D_{f}}{4(2-D_{f})}(\lambda_{\max}-\Delta \lambda)^{2}\left[1-\left(\frac{\lambda_{\min}}{\lambda_{\max}-\Delta\lambda} \right)^{2-D_{f}}\right]/A \tag{9}\]  
where \(A_{p0}\) and \(A_{pt}\) are the pore area of the injected medium on a certain cross-section before and after percolation, respectively; \(\Delta\lambda\) is the reduction in the diameter of the pore channels.