# Model Test on Grouting Properties of Alluvial Filler Soil
## 6 Engineering Characteristic Analysis of Grouting Soil  
### Soil Strength Analysis  
The physical and mechanical properties of the soil would be changed after grouting. The shear strength parameters, namely cohesion \(c\) and internal friction angle \(\Phi\) which can best reflect the strength of soil mass, would be improved. If the improved values are \(\Delta c\) and, \(\Delta\Phi\) respectively, the shear strengths of the grouting soil can be obtained by Taylor series, and the expression is as follows [21]:  
\[ \begin{array}{l}f(c_{0}+\Delta c,\ \phi_{0}+\Delta\phi)=f(c_{0},\ \phi_{0})+\left(\Delta c \frac{\partial}{\partial c}+\Delta\phi\frac{\partial}{\partial\phi}\right)f( c_{0},\ \phi_{0})+\frac{1}{2}\left(\Delta c\frac{\partial}{\partial c}+\Delta\phi\frac{ \partial}{\partial\phi}\right)^{2}f(c_{0},\ \phi_{0})\\ +\frac{1}{3!}\left(\Delta c\frac{\partial}{\partial c}+\Delta\phi\frac{ \partial}{\partial\phi}\right)^{3}\!f(c_{0},\ \phi_{0})+\cdots+\frac{1}{n!}\left(\Delta c\frac{ \partial}{\partial c}+\Delta\phi\frac{\partial}{\partial\phi}\right)^{n}\!f(c _{0},\ \phi_{0})+\Delta\\ \end{array}  \tag{4}\]  
where \(c_{0}\) and \(\phi_{0}\) are the initial shear strength values of the soil before grouting. For convenience of calculation, consider only the first three items in the formula. Equation (4) can be written as follows:  
\[ \begin{array}{l}f(c_{0}+\Delta c,\ \phi_{0}+\Delta\phi)=f(c_{0},\ \phi_{0})+\Delta c \frac{\partial f(c_{0},\ \phi_{0})}{\partial c}+\Delta\phi\frac{\partial f(c_{0},\ \phi_{0})}{ \partial c}\\ +\Delta c\frac{\partial^{2}f(c_{0},\ \phi_{0})}{\partial c^{2}}+\Delta\Phi \frac{\partial^{2}f(c_{0},\ \phi_{0})}{\partial\phi^{2}}+\Delta c\Delta\phi\frac{ \partial^{2}f(c_{0},\ \phi_{0})}{\partial c\phi}\end{array}  \tag{5}\]  
Equation (4) is used to calculate the shear strengths of the grouting soil after the grouting process. It is derived from Taylor series expansion and the Mohr-Coulomb yield criterion. By analyzing the shear strength parameters of the soil before and after grouting, the incremental values of cohesion (\(\Delta c\)) and internal friction angle (\(\Delta\Phi\)) can be obtained. Equation (5) provides a more straightforward expression for calculating the incremental shear strength values (\(\Delta c\) and \(\Delta\Phi\)) of the grouting soil. According to the Mohr-Coulomb yield criterion, the yield function of soil-rock mass can be expressed as [31]:  
\[\left\{ \begin{array}{l}f(c,\phi)=\sigma_{1}-\sigma_{3}k-2c\sqrt{k}\\ k=\frac{1+\sin\phi}{1-\sin\phi}\end{array} \right. \tag{6}\]  
where \(k\) is the coefficient of soil pressure, \(\sigma_{1}\) and \(\sigma_{3}\) are the minimum and maximum principal stresses, respectively. \(f(c,\phi)\) is the yield-critical function of the soil. Subtracting Equation (5) from Equation (6), the yield function can be expressed in the following expression:  
\[\begin{array}{l}f(c,\phi)=\sigma_{1}-\sigma_{3}\frac{1+\sin\phi}{1-\sin\phi }-2c_{0}\sqrt{\frac{1+\sin\phi}{1-\sin\phi}}-2\Delta c\sqrt{\frac{1+\sin\phi} {1-\sin\phi}}-\Delta\phi\frac{2c_{0}\cos\phi_{0}}{(1+\sin\phi_{0})^{2}}- \Delta\phi\sigma_{3}\frac{2\cos\phi_{0}}{(1-\sin\phi_{0})^{2}}\\ -2\Delta\phi\leftwhere \(\Delta c\), \(\Delta\Phi\) are the incremental values of the cohesion force and the internal friction of the soil, respectively. Equation (6) represents the Mohr-Coulomb yield criterion, which is a widely used model in geotechnical engineering to assess the failure of soil and rock materials. It relates the shear strength of the material to the principal stresses acting on it. The yield function \(f(c,\phi)\) is defined as zero when the material is on the verge of failure. Equation (7) incorporates the incremental shear strength values (\(\Delta c\) and \(\Delta\Phi\)) to describe the behavior of the soil with improved strength characteristics. Because the test soil is compacted artificially, it is difficult to ensure that the soil is completely homogeneous. In the proposed model, according to the results of the geo-radar test, the area of slurry diffusion can be obtained.