# Experimental Study on the Permeability of Microbial-Solidified Calcareous Sand Based on MICP
## 3 Results and Discussion
### Correlational Analyses  
According to the results of the permeability tests, the correlation of the permeability coefficient (\(k\)) with the uniformity coefficient (\(C_{u}\)) and curvature coefficient (\(C_{c}\)) was analyzed. To reduce the influence of other factors, the effective particle size (\(d_{10}\)) of each sand sample was controlled to be a constant value. Therefore, \(d_{10}\) was introduced into the fitting equation, based on the experience of the expression of the permeability coefficient established by some scholars [53]. The analysis results are shown in Figure 11, where MT = 1 means the MICP treatment is performed once, and so on.  
For Figure 11a-c, the approximate expression for different \(C_{u}\) is given as the following equation:  
\[k=ae^{-C_{u}/\text{lg}d_{10}}+b \tag{8}\]  
Figure 11: Correlation of permeability coefficient (k) with uniformity coefficients (Cu) and curvature coefficient (Cc), respectively, for sand samples with a varied number of MICP treatments (MT) and relative densities (\(D_{r}\)): in Figure (**a–c**), Cu =2.0, 6.0 and 8.0 for samples A, E and B, respectively; in Figure (**d–f**), Cc =0.5, 1.5 and 2.0 for samples D, B and C, respectively.  
where \(k\) is the permeability coefficient of the sand sample, cm/s; \(C_{u}\) is the uniformity coefficient; \(d_{10}\) is the effective particle size, \(d_{10}=0.25\) mm; and \(a\) and \(b\) are the relative coefficients.  
For Figure 11d-f, the approximate expression for different \(C_{c}\) is given as the following equation:  
\[k=-a^{\prime}e^{C_{c}/\mathrm{lg}d_{10}}+b^{\prime} \tag{9}\]  
where \(k\) is the permeability coefficient of the sand sample, cm/s; \(C_{c}\) is the curvature coefficient; \(d_{10}\) is the effective particle size, \(d_{10}=0.25\); and \(a^{\prime}\) and \(b^{\prime}\) are the relative coefficients.  
Equations (8) and (9) show that the permeability coefficient is an exponential function of \(C_{u}\) and \(C_{c}\), respectively. Overall, \(k\) is roughly proportional to \(C_{u}\) and \(C_{c}\)[53], and most curves show a high fitting degree (\(R^{2}>90\%\)). Notably, with the case of MT \(<5\), the curves involving \(C_{u}\) showed a concave trend, while the curves involving \(C_{c}\) showed a convex trend. This result indicates that when increasing the MICP treatment cycles, the higher the \(C_{u}\) (\(C_{u}\geq 6.0\)) and the lower the \(C_{c}\) (\(2.0>C_{c}>0.5\)), more of a decrease in \(k\) will be observed. Indeed, larger \(C_{u}\) and smaller \(C_{c}\) often imply well grain gradation. Therefore, it can be inferred that only fewer MICP treatment cycles are needed to achieve the target permeability coefficient when the calcareous sand shows a well grain gradation.  
In addition, Figure 11 also presents an interesting result in that the curves with poor fitting results (\(R^{2}<90\%\)) are mainly concentrated on MT = 5-7. It is deduced that after a number of cycles of MICP treatment (MT \(\geq 5\)), the interparticle pores of the calcareous sand are basically filled by the induced CaCO\({}_{3}\), and the difference in the pore structure caused by the different particle gradations (\(C_{u}\) and \(C_{c}\)) has been weakened. Therefore, the permeability coefficient does not correlate well with \(C_{u}\) or \(C_{c}\) after a number of cycles of MICP treatment (MT \(\geq 5\)).  
Finally, to attempt to observe a theoretically more general fitting curve, the values of \(C_{u}\) and \(C_{c}\) are extended in Figure 11b,e, respectively. The results are shown in Figure 12a,b. It can be seen that for \(C_{u}\geq 6.0\), the decrease in \(k\) becomes increasingly obvious with the increase in \(C_{u}\) and MT; for \(2.0>C_{c}>0.5\), the decrease in \(k\) keeps increasing with the increase in \(C_{c}\) and MT; however, when \(C_{c}>2.0\), the decrease in \(k\) remains stable. However, the validity of the reacquired curves need to be verified by repeating permeability tests with a wider range of grain gradations in the future.