## 3 Experimental Results
### Initial Shear Modulus, \(G_{0}\)  
The initial shear modulus \(G_{0}\) is defined as the shear modulus at a small strain amplitude of \(10^{-4}\%\).  
Figure 6a illustrates the effect of the confining stress on the initial shear modulus. The relationship between the initial shear modulus and the confining stress can be satisfactorily expressed by a power function of Equation (6) proposed by [PERSON] et al. [49]:  
\[G_{0}=\alpha\cdot\sigma_{\mathrm{c}}^{\beta} \tag{6}\]  
where \(G_{0}\) is the initial shear modulus, \(\alpha\) and \(\beta\) are the fitting parameters related to cement level, void ratio, and other factors, and \(\sigma_{\mathrm{c}}\) is the confining stress. The _R-square_ is greater than 0.96 in all cases. As the treatment time increased from 12 h to 48 h, \(\alpha\) rose from 10.9 to 31.8, and \(\beta\) decreased from 0.51 to 0.36.  
Figure 6b reveals the relationship between the initial shear modulus and the treatment duration. It indicates that the increasing treatment time contributed to the increase in \(G_{0}\), which can be attributed to the calcite cementation gradually precipitating with the increasing time. The shear modulus of the untreated sands at confining stress of 25 kPa, 50 kPa, and 100 kPa was 58 MPa, 68 MPa, and 95 MPa, respectively, and it was about 98 MPa, 133 MPa, and 164 MPa after being treated for 48 h.  
When the MICP treatment time was less than 18 h, the difference between the dynamic shear modulus of the treated samples under a low confining stress of 25 kPa and untreated samples was not obvious. Under a confining stress of 100 kPa, the dynamic shear modulus of the treated samples was much higher than that of the untreated ones. This is because the calcareous sands used in the tests are loose and porous, making it challenging for calcium carbonate precipitated in the pores to form effective contacts in cases with short treatment time. The neighboring particles become clustered under a higher confining pressure, and the calcium carbonate particles precipitated in the pores, which do not contribute to the shear strength, have the potential to become effective contact points, thereby improving the initial dynamic shear modulus of the samples. [PERSON] et al. [8] reported that a lower content of CaCO\({}_{3}\) (C\({}_{\mathrm{CaCO_{3}}}\) = 3.5%, which is consistent with the content of CaCO\({}_{3}\) in the specimen treated for 12 h in this study) had no significant effect on the strength or stiffness properties of sand compared to untreated sand.  
Researchers have stated that a reliable correlation can be established between the \(G_{0}\) and \(q_{\mathrm{ucs}}\) for the cemented soil [49]. This study uses the average peak stress of two replicate specimens to represent the unconfined compressive strength. Figure 6c depicts the variation in \(G_{0}\) with \(q_{\mathrm{ucs}}\) for bio-treated samples under different confining stresses. The \(G_{0}\) increases with the \(q_{\mathrm{ucs}}\), and the relationship between them can be defined by a power function presented by [PERSON] et al. [50]:  
\[G_{0}=m\times q^{n}_{\mathrm{ucs}} \tag{7}\]  
where \(G_{0}\) is described in MPa, \(q_{\mathrm{ucs}}\) is described in kPa, and constants \(m\) and \(n\) are the fitting parameters in the formula. The _R-square_ exceeds 0.98 in all cases. The \(G_{0}\) of the treated sands rises nonlinearly with an increase in \(q_{\mathrm{ucs}}\), which is related to cementation between particles.