# Experimental Study on the Pre-Peak Mechanical and Seepage Characteristics of Granite
## 3 Analysis of Test Results
### Effective Stress Analysis under Stress-Seepage Coupling  
The effective stress principle is essential in stress-seepage field coupling [39]. A great deal of research has been conducted on this: [PERSON] [40] first proposed the concept of effective stress to describe the geotechnical medium via the coupling of external stress and pore water pressure. The formulation is as follows:  
\[\sigma^{\prime}=\sigma-p \tag{4}\]  
where \(\sigma\) is the stress acting on the entire geotechnical medium, \(\sigma^{\prime}\) is the stress acting on the particles of the geotechnical medium, and \(p\) is the fluid pressure between particles. The formulation is widely used in the field of geotechnical solid-flow coupling. Researchers have extended the formulation with further improvements: [PERSON] [41] introduced an effective stress factor \(\alpha\), which corrects the equation as follows:  
\[\sigma^{\prime}=\sigma-\alpha p \tag{5}\]  
A big controversy exists for the effective stress coefficient \(\alpha\) value. The former also proposed a number of calculation methods. The researchers of [42] consider the effective stress factor \(\alpha\) to be equal to the porosity of the porous medium \(\varphi\). [PERSON] [43] and [PERSON] [44] proposed that the effective stress coefficient is related to the equivalent bulk modulus (\(K\)) and grain bulk modulus (\(K_{s}\)) of the porous medium. The literature indicates that the effective stress coefficient does not directly depend on the porosity \(\varphi\), and after rigorous formulation derivation and experimental validation, the following expression is obtained:  
\[\alpha=1-K/K_{s} \tag{6}\]  
In this paper, the volumetric modulus method proposed in the literature [45] was applied. Based on the known volumetric modulus of the rock and the equivalent volumetric modulus of the fractured rock mass, combined with the effective stress coefficients in Equation (6) and substituted into Equation (5) to derive the effective stresses, the curves between the permeability and the effective stresses are plotted in Figure 10.  
Many researchers have carried out many experiments and studies to investigate the relationship between permeability and effective stress in porous media and have put forward different views and modeling relationships. The literature [40] suggests that an exponential model can be used to describe the relationship between permeability and effective stress in porous rocks:  
\[K=K_{0}\mathrm{e}^{-\beta\mathrm{P_{c}}} \tag{7}\]  
where \(K\) is the permeability of the porous rock, \(K_{0}\) is the initial permeability of the porous rock, \(\beta\) is the regression coefficient, and \(\mathrm{P_{c}}\) is the effective stress acting on the porous rock. [PERSON] et al. [46] and [PERSON] et al. [47] used a power law relationship to characterize the permeability of porous rocks as a function of the confining pressure:  
\[K=A(\sigma_{c})-B+C \tag{8}\]  
where \(A\), \(B\), and \(C\) are the regression coefficients, and \(\sigma_{c}\) is the confining pressure. [PERSON] et al. [48] found that the relationship between permeability and confining pressure for cohesive sandstones can be expressed as a polynomial:  
\[K=a\sigma_{c}^{2}+b\sigma_{c}+C \tag{9}\]  
where \(a\), \(b\), and \(c\) are the regression coefficients. [PERSON] et al. [49] proposed the use of a logarithmic model to characterize the relationship between permeability and effective stress:  
\[\left(K/Kref\right)^{1/3}=1-Sln\left(\sigma_{eff}/\sigma_{eff0}\right) \tag{10}\]  
where \(\sigma_{eff}\) and \(Kref\) are the initial effective stress and corresponding initial permeability, respectively, and \(\sigma_{eff}\) and \(K\) are the effective stress and related permeability, respectively. The literature [50] gives \(s=\sqrt{2}h/\alpha_{ref}\) via the derivation of the formula, where h is the height of the convex body on the microcracks. Combining the data obtained from the tests and the bulk modulus method described in the previous section, the permeability was calculated and plotted against the effective stresses.