# Analysis of Damage and Permeability Evolution of Sandstone under Compression Deformation
## 2 Coupling Analysis of Rock Damage and Permeability
### Damage and Deformation  
The definition of the damage variable is not unique in damage mechanics theory. Since stress-induced damage is a permanent defect for rocks that is usually unrecoverable and irreversible, the main dependent variable of damage expression is related to plastic strain [17; 27; 28]. This relationship between the damage variable and plastic strain can be verified by rock damage experiments. Figure 2 shows the damage-strain and stress-strain curves of sandstone obtained from acoustic emission tests under conventional triaxial compression [29; 30]. The test results indicate that sandstone enters the stage of stable crack growth when the differential stress (\(\sigma_{1}\)-\(\sigma_{3}\)) exceeds the elasticity limit \(\sigma_{e}\), at which time the sandstone has undergone irreversible plastic deformation and damage. When the differential stress continues to increase, the rock enters the unstable crack growth, failure, and residual deformation stages in sequence. According to the evolution law of rock damage, the damage-strain curve can be divided into three stages, namely, slow growth, accelerated growth, and balance. The rapid accumulation of damage is mainly concentrated in the period of growth and propagation of rock new cracks, that is, unstable crack growth and failure stages in the stress-strain curve. Therefore, rock damage is closely related to its unrecoverable deformation and crack development. Figure 1: Coupling mechanism of rock damage and permeability under stress and seepage fields. Based on this, the following unified expression can be used to define the damage variable. It is important to note that the convention of considering compressive stress and strain as positive, and tensile stress and strain as negative, is adopted here. \[\frac{dD}{d\varepsilon_{p}}= \begin{cases}f\big{(}\varepsilon_{p}\big{)}&\big{(} \varepsilon_{p}>0\big{)}\\ 0&\big{(}\varepsilon_{p}=0\big{)}\end{cases}  \tag{1}\]  
where \(D\) and \(\varepsilon_{p}\) are the damage variable and plastic strain, and \(f(\varepsilon_{p})\) denotes the probability density function (PDF) of damage. The form of the damage PDF should be determined according to the damage mechanism of the rock. Commonly, there are normal distribution functions and Weibull distribution functions [29]. In fact, the Weibull distribution is related to other distributions, such as Pareto, exponential, and normal distributions, depending on its shape parameter \(m\). Figure 3 shows the PDF and cumulative distribution function (CDF) of the Weibull distribution with different shape parameters. Comparing Figures 2 and 3, it is obvious the evolution law of the CDF of the Weibull distribution with a shape parameter greater than one is more in line with that of rock damage evolution. This is because a value of the shape parameter greater than one indicates that the failure rate increases gradually with time, which is often used to describe aging or defect processes. In addition, the PDF of the Weibull distribution (\(m>1\)) is also consistent with the acoustic emission test results, which fully demonstrates the superiority of using the Weibull distribution to describe rock damage. Based on the above analysis and theoretical derivation, the PDF of rock damage and its CDF were proposed in our previous studies [30]. The applicability of this damage variable model was demonstrated through experimental comparisons. \[D=1-\exp\left\{-\left(\frac{\varepsilon_{ep}}{\eta}\right)^{m}\right\} \tag{2}\]  
\[\left\{ \begin{array}{l}\omega=\frac{E\cdot e_{ep}^{p}}{\sigma_{p}-\left( \alpha_{2}+\sigma_{3}\right)}\\ m=\frac{1}{1-\ln(\omega)}\\ \eta=e_{ep}^{p}\cdot\left\{\ln(\omega)\right\}^{\ln(\omega)-1}\end{array} \right. \tag{3}\]  
Figure 2: Damage–strain and stress–strain curves of sandstone at a confining pressure of 10 MPa; \(\sigma_{c}\), \(\sigma_{d}\), and \(\sigma_{p}\) are the elasticity limit, dilatancy stress, and peak stress [30].