# Analysis of Damage and Permeability Evolution of Sandstone under Compression Deformation
## 2 Coupling Analysis of Rock Damage and Permeability
### Coupling Equation  
Establishing the damage-permeability evolution equation is the key to coupled analysis. It is well known that the volumetric strain of rock consists of elastic and plastic volumetric strain. For porous media such as rock and soil, the elastic volumetric strain can be derived from [PERSON]'s poroelasticity theory [34]. \[\varepsilon_{V}^{\epsilon}=-\frac{1}{K}(\sigma_{m}-\alpha_{b}\cdot p)=-\frac{ \sigma_{\epsilon}}{K} \tag{12}\]  
Figure 6: Comparison results between theoretical model and experimental data in terms of porosity and permeability. where \(\varepsilon_{V}^{e}\) denotes the elastic volumetric strain of the rock, \(\alpha_{b}\) and \(p\) are [PERSON]'s coefficient and the pore pressure, \(\sigma_{m}\) and \(\sigma_{e}\) are the mean stress and effective mean stress of the rock, and \(K\) denotes the bulk modulus of the rock. Equation (2) can be transformed into the following form:  
\[\varepsilon_{V}^{p}=\Phi\cdot\eta\cdot\left(\ln\biggl{(}\frac{1}{1-D}\biggr{)} \right)^{1/m} \tag{13}\]  
where \(\varepsilon_{V}^{p}\) denotes the plastic volumetric strain of the rock, and \(\Phi\) is used to describe the relationship between the equivalent plastic strain and plastic volumetric strain. Since the volume change in the rock matrix in the compressed state was much smaller than that in the void (\(\varepsilon_{Vm}<<\varepsilon_{V}\)), the volumetric strain of the rock matrix was neglected in the subsequent theoretical analysis. Substituting Equations (12) and (13) into Equation (11), the damage-permeability evolution equation can be derived. \[k=k_{0}\biggl{\{}1-\frac{\sigma_{e}}{K\phi_{0}}+\frac{f(D)}{K\phi_{0}}\biggr{\}} ^{3} \tag{14}\]  
\[f(D)=\Phi\cdot\eta K\biggl{\{}\ln\biggl{(}\frac{1}{1-D}\biggr{)}\biggr{\}}^{1/m} \tag{15}\]  
The value of function \(f(D)\) is always positive due to the damage variable varying from zero to one. In the case of determining the bulk modulus, scale, and shape parameters, the relationship between \(f(D)\) and the damage variable can be obtained according to Equation (15). As shown in Figure 7, \(f(D)\) is an \"S-shaped\" curve with slow growth in the early stage and rapid growth in the later stage. The permeability evolution of rock throughout the compression process can be described based on the relationship between \(f(D)\) and the effective mean stress. For example, when the growth rate of \(f(D)\) is lower than that of the effective mean stress, the rock is in the compression stage and the permeability decreases; when the growth rate of \(f(D)\) is equal to that of the effective mean stress, the permeability and volume of the rock reach the minimum; and when the growth rate of \(f(D)\) is higher than that of the effective mean stress, the rock is in the dilatancy stage, and the permeability increases. Therefore, the derived evolution equation of damage-permeability conforms to the experimental results and phenomena and is reasonable and reliable. Figure 7: The curve of \(f(D)\) with the damage variable. The coupled seepage equation can be derived by introducing the evolution equation of porosity and permeability Equations (5) and (11) into the continuity equation of fluid seepage in porous media:  
\[\frac{\partial p}{\partial t}\cdot\left\{\frac{2\phi\cdot(\alpha_{b}-1)+1}{K} \right\}_{\varepsilon_{V}}+\frac{\partial\varepsilon_{V}}{\partial t}=\
abla \cdot\left\{\left(\frac{k}{\mu}\right)_{\varepsilon_{V}}\
abla p\right\} \tag{16}\]  
\[ \begin{cases}\frac{\frac{\partial\tau_{V}^{*}}{\partial x}+\frac{\partial \sigma_{V}^{*}}{\partial y}+\frac{\partial\tau_{V}^{*}}{\partial z}+F_{x}- \alpha_{b}\frac{\partial p}{\partial x}=0\\ \frac{\partial\tau_{V}^{*}}{\partial x}+\frac{\partial\tau_{V}^{*}}{\partial y} +\frac{\partial\tau_{V}^{*}}{\partial z}+F_{y}-\alpha_{b}\frac{\partial p}{ \partial y}=0\\ \frac{\partial\tau_{V}^{*}}{\partial x}+\frac{\partial\tau_{V}^{*}}{\partial y }+\frac{\partial\tau_{V}^{*}}{\partial z}+F_{z}-\alpha_{b}\frac{\partial p}{ \partial z}=0\end{cases}