## 2 Characterization of Damage Model and Hydro-Mechanical Properties Model
### Hydro-Mechanical Properties Model  
There is a certain correlation between the changes in volume porosity caused by rock expansion and the plastic deformation generated by the rock after entering the internal damage expansion and deformation failure stages. Worldwide, some researchers, such as [PERSON] [25], and [PERSON] [26], have proposed this viewpoint in this regard. Among them, [PERSON] and other scholars proposed the relationship equation between the change rate of expanded porosity and strain increment through their research on the mechanical model and numerical simulation of viscoelastic plastic materials in porous media. \[\mathrm{d}\phi_{dil}=\frac{m_{d}q\big{(}\phi_{dil}^{\mathrm{max}}-\phi_{dil} \big{)}\mathrm{d}\varepsilon_{p}}{p}, \tag{9}\]  
In Equation (9), \(\phi_{dil}^{\mathrm{max}}\) is the maximum threshold for expanding porosity, \(m_{d}\) is the parameter characterizing the rock expansion rate, and \(p\) and \(q\) are the average stress and deviatoric stress of the rock, respectively. It can be seen that in the linear elastic stage of the rock hydro-mechanical properties mechanical characteristics test, the rock does not exhibit the expansion phenomenon and plastic deformation characteristics and its expansion porosity is 0. When entering the stage of plastic deformation, namely internal damage propagation and deformation failure, [PERSON]'s law is no longer applicable to the law of rock seepage at present, the increment of expanded porosity shows a significant increase with the increase of plastic deformation increment and deviator stress. Based on the theory of porous continuous media, the instantaneous permeate flow rate inside rocks can be expressed as follows:  
\[\Delta V_{l}=\frac{A_{s}\Delta p\Delta t}{\mu_{l}L_{s}}K, \tag{10}\]  
In Equation (10), \(\Delta V_{l}\) is the volume of infiltration water in the rock sample at the instantaneous moment (m\({}^{3}\)); \(A_{s}\) is the cross-sectional area of the rock sample (m\({}^{2}\)); \(\Delta P\) is the difference of pore pressure applied at the upper and lower ends of the rock sample (Pa); \(\Delta t\) is the time interval (s); \(K\) is the permeability of rock (m\({}^{2}\)); \(\mu\) is the dynamic viscosity coefficient of water at 20 \({}^{\circ}\)C (Pa \(\cdot\) s), \(\mu\) = 1 \(\times\) 10\({}^{-3}\) Pa\(\cdot\)s; \(L_{\rm s}\) is the height of the rock sample (m). According to the study by [PERSON] et al. [27] on the creep test of rock hydro-mechanical property characteristics, the relationship equation between permeability and the volumetric strain of rocks under hydro-mechanical properties can be obtained. The volumetric strain can be used to characterize the instantaneous infiltration amount of water into the rock sample, as follows:  
\[K=\left\{ \begin{array}{ll}y=a_{1}\exp(-b_{1}\varepsilon_{v})+c_{1}&(t<t_{ A})\\ L&(t_{A}\leq t<t_{B}),\\ y=a_{2}\exp[b_{2}(\varepsilon_{vB}-\varepsilon_{v})]+c_{1}&(t\geq t_{B})\end{array} \right. \tag{11}\]  
In Equation (11), \(\varepsilon_{\rm vB}\) is the volumetric strain at the volume strain dilatancy point, and \(\varepsilon_{\rm v}\) is the volumetric strain. \(t_{B}\) is the time at the volume strain dilatancy point, and the permeability in the volume compression stage is expressed in the form of exponential function, while the permeability in the volume dilatancy stage is expressed in the form of a logarithmic function. The list of experimental parameters in reference [27] is shown in Table 1.  
The effect of permeable water on rocks is mainly reflected in the water pressure in the microcracks and pores within the rock. Therefore, for saturated rocks that were fully saturated with water in a vacuum before the test, the ratio of the instantaneous seepage flow rate to the instantaneous pore water volume can be used to reflect the influence of permeable water on the internal stress field changes of the rock. We define a coupled variable of hydro-mechanical properties \(\xi\) to characterize the influence of instantaneous infiltration of water on the rock stress field.