# Study on Experimental and Constitutive Model of Granite Gneiss under Hydro-Mechanical Properties
## 4 Numerical Simulation  
### Proposal of Coupling Variables  
During the rock deformation and failure stage, there is relevance in the sample plastic deformation and the rock porosity changes. Entering the internal damage expansion, deformation, and failure stage, a relevance between the rock expansion caused volume porosity variations and the rock-generated plastic deformation. [PERSON] et al. established a correlation formula between plastic strain increment and expansion porosity increment [30,31]. An equation was proposed for the relation in the variation rate on expansion porosity and strain increment. This equation applies to porous media:  
\[\mathrm{d}\phi_{dil}=\frac{m_{d}q\big{(}\phi_{dil}^{\mathrm{max}}-\phi_{dil} \big{)}\mathrm{d}\mathrm{e}_{p}}{p}, \tag{5}\]  
In Equation (5), \(\phi_{dil}^{\mathrm{max}}\) is the expanding porosity maximum threshold, \(m_{d}\) is the rock expansion rate, \(q\) is the rock deviatoric stress, \(p\) is the average stress. Rock expansion porosity is 0 in the linear elastic stage. During the plastic deformation stage, the deviatoric stress and plastic deformation increments are positively correlated with the expansion porosity increment.  
In accordance with the porous media theory, the instantaneous water infiltrating volume permeate \(\Delta V_{l}\) (m\({}^{3}\)) inner the rock is obtained as the Formula (6):  
\[\Delta V_{l}=\frac{A_{s}\Delta p\Delta t}{\mu_{l}L_{s}}K, \tag{6}\]  
Figure 4: Triaxial compression tests permeability–volumetric strain curves: (**a**) section OB; (**b**) section BC.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Rock**} & \multicolumn{5}{c}{**Fitting Results**} \\ \cline{2-6}  & \(\mathbf{a_{1}}\) & \(\mathbf{b_{1}}\) & \(\mathbf{c_{1}}\) & \(\mathbf{a_{2}}\) & \(\mathbf{b_{2}}\) \\ \hline R1 & 0.912 & 2.592 & 0.389 & 2.142 & 1.645 \\ R2 & 2.082 & 6.452 & 1.165 & 9.331 & 1.423 \\ R3 & 0.692 & 5.268 & 0.077 & 0.618 & 0.870 \\ R4 & 0.991 & 4.599 & 0.499 & 1.532 & 1.253 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Permeability–volumetric strain curves fitting results.  
In line with the relevance equation between volume strain and permeability in the granite gneiss hydro-mechanical properties test, the volume strain can be used to characterize permeate volume immersed into the rock specimen at any instant, so Formula (6) can be rewritten as Formula (7):  
\[\Delta V_{l}=\left\{\begin{array}{ll}\frac{A_{s}\Delta p\Delta t}{\mu_{l}L_{s}} (a_{1}\exp(-b_{1}\varepsilon_{v})+c_{1})&(t<t_{B})\\ \frac{A_{s}\Delta p\Delta t}{\mu_{l}L_{s}}a_{2}\ln(\varepsilon_{v}^{\max}- \varepsilon_{v}+b_{2})&(t\geq t_{B})\end{array}\right., \tag{7}\]  
where \(t_{B}\) is the dilatancy point of volume strain and the permeability expressed in exponential function and logarithmic function in the volume compression and dilatancy stage.  
The permeable water affection is primarily reflected in water pressure on rock specimens' internal microcracks and pores. With regard to saturated rocks, the water influence on the rock stress field can be expressed by the ratio of permeability rate to pore water volume, defined as coupled variable \(\xi\) with the aim of demonstrating the impact on water instantaneous infiltration, as shown in Equation (8):  
\[\xi=\frac{\Delta V_{l}}{A_{s}L_{s}\phi}, \tag{8}\]  
Formula (8) is the coupling variable expression we define. The rock volume strain, plastic strain, and deviatoric stress changes directly affect the coupled variable. In the meanwhile, the coupled variable under these influences will, in turn, transform the volume strain, plastic strain, and other variables, as well as achieve its coupling characteristic.