# Numerical Simulation of the Influence of the Baihetan Reservoir Impoundment on Regional Seismicity
## 2 Pore-Viscoelastic Method  
Considering the long-term use of the reservoir after construction and the viscosity of the crustal medium, it is necessary to construct fully coupled pore-viscoelasticity equations to study the impact of reservoir impoundment on strata. According to the instantaneous elastic effect caused by water storage and the corresponding viscous relaxation effect for a long time, we derived a pore-viscoelasticity constitutive equation describing the evolution of underground media with time from fluid-saturated elastic porous media equations [25]. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Fault** & **Dip** & **Strike** & **Dip Direction** & **Reference** \\ \hline F1 & ZMHF & 60\({}^{\circ}\)–70\({}^{\circ}\) & 330\({}^{\circ}\) & NE & [18] \\ F2 & DLSF & - & 330\({}^{\circ}\)–360\({}^{\circ}\) & E/W & [18] \\ F3 & LFF & - & 45\({}^{\circ}\) & NW & [19] \\ F4 & NHF & - & 45\({}^{\circ}\) & SE & - \\ F5 & PDHF & 75\({}^{\circ}\)–80\({}^{\circ}\) & 20\({}^{\circ}\)E & E & - \\ F6 & XJF1 & 50\({}^{\circ}\)–70\({}^{\circ}\) & 340\({}^{\circ}\)–345\({}^{\circ}\) & E/W & [20] \\ F7 & XJF2 & 50\({}^{\circ}\)–70\({}^{\circ}\) & 340\({}^{\circ}\)–345\({}^{\circ}\) & E/W & [20] \\ F8 & ZLF & 30\({}^{\circ}\)–50\({}^{\circ}\) & 30\({}^{\circ}\)–40\({}^{\circ}\) & NW & [21] \\ \hline \end{tabular}
\end{table} 
Table 1: Faults geometry near the Baihetan Reservoir. Therefore, we can better analyze the early elastic response and long-term relaxation effect of the crust. In [PERSON] and [PERSON]'s theory, the diffusion and the elastic constitutive equations are as follows:  
\[\sigma_{ij}=2G\epsilon_{ij}+\frac{v}{1+v}(\sigma_{kk}+3p)\delta_{ij}-\frac{3(v_{ u}-v)}{B(1+v)(1+v_{u})}p\delta_{ij} \tag{1}\]  
\[c\
abla^{2}p=\frac{\partial}{\partial t}\left(\sigma_{kk}+\frac{3}{B}p\right) \tag{2}\]  
\[c=\frac{1}{\kappa}\left[\frac{3(v_{u}-v)}{2 GB(1+v)(1+v_{u})}\right] \tag{3}\]  
where \(\sigma_{ij}\) denotes the stress tensor, \(\epsilon_{ij}\) denotes the strain tensor, G is the shear modulus, and \(c\) is the diffusivity related to Poisson's ratio \(\
u\), undrained Poisson's ratio \(\
u_{u}\), Skempton coefficient \(B\), and permeability \(\kappa\). Equation (2) reflects the diffusion of the pore pressure with time. The term \(2G\epsilon_{ij}+\frac{p}{1+v}(\sigma_{kk}+3p)\delta_{ij}\) in Equation (1) is the elastic stress caused by strain. We replaced it with a viscoelastic integral form constitutive equation based on the Boltzmann superposition principle (4), so we get the pore-viscoelastic constitutive equation with an integral form (5):  
\[\sigma_{ij}=2\int_{-\infty}^{t}g(t-\xi)\frac{\partial\epsilon_{ij}(\xi)}{ \partial\xi}d\xi+\left\{\int_{-\infty}^{t}\left[K(t-\xi)-\frac{2}{3}g(t-\xi) \right]\frac{\partial\epsilon_{kk}(\xi)}{\partial\xi}d\xi\right\}\delta_{ij} \tag{4}\]  
\[\sigma_{ij}=2\int_{-\infty}^{t}g(t-\xi)\frac{\partial\epsilon_{ij}(\xi)}{ \partial\xi}d\xi+\left\{\int_{-\infty}^{t}\left[K(t-\xi)-\frac{2}{3}g(t-\xi) \right]\frac{\partial\epsilon_{kk}(\xi)}{\partial\xi}d\xi\right\}\delta_{ij}+ \left[-\frac{3(v_{u}-v)}{B(1+v)(1+v_{u})}p\right]\delta_{ij} \tag{5}\]  
where \(\delta_{ij}\) is a Kronecker delta function and K is the constant of bulk modulus, and \(g(t)=Ge^{-\frac{G}{\eta}t}\) for maxwell viscoelastic medium, and \(g(0)=G\) when \(t=0\), in which \(G\) and \(\eta\) are the constants of shear modulus and viscosity, respectively.