# Analysis of Cement Sheath-Rock Damage Mechanism--A Case Study on Water Injection Wells
## 2 Cement-Rock Bond Interface Damage Model
### Darcy's Seepage Equation  
The Darcy velocity represents the relative velocity between the fluid and the rock skeleton particles. In the casing water channeling problem, only single-phase fluid is considered. According to [PERSON]'s law, its expression can be derived as Equation (16):  
\[\mathrm{v}=\frac{-\mathrm{K}}{\Phi\dot{\Phi}}(\
abla\mathrm{P_{w}}+\rho_{w} \mathrm{g}) \tag{16}\]  
where the permeability of the porous medium is denoted as \(\mathrm{K}\), the porosity is represented by \(\Phi\), the viscosity of the fluid is \(\mathrm{\theta}\), the fluid pressure is \(\mathrm{P_{w}}\), and the fluid density is \(\rho_{w}\).  
The continuity equation for single-phase fluid is used to describe the water channeling phenomenon outside the casing in wells and can be expressed as Equation (17):  
\[\frac{\partial(\rho_{w}\Phi)}{\partial\mathrm{t}}+\
abla\left(-\frac{\rho_{w} }{\theta_{w}}\
abla\mathrm{P}\right)+\rho_{w}\alpha\frac{\partial\varepsilon _{v}}{\partial\mathrm{t}}=\mathrm{Q_{m}} \tag{17}\]  
\[\frac{\partial(\rho_{w}\Phi)}{\partial\mathrm{t}}=\rho_{w}\mathrm{S}\frac{ \partial\mathrm{P_{w}}}{\partial\mathrm{t}} \tag{18}\]  
where \(\varepsilon_{v}\) is the volumetric strain, the variable \(\mathrm{t}\) represents time, \(\mathrm{u}\) represents fluid velocity, \(\mathrm{Q_{m}}\) represents the mass source term, and \(\mathrm{S}\) represents the storage coefficient.  
The relationship between the evolution of porosity in rock bonding interfaces and damage progression during stress-strain changes [28].  
\[\mathrm{\phi}=(\phi_{0}-\phi_{\mathrm{R}})\exp(\alpha_{\mathrm{t}}\sigma_{ \mathrm{m}})+\phi_{\mathrm{R}} \tag{19}\]  
In the equation, initial porosity is denoted as \(\phi_{0}\), porosity after damage is represented by \(\phi_{\mathrm{R}}\), the stress sensitivity coefficient is denoted as \(\alpha_{\mathrm{t}}\) with a value of 5.0 \(\times\) 10\({}^{-8}\)/Pa, and \(\sigma_{\mathrm{m}}\) is the effective mean stress.  
\[\sigma_{\mathrm{m}}=\frac{\sigma_{1}+\sigma_{2}+\sigma_{3}}{3} \tag{20}\]  
The evolution equation of permeability can be obtained through the porosity solution formula [21]:  
\[\mathrm{k}=\mathrm{k}_{0}(\phi_{0}/\phi_{\mathrm{R}})^{3}\exp(\alpha_{ \mathrm{t}}\mathrm{D}) \tag{21}\]  
In the equation, \(\mathrm{k}_{0}\) is the initial permeability and \(\mathrm{D}\) is the damage parameter of the rock bonding interface. The damage along the tangent direction of the bonding interface crack is denoted as \(\mathrm{D}\), according to the rock bonding interface damage law, and the permeability in the tangent and normal directions of the rock bonding interface is represented by parameters \(\mathrm{k}_{1}\) and \(\mathrm{k}_{2}\), respectively. Assuming the principal stress direction and the \(x\)-axis form an angle \(\beta\), Equation (22) provides a mathematical expression for calculating the permeability \(\mathrm{k}\) of the rock bonding interface [29]:  
\[\mathrm{k}=\begin{bmatrix}\mathrm{k}_{1}\cos^{2}\beta+\mathrm{k}_{2}\sin^{2} \beta&(\mathrm{k}_{1}-\mathrm{k}_{2})\mathrm{cos}\beta\sin\beta&0\\ (\mathrm{k}_{1}-\mathrm{k}_{2})\mathrm{cos}\beta\sin\beta&\mathrm{k}_{1}\cos^ {2}\beta+\mathrm{k}_{2}\sin^{2}\beta&0\\ 0&0&\mathrm{k}_{2}\end{bmatrix} \tag{22}\]where: \(\mathrm{k_{1}=k_{0}(\phi_{0}/\phi_{\mathrm{R}})^{3}\exp(\alpha_{\mathrm{t}}D_{1})}\), \(\mathrm{k_{2}=k_{0}(\phi_{0}/\phi_{\mathrm{R}})^{3}\exp(\alpha_{\mathrm{t}}D)}\).