# Analysis of Cement Sheath-Rock Damage Mechanism--A Case Study on Water Injection Wells
## 2 Cement-Rock Bond Interface Damage Model
### Constitutive Model Considering Damage Parameters for Bonded Rock at the Bond Interface  
During the actual oilfield development process, when the rock material on the bonding surface undergoes damage, the trajectory of damage evolution gradually expands along the interface. This progressive process ultimately leads to the occurrence of the cement sheath's failure, triggering a chain reaction that results in casing damage, water channeling, and other well-related failures. It is important to note that throughout the entire damage process, the bonded surface rock is subjected to both tensile and shear stress-induced damage. In the bonded surface rock and reservoir rock, when the bonded surface rock encounters peak shear or tensile stress, the damage initiation occurs. The damage to the bonded surface rock can be categorized into two distinct aspects. Firstly, the damage caused by normal stress primarily relates to the tensile strength of the adhered rock at the bonding interface. Secondly, another small plane emerges along the tangential direction of the bonding surface. This component encompasses the collective impact of cleavage damage caused by the bonded surface, interwoven with the combined effect of shear stress. Taking into account the damage effect during the deformation of the bond interface, the stress-strain equation with damage parameters can be expressed as Equation (10):  
\[\sigma_{\mathrm{ij}}{}^{\prime}=(1-\mathrm{D})\sigma_{\mathrm{ij}}=C_{ \mathrm{ij}}:\epsilon_{\mathrm{ij}} \tag{10}\]  
The degradation constitutive relationship matrix of bonded rock at the bond interface, considering damage parameters can be expressed as Equation (11) [27]:  
\[C_{\mathrm{ij}}=\frac{1}{\mathrm{E}} \begin{bmatrix}\mathrm{d}_{1}&-\mathrm{d }_{2}\mu&0\\ -\mathrm{d}_{2}\mu&\mathrm{d}_{1}&0\\ 0&0&\mathrm{d}_{2}/\mathrm{G}\end{bmatrix}  \tag{11}\]  
where \(\mathrm{d}_{1}=(1-\mathrm{D1})\) and \(\mathrm{d}_{2}=(1-\mathrm{D2})\). \(\mathrm{D1}\) represents the damage value in the normal direction, while \(\mathrm{D2}\) represents the damage value in the tangential direction on the bonded rock at the bond interface. The shear cracking damage \(\mathrm{D21}\) of the bond interface can be expressed as Equation (12):  
\[\mathrm{D21}=1-\left(\frac{\delta_{t}}{\delta}\right)\left[1-\frac{1-\exp\! \left(-\alpha\mathrm{l}\!\left(\frac{\delta-\delta_{t}}{\delta_{c}-\delta_{t}} \right)\right)}{1-\exp\!\left(-\alpha\mathrm{l}\right)}\right] \tag{12}\]  
In the equation, The shear displacement corresponding to the peak shear stress can be denoted as \(\delta_{t}\), the displacement at the point of residual shear stress can be represented as \(\delta_{c}\), and a1 represents a material property, which is determined to be 5 based on the experimental study in reference [19]. The damage caused by the bonding force can be expressed using statistical damage parameters as Equation (13):  
\[\mathrm{D22}=1-\exp\!\left[-\!\left(\frac{\mathrm{F}}{\mathrm{F}\mathrm{0}} \right)^{\mathrm{m}}\right] \tag{13}\]  
where m and F0, respectively, represent the influence of the Weibull distribution function on the shape and size of rock microelements, and F represents the distribution variable of microelement strength. By coupling Equations (12) and (13), we can obtain Equation (14):  
\[\mathrm{D2}=\mathrm{D21}+\mathrm{D22}-\mathrm{D21}\times\mathrm{D22} \tag{14}\]  
According to porous media theory, the constitutive relationship of rock at the bonded interface follows the generalized Hooke's law.