# Analysis of Cement Sheath-Rock Damage Mechanism--A Case Study on Water Injection Wells
## 2 Cement-Rock Bond Interface Damage Model  
### Stress Analysis of the Bond Interface Rocks  
The cement-rock bond interface can be effectively represented as a transitional layer between the rock and cement sheath. To explore the damage mechanism of the bonded interface rocks, a simplified model was utilized. In this model, the rock, cement sheath, and bond interface rock portion were considered as homogeneous media with uniform properties, as depicted in Figure 1. Additionally, it is assumed that each medium satisfies the Mohr-Coulomb strength criterion and is not influenced by the surface shape of the bond interface rocks. When the stress on the interface between the cement and rock shifts, it not only affects the interface strength but also causes modifications to the bonding strength between the cement-rock interface and the cement sheath. The stress and strain formulas for the cement-rock interface can be represented as (1) and (2):  
\[\{\sigma\}=\big{[}\sigma_{\mathrm{x}},\sigma_{\mathrm{y}},\tau\big{]}^{ \mathrm{T}} \tag{1}\]  
\[\{\varepsilon\}=\big{[}\varepsilon_{x},\varepsilon_{y},\gamma\big{]}^{ \mathrm{T}} \tag{2}\]  
where \(\sigma_{\mathrm{x}}\) and \(\sigma_{\mathrm{y}}\) represent the stress components in the x and y directions, respectively, while \(\tau\) denotes the shear stress component. The variables \(\varepsilon_{\mathrm{x}}\) and \(\varepsilon_{y}\) denote the strain components in the x and y directions, respectively, and \(\gamma\) represents the shear strain component. Figure 1: Schematic diagram of the composite rock specimen. When damage occurs along the cement-rock interface, the total shear stress on the interface can be expressed as the sum of the shear stress and bond strength, as indicated by Equation (3):  
\[\tau=\tau_{\text{c}}+\tau_{\text{f}} \tag{3}\]  
\[\tau_{\text{c}}=\sigma_{\text{n}}\text{tan}\;\theta+\text{C}_{\text{j}} \tag{4}\]  
where \(\tau_{\text{c}}\) represents the shear stress on the bond interface rocks, \(\tau_{\text{f}}\) represents the bonding force on the bond interface rocks, \(\sigma_{\text{n}}\) represents the normal stress on the bond interface, \(\theta\) represents the internal friction angle of the bond interface rocks, and \(\text{C}_{\text{j}}\) represents the cohesion of the bond interface rocks. Equation (5), proposed in previous studies, establishes a relationship between shear strength and microhardness based on strain gradient plasticity [25]. However, this formula does not take into account the bonding quality between concrete and rock or the adhesive forces present at the interface. \[\tau_{\text{f}}=\frac{\sqrt{3}\iint\text{hvds}}{9\text{A}} \tag{5}\]  
where hv represents the failure strength of the bonding points, A represents the projected surface area covered by the rock-cement sheath combination on the plane, and s represents the contact area between the rock and cement. To accurately describe the influence of bond strength, the cement-rock interface is divided into Ni individual cells with N bonding points. The bond strength between these locations signifies the strength at which the cement-rock interface fails. When the cement-rock interface undergoes shear and experiences cracking, the bond strength is influenced by the diffusion and solidification processes occurring on the surfaces of both the rock and the cement sheath. Hence, the formula for bond strength can be expressed as Equation (6). \[\tau_{\text{f}}=\frac{\text{N}\int_{0}^{\text{n}}(\text{hv}+\tau_{\text{cf}}) \text{ds}}{\text{N}_{\text{i}}\text{B}}(1+\text{k}1/\text{k}2) \tag{6}\]  
where k1 and k2 are the normal permeabilities of the rock and the cement-rock interface, and B is an experimental coefficient.