# Study on the Mechanism of Stress Sensitivity Changes in Ultra-Deep Carbonate Reservoirs
## 4 Example Application
### Establishment of Dual-Porosity Model  
Considering the strong heterogeneity of the reservoir, the fluid flow behavior, and the current development status, we determine the parameters of the carbonate reservoir model (Table 5). The size of the established model is 1680 m \(\times\) 756 m \(\times\) 900 m with the grid dimensions of 15 m \(\times\) 2 m \(\times\) 10 m.  
For the equations in Table 5, CDARCY is the Darcy constant; K is the rock permeability; V is the matrix volume per rock unit; \(l\)x is the distance between fractures in the X direction; \(ly\) is the distance between fractures in the Y direction; and \(lz\) is the distance between fractures in the Z direction. Figure 6: Characteristics of the resistivity difference display. \begin{table}
\begin{tabular}{c c c c} \hline
**Model Attribute** & **Value** & **Model Attribute** & **Value** \\ \hline Matrix model grid & \(\text{X}=112\text{~{}XY}=378\text{~{}NZ}=90\) & Fracture model grid & \(\text{N}=112\text{~{}NY}=378\text{~{}NZ}=90\) \\ Model geometry & 160 m \(\star\) 756 m \(\star\) 900 m & Model grid step size & \(\text{DX}=15\text{~{}DY}=2\text{~{}DZ}=10\) \\ Mean matrix permeability & 37.613 mD & Mean fracture permeability & 152.818 mD \\ Mean matrix porosity & 18.92\% & Mean fracture porosity & 1.092\% \\ Middle depth of reservoir & 6743 m & Matrix fracture conductivity & \(\text{Tr}=\text{CDARCY}\ast\text{K}\ast\text{V}\ast\sigma\) \\ Reservoir pressure & 87.75 MPa & Matrix to fracture cross-flow coefficient & \(\sigma=4\{\frac{1}{l\text{x}^{2}}+\frac{1}{l\text{y}^{2}}+\frac{1}{l\text{z} }\}\) \\ \hline \end{tabular}
\end{table} 
Table 5: Parameters for the dual-porosity model in the simulator. Figure 7: Structural stress field-simulated boundary conditions. According to the mechanical parameters of rock samples (Table 6), [PERSON]'s modulus of the matrix rock is 53.41 GPa and Poisson's ratio is 0.201, and Young's modulus of the fracture is 5.041 GPa and Poisson's ratio of the fracture is 0.33. According to the stress deformation method of the matrix-fracture dual-media model, the change in core porosity and permeability loss caused by effective stress is  
\[\text{K}=K_{0}*e^{-0.125*(\frac{\Delta\text{P}}{\Delta\text{T}})} \tag{13}\]  
When the effective stress \(\Delta\)P is 0, K = K0, i.e., no stress sensitivity occurs, which is realistic. For rock samples in the same state, at this time with the increase in effective stress, the reservoir permeability loss occurs, and the smaller the Young's modulus, the greater the difference in stress sensitivity. Due to the presence of an underlying aquifer, which supplies the energy lost from the production process, the current model has adopted the use of the initiation of pressure gradients to limit this energy supply and maintain balance during the production process. This allows for the better control of fluid movement between adjacent layers through the creation of a minimum pressure difference between neighboring lower-pressure layers. By maintaining this pressure gradient within the threshold specified by Table 7, one can limit any lateral movements between the cells and help calibrate the minimum boundary pressure for initiating these pressure gradients within the simulated volume. This method could also be used to study the changes in production indicators as a result of various pressure gradients initiated between different zones. (The unit measured for the flow threshold pressure is \"bar\"). Iniliation of pressure gradient modelling for permeability velocity is as follows:  
\[\left\{ \begin{array}{ll}\text{V}=0&\frac{\Delta\text{P}}{\Delta\text{P}}<0\\ \text{V}=\frac{\text{K}}{\mu}(\frac{\Delta\text{P}}{L}-C)&\frac{\Delta\text{P} }{L}\geq 0\end{array}