## 1 Introduction
### Finite Element Model  
When calculating productivity, it is assumed that the fluid flow within the reservoir is isothermal and in a steady state, adhering to [PERSON]'s law. The flow equation can be expressed as follows:  
\[\frac{\partial}{\partial x}\left(K_{x}\frac{\partial p}{\partial x}\right)+ \frac{\partial}{\partial y}\left(K_{y}\frac{\partial p}{\partial y}\right)+ \frac{\partial}{\partial z}\left(K_{z}\frac{\partial p}{\partial z}\right)=0, \tag{3}\]  
where \(p\) is the fluid pressure, Pa; and \(K_{x}\), \(K_{y}\), and \(K_{z}\) are the permeabilities in the \(x\), \(y\), and \(z\) directions, respectively, m\({}^{2}\).  
Due to the heterogeneity of the formation, the size and shape of the perforation holes created by the same type of perforating gun may not be identical. Additionally, the fragments generated during perforation can obstruct the holes, resulting in an unevenly compacted zone around the holes. These uncertainties pose challenges in modeling perforated wells. To simplify the modeling process and facilitate subsequent analysis, the following simplifications and assumptions are made regarding the perforation holes: (1) Perforation holes created by the same type of perforating gun are assumed to be identical. (2) The geometric shape of the perforation holes is considered to be a regular cylinder. (3) All the holes are assumed to be unobstructed, and the thickness and compaction degree of the compacted zone around the holes are assumed to be uniform. (4) The shape of the drilling mud-contaminated zone is assumed to be a cylindrical annulus, uniformly surrounding the wellbore, and the degree of drilling mud contamination is also assumed to be uniform. Based on these assumptions, a 3D geometric model of a perforated well, encompassing the wellbore, perforation holes, compacted zone, and contamination zone, is established as depicted in Figure 1.  
Figure 1: Geometric model of a perforated well.  
The mesh partitioning of geometric models in the context of wellbore simulation requires careful consideration of both the computational accuracy and efficiency. In this study, a two-step approach is employed to achieve an optimal mesh refinement. Firstly, an adaptive mesh refinement method is utilized to refine the mesh around the wellbore region, taking into account the significant difference in size between the formation and the wellbore. Subsequently, a mesh size sensitivity analysis is conducted to determine the final mesh partitioning method, as illustrated in Figure 2. The maximum element size is set as follows: 0.005 m for the compacted zone around the perforation hole, 0.02 m for the contaminated zone around the wellbore, 0.05 m for the near-wellbore formation region, and 0.5 m for the peripheral formation region.  
To simulate the flow field of the reservoir during production from the perforated well, pressure boundary conditions are applied. Specifically, a pressure difference of 1 MPa is assumed, with 1 MPa and 2 MPa applied as pressure boundary conditions to the inner wall of the wellbore and the outer boundary of the reservoir, respectively. The streamline distribution around the wellbore obtained from the numerical simulation is shown in Figure 3. The results indicate that the reservoir fluid primarily enters the wellbore through the perforation holes. By integrating the flow rate at each perforation boundary, the productivity \(Q\) of the well can be determined. Additionally, by substituting the same parameters into Equation (2), the productivity of an ideal open-hole well without pollution and perforations can be calculated. Finally, Equation (1) is employed to calculate the productivity ratio of the perforated well. The accuracy of the aforementioned finite element numerical simulation methods was validated by comparing the productivity ratio data with reference [7] under the given parameters.  
Figure 3: Streamline distribution around the wellbore.  
Figure 2: Mesh grids.