# Scalability Analyses of Premium Connections Characterized by a Surface Fractal Function
## 3 Contact Analysis of Premium Connections
### Analysis of Influencing Factors  
#### 3.3.1 The Fractal Dimension  
Nephograms of the premium connections' Von Mises stress with different fractal dimensions \(D\) under the action of make-up torque are shown in Figure 9. When the fractal dimension \(D\) increases, there is a decrease in the maximum Von Mises stress at the sealing surface, which is accompanied by an increase in the area of plastic deformation. This is because the number of asperities on the fractal surface increases with the increase in the fractal dimension \(D\). The sealing surface becomes smoother, and the distribution of stress more uniform, resulting in a reduction in Von Mises stress. When the fractal dimension \(D\) increases, the volume and stiffness of a single asperity decrease. Under the same make-up torque, there is an increase in both the proportion and area of plastic deformation of the asperities on the fractal surface.  
The distribution of contact pressure with different values of the fractal dimension \(D\) on the sealing surface is shown in Figure 10. With an increase in the fractal dimension \(D\), the effective contact length remains constant at 9.08 \(\times\) 10\({}^{-3}\) m. The overall contact pressure on the sealing surface decreases. The distribution of the contact pressure becomes more uniform. According to Equation (17), with an increase in the fractal dimension, there is an augmentation in the actual contact area of the sealing surface. This causes a decrease in the contact pressure at the same pressure level, and the distribution of contact pressure becomes more uniform. It can be concluded from the above analysis that the larger the fractal dimension \(D\), the smaller the Von Mises stress on the sealing surface. The more uniform the distribution of contact pressure, the more favorable the sealing.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mesh Size (m)** & **Number of Element** & \
\begin{tabular}{c} **Maximum Contact** \\ **Pressure (Pa)** \\ \end{tabular} &
\begin{tabular}{c} **Change Rate of** \\ **Contact Pressure (\%)** \\ \end{tabular} \\ \hline \(6\times 10^{-6}\) & 4319 & \(9.75\times 10^{8}\) & \\ \(5\times 10^{-6}\) & 5451 & \(1.06\times 10^{9}\) & 8.72 \\ \(4\times 10^{-6}\) & 6902 & \(1.15\times 10^{9}\) & 8.49 \\ \(3\times 10^{-6}\) & 10,608 & \(1.20\times 10^{9}\) & 4.35 \\ \(2\times 10^{-6}\) & 21,584 & \(1.14\times 10^{9}\) & 3.33 \\ \(1\times 10^{-6}\) & 81,812 & \(1.21\times 10^{9}\) & 4.31 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mesh independence verification.  
Figure 9: Nephograms with different fractal dimensions. (**a**) \(D\) = 2.1; (**b**) \(D\) = 2.5; (**c**) \(D\) = 2.9.  
A premium connection should make up according to the torque and speed recommended by the manufacturer. If the recommended value is exceeded, the stress on the sealing surface will sharply increase, which can easily cause gluing, leading to sealing failure. As shown in Table 4, the maximum Von Mises stress and maximum contact pressure on the sealing surface significantly exceeded the ultimate strength of the material when the fractal dimension \(D\) was 2.1 and 2.5, and the sealing surface may have required gluing in order to avoid the sealing failing before the premium connection enters the well, which would interfere with the results in the next sections for the analysis of the load factor. Fractal dimensions \(D\) of 2.1 and 2.5 will not be considered below. Meanwhile, to reduce the computation time, according to the conclusion in Section 2.4, changes in the contact pressure tend to be stable when the fractal dimension \(D\) is greater than 2.7. In the following sections, the fractal dimension \(D\) of the fractal model of the connection is taken as 2.7.