## 2 Materials and Methods
### Method of Energy Dissipation and Sealing Performance Analysis of the Premium Connection
#### 2.4.2 Sealing Performance Analysis Method  
To quantitatively assess the sealing performance of the premium connection, the contact pressure over the effective sealing length of the seal surface is calculated and integrated. The ability of the metal-to-metal sealing structure to prevent fluid flow, based on this integral result, is referred to as the sealing performance evaluation index, \(W_{a}\), defined by Equation (13) [30].  
\[W_{a}=\int_{L_{yx}}p_{c}^{x}(L)dL \tag{13}\]  
Figure 9: Sealing surface force displacement hysteresis curve.  
In the equation:  
\(W_{a}\)--The sealing performance evaluation indexmm\(\cdot\)MPa;  
\(L_{yx}\)--Effective sealing length, mm;  
\(p_{c}\)--Normal contact pressure distribution on the sealing surface, MPa;  
\(x\)--Sealing parameters.  
[PERSON] determined the sealing parameter \(x\) in Equation (13) through experimental methods. When thread lubricant is present in the metal-metal sealing structure, the sealing parameter \(x\) is 1.2; when thread lubricant is absent, the sealing parameter \(x\) is 1.4, as detailed in Equation (14) [31]:  
\[W_{a}=\left\{\begin{array}{c}\int_{L_{yx}}p_{c}^{1.2}(L)dL\;(\text{The presence of thread sealant, mm}\cdot\text{MPa}^{1.2})\\ \int_{L_{yx}}p_{c}^{1.4}(L)dL\;(\text{The absence of thread sealant, mm}\cdot\text{MPa}^{1.4})\end{array}\right. \tag{14}\]  
In the premium connection, the metal-metal sealing structure includes two types: the cone-to-cone sealing surface and the torque shoulder structure. Therefore, Equation (15) is revised in this paper to the following:  
\[W_{a}=\int_{L_{sal}+L_{tor}}p_{c}^{x}(L)dL \tag{15}\]  
In the equation:  
\(L_{seal}\)--Effective sealing length of the sealing surface;  
\(L_{tor}\)--Effective sealing length of the torque shoulder.  
When the maximum leakage rate of the metal-to-metal sealing structure is 15 min/ 0.025 cm\({}^{3}\)[30], a critical sealing index, denoted as \(W_{ac}\), is proposed, as shown in Equation (16). By comparing the sealing performance evaluation index \(W_{a}\) with the critical sealing index \(W_{ac}\), the sealing performance can be assessed. When \(W_{a}<W_{ac}\), it indicates that leakage may occur, and preventive measures should be taken; when \(W_{a}>W_{ac}\), it indicates that the sealing surface is safe and there is no risk of leakage, ensuring a reliable sealing performance.  
\[W_{ac}=\left\{\begin{array}{c}1.843\times\left(\frac{p_{c}}{p_{a}}\right)^{1.177}(\text{The presence of thread sealant, mm}\cdot\text{MPa}^{1.2})\\ 103.6\times\left(\frac{p_{c}}{p_{a}}\right)^{0.838}(\text{The absence of thread sealant, mm}\cdot\text{MPa}^{1.4})\end{array}\right. \tag{16}\]  
In the equation:  
\(p_{g}\)--Sealing pressure, MPa;  
\(p_{a}\)--Standard atmospheric pressure, MPa, is typically taken as 0.1 MPa in normal conditions.  
Based on the analysis in Figure 10, it can be observed that under the same gas pressure, the leakage potential of the premium connection with thread sealant is significantly lower than that without thread sealant. This indicates the critical role of thread sealant in the gas sealing performance.