# Scalability Analyses of Premium Connections Characterized by a Surface Fractal Function
## 1 Introduction  
In high-pressure, high-temperature (HPHT) wells with pressures greater than 7 \(\times\) 10\({}^{7}\) Pa and temperatures greater than 150 \({}^{\circ}\)C, premium connections are used to connect the tubing to several kilometers of string [1], as shown in Figure 1, forming a underground channel for oil and gas. The heavy downhole loads greatly threaten the sealability of premium connections. A premium connection, as shown in Figure 2, contains threads, sealing surfaces and a torque shoulder. The sealability of a premium connection is guaranteed by two metal sealing surfaces with an interference fit. The sealing surfaces consist of a large number of asperities with different diameters. When in contact, the actual contact area between the two sealing surfaces is significantly lower than the nominal contact area [2]. Additionally, the contact pressure on the surface also changes significantly, which affects the sealability of the premium connection. Previous studies on the sealability of premium connections have mostly been macroscopic, assuming that the sealing surface is smooth in analyzing the stress, contact pressure, contact area or contact length. Considering the roughness of premium connections, [PERSON] established a gas leakage rate model of a cone-cone premium connection, and investigated the impact of sealing surface roughness on the leakage rate of the premium connection [3]. [PERSON] and [PERSON] et al. established a two-dimensional (2D) finite element model (FEM) of premium connections with two sealing structures, and verified the accuracy of the simplified symmetry model through experiments [4; 5]. [PERSON] et al. carried out tests on a full-size premium connection and determined the optimal structural size according to the wear resistance, internal pressure strength and outer pressure resistance [6]. [PERSON] et al. established a three-dimensional (3D) FEM of premium connections [7; 8] that consider the structure of the sealing surface, downhole conditions, and loads, and obtained the stress, contact pressure, and contact length. In addition, [PERSON] suggested a method to assess the sealing performance via the sealing strength [9]. Through physical tests and numerical simulation methods, the sealing strength and critical sealing strength were derived via the functional relationship between sealability and contact pressure [10; 11; 12; 13]. Previous research has neglected the effect of changes in the real contact area and contact pressure due to elastic and plastic deformation of asperities on the sealability of the sealing surface upon contact. The metal sealing surface is composed of a great number of asperities of different sizes. The profile of the rough surface can be described using either a statistical model or a fractal model. The statistical model mainly depends on the statistical parameters of the rough surface. Additionally, the statistical parameters are affected by the instrument's resolution and the sampling length, so they cannot accurately characterize all the features of the rough surface. However, this problem can be solved using a fractal model. The rough surface described by the fractal model is characterized by continuity, scale independence and self-similarity; thus, it objectively represents the characteristics of the rough surface [14]. The concept of fractals was first proposed by [PERSON], who discovered fractal geometry,  
Figure 1: Schematic diagram of underground tubing string. Figure 2: Cone-cone premium connection. which applies to objects with natural forms, no specific proportions and sizes, and infinite detail. Since then, the Weierstrass-Mandelbrot fractal function (W-M function) has been widely used in the study of tribology [15]. [PERSON] and [PERSON] applied the W-M function to analyze the contact behavior of two rough surfaces, established Majumdar-Bhushan (M-B) fractal contact theory, and obtained expressions for the contact load and contact area using fractal parameters [16]. [PERSON] et al.