Effects of Make-Up Torque on the Sealability of Sphere-Type Premium Connection for Tubing and Casing Strings

: The present investigations on sealability evaluation for tubing and casing premium connections depend on the FEM with testing. This paper proposed a theoretical model to evaluate the sealability of a sphere-type premium connection based on make-up torque, which combines Hertz contact pressure and the von Mises yield criterion for calculating elastic–plastic contact pressure distribution on sealing interface and adopts the gas sealing criterion obtained from Murtagian’s experimental results for deducing gas sealing capacity. With the proposed model, the effects of additional make-up torque from the sealing interface on the sealing contact pressure distribution and key sealability parameters, including contact width, yield width, average contact pressure and gas sealing capacity, were analyzed and compared. The results show that additional make-up torque from the sealing interface closely inﬂuenced sealability parameters’ variation and gas sealing capacity. The gas sealing index based on the sealing contact energy theory should be recommended for sealability evaluation other than average contact pressure on the sealing interface. For improving gas sealability, make-up torque should be controlled accurately for ensuring enough average contact pressure and contact width but a proper yield width, and a lager sphere radius should be selected for reducing the risk of yield sticking.


Introduction
Premium connection has become one of the key technologies for wellbore integrity and safety in harsh serving wells, such as high-pressure high-temperature (HPHT) gas wells, thermal recovery wells and shale gas wells [1,2].Differently from the API casing and tubing connection that only depends on thread interference contact sealing and a high-quality thread compound filled in the leakage path between the pin and coupling threads to meet a lower requirement of gas sealability, premium connection adopts a special metal-to-metal seal structure to attain high gas leakage resistance and a torque shoulder to control make-up torque and provide additional sealability, and it also usually adopts American Petroleum Institute (API) buttress threads or modified buttress threads for high joint strength, shown in Figure 1.
The general metal-to-metal seal structures for a premium connection can be mainly divided into four types, namely cone-to-cone seal, sphere-to-cone seal, sphere-to-cylinder seal and sphere-to-sphere seal; the latter three are collectively referred to as sphere-type seals, shown in Figure 2. Compared with a cone-to-cone seal, a sphere-type seal is a typical non-coordinated contact static seal, which tends to have higher contact pressure distribution and also yield more easily on a sealing interface.A certain yield width on the sealing interface can be conducive to blocking the micro leakage channel and forming an The general metal-to-metal seal structures for a premium connection can be mainly divided into four types, namely cone-to-cone seal, sphere-to-cone seal, sphere-to-cylinder seal and sphere-to-sphere seal; the latter three are collectively referred to as sphere-type seals, shown in Figure 2. Compared with a cone-to-cone seal, a sphere-type seal is a typical non-coordinated contact static seal, which tends to have higher contact pressure distribution and also yield more easily on a sealing interface.A certain yield width on the sealing interface can be conducive to blocking the micro leakage channel and forming an excellent gas seal rapidly, while a relevant larger yield width can easily lead to yield bonding and damage on the sealing surface.Consequently, accurate and efficient acquirement of the elastic-plastic contact pressure distribution on the sealing interface gives great importance to sealability evaluation for sphere-type premium connections.
(a) (b) (c) Figure 2. Metal-to-metal seal structures of a sphere-type premium connection, the sphere-to-cone type is for (a), the sphere-to-cylinder type is for (b) and the sphere-to-sphere is for (c).
Up to now, the sealability of premium connections was investigated in many previous references, whose methods can mainly be divided into three categories.The first category is based on experimental testing.Payne and Schwind proposed an international standard for casing/tubing connection testing [3].Based on those testing procedures, a lot of leak-proof tests have been carried out.And new sealing structures have been developed for premium connections.Salzano et al. investigated the sealing mechanism and structural integrity of a premium connection under mechanical and pressure loads and exposed to extremely low temperature in the experiment [4].Hamilton   The general metal-to-metal seal structures for a premium connection can be mainly divided into four types, namely cone-to-cone seal, sphere-to-cone seal, sphere-to-cylinder seal and sphere-to-sphere seal; the latter three are collectively referred to as sphere-type seals, shown in Figure 2. Compared with a cone-to-cone seal, a sphere-type seal is a typical non-coordinated contact static seal, which tends to have higher contact pressure distribution and also yield more easily on a sealing interface.A certain yield width on the sealing interface can be conducive to blocking the micro leakage channel and forming an excellent gas seal rapidly, while a relevant larger yield width can easily lead to yield bonding and damage on the sealing surface.Consequently, accurate and efficient acquirement of the elastic-plastic contact pressure distribution on the sealing interface gives great importance to sealability evaluation for sphere-type premium connections.

et al. presented adopting
(a) (b) (c) Figure 2. Metal-to-metal seal structures of a sphere-type premium connection, the sphere-to-cone type is for (a), the sphere-to-cylinder type is for (b) and the sphere-to-sphere is for (c).
Up to now, the sealability of premium connections was investigated in many previous references, whose methods can mainly be divided into three categories.The first category is based on experimental testing.Payne and Schwind proposed an international standard for casing/tubing connection testing [3].Based on those testing procedures, a lot of leak-proof tests have been carried out.And new sealing structures have been developed for premium connections.Salzano et al. investigated the sealing mechanism and structural integrity of a premium connection under mechanical and pressure loads and exposed to extremely low temperature in the experiment [4].Hamilton et al. presented adopting ultrasonic techniques to accurately examine seal surface contact pressure in premium Figure 2. Metal-to-metal seal structures of a sphere-type premium connection, the sphere-to-cone type is for (a), the sphere-to-cylinder type is for (b) and the sphere-to-sphere is for (c).
Up to now, the sealability of premium connections was investigated in many previous references, whose methods can mainly be divided into three categories.The first category is based on experimental testing.Payne and Schwind proposed an international standard for casing/tubing connection testing [3].Based on those testing procedures, a lot of leak-proof tests have been carried out.And new sealing structures have been developed for premium connections.Salzano et al. investigated the sealing mechanism and structural integrity of a premium connection under mechanical and pressure loads and exposed to extremely low temperature in the experiment [4].Hamilton et al. presented adopting ultrasonic techniques to accurately examine seal surface contact pressure in premium connections [5].Ernens et al. adopted an experimental setup and a stochastic numerical sealing model to describe the mechanisms of sealing metal-to-metal seals [6].Keita et al. investigated the influence of grease on high-pressure gas tightness by metal-to-metal seals of premium connections [7].There is no doubt that implementing a full-scale experiment is the most direct and effective mothed for sealability evaluation of a premium connection; however, it is expensive and time-consuming.The second category is the theoretical method, which is on the basis of macroscopic or microscopic contact mechanics.The former gets the sealing contact pressure distribution first and then uses average sealing contact pressure or a sealing index integrated by sealing width and sealing pressure to evaluate the sealability [8], while the latter can calculate the gas leakage rate of a premium connection directly [9].Although the theoretical method is convenient for usage, the stringent assumptions of the elastic material and contact pressure make its application limited.The third method is the finite element method (FEM), which is an effective way to deal with material nonlinearity, contact nonlinearity and complex load conditions for premium connections and has been taken as the most popular and necessary tool for new product development [10][11][12][13].Chen et al. investigated the sealing mechanism of tubing and casing premium threaded connections under complex loads [14].Kim et al. analyzed the effects of stabbing flank angle, upper stabbing flank corner radius on the von Mises stress of premium connection and presented the design criteria [15].Zhang et al. adopted the viscoelastic finite element model to predict the relaxation of contact pressure on the premium connections' sealing surface versus time under different temperatures [16].Yu et al. have analyzed the effect of energy dissipation on the sealing surface of premium connections with a microslip shear layer model [17].Dou et al. have carried out a sealing ability simulation for a premium connection based on ISO 13679 CAL IV tests with FEM [18].Generally, a preset interference between the sealing interfaces is made to produce sealing contact pressure.However, the preset interference on the sealing interface in FEM models differs mostly from the actual produced interference because of improper operation during make-up torque, thus leading to inaccurate simulated results of sealing contact pressure.As a result, it is hard to evaluate the sealability exactly and design the sealing parameters optimally.Xu and Yang have theoretically investigated the effect of make-up torque on the joint strength of premium connections but not focused on its sealability [19].In fact, the sealing contact pressure distribution depends closely on the real additional make-up torque from the sealing interface, so we can calculate the sealing contact pressure on the basis of the real make-up toque curve.
In this paper, a theoretical model for evaluating the sealability of a sphere-type premium connection is firstly proposed based on the additional make-up torque from the sealing interface, which combines Hertz contact pressure and the von Mises yield criterion for calculating elastic-plastic contact pressure distribution on the sealing interface and adopts the gas sealing criterion obtained from Murtagian's experimental results [20] for deducing gas sealing capacity.Then, based on the proposed model, the effects of additional make-up torque from the sealing interface on the contact pressure distribution and key sealing parameters, including contact width, yield width, average contact pressure and gas sealing capacity for sphere-type premium connection are analyzed and also compared for three kinds of sphere-type premium connections.In addition, some measures for ensuring the gas sealability of sphere-type premium connections for tubing and casing strings are proposed; those results can provide a useful guidance for sealability evaluation, parameter design and make-up torque control for sphere-type premium connections.

Model Development
The sealing structure of a sphere-type premium connection belongs to a typical noncoordinated radial interference contact seal.According to the equivalent principle of contact mechanics [21], sphere-to-sphere contact can be regarded as sphere-of-equivalentradius-to-cone contact, and sphere-to-cylinder contact can also be taken as sphere-to-cone with zero taper.Consequently, sphere-to-cone contact was selected as an archetype to develop the sealability evaluation model for a sphere-type premium connection.Figure 3 shows the sealing formation process for a sphere-to-cone contact premium connection.An initial circular line contact lies between the sphere pin and the cone coupling when the sealing surface starts to contact at the point of O.With make-up torque, normal interference is produced between the sphere pin and cone coupling, thus leading to normal contact pressure of p sN (x) on the contact interface.When make-up torque operation completes, a circumferential sealing interface with a certain contact width of 2w s and yield width of 2w y can form to meet the required gas sealing capacity for the premium connection.Figure 3a,b show, respectively, the initial sealing contact and final sealing contact for a sphere-to-cone premium connection.
with zero taper.Consequently, sphere-to-cone contact was selected as an archet develop the sealability evaluation model for a sphere-type premium connection.Fi shows the sealing formation process for a sphere-to-cone contact premium connectio initial circular line contact lies between the sphere pin and the cone coupling wh sealing surface starts to contact at the point of O.With make-up torque, n interference is produced between the sphere pin and cone coupling, thus lead normal contact pressure of psN(x) on the contact interface.When make-up torque ope completes, a circumferential sealing interface with a certain contact width of 2ws and width of 2wy can form to meet the required gas sealing capacity for the pre connection.Figure 3a,b show, respectively, the initial sealing contact and final s contact for a sphere-to-cone premium connection.To evaluate the sealability of a premium connection, the first key is to acqui elastic-plastic contact pressure distribution psN(x) on the sealing interface and to g main sealability parameters, including average contact pressure, contact width and width on the sealing interface, as well as gas sealing capacity for the sphere-type pre connection.During practical making-up of torque for the premium connection, the c pressure on the sealing interface brings out additional make-up torque, which c easily accessed by the make-up torque curve.Consequently, it is more accurat convenient or calculating the psN(x) based on the real make-up torque data.

The Model of Additional Make-Up Torque from Sealing Interface for Sphere-Type Prem Connection
Figure 4 shows the typical curve between make-up torque and turns for pre connection.It can be easily found that the final make-up torque TD involves mainl parts, namely the leading torque TA, the thread interference torque Tti, the add make-up torque from sealing interface Tse and the additional make-up torque shoulder Tsh, as follows [22]: To evaluate the sealability of a premium connection, the first key is to acquire the elastic-plastic contact pressure distribution p sN (x) on the sealing interface and to get the main sealability parameters, including average contact pressure, contact width and yield width on the sealing interface, as well as gas sealing capacity for the sphere-type premium connection.During practical making-up of torque for the premium connection, the contact pressure on the sealing interface brings out additional make-up torque, which can be easily accessed by the make-up torque curve.Consequently, it is more accurate and convenient or calculating the p sN (x) based on the real make-up torque data.

The Model of Additional Make-Up Torque from Sealing Interface for Sphere-Type Premium Connection
Figure 4 shows the typical curve between make-up torque and turns for premium connection.It can be easily found that the final make-up torque T D involves mainly four parts, namely the leading torque T A , the thread interference torque T ti , the additional makeup torque from sealing interface T se and the additional make-up torque from shoulder T sh , as follows [22]: Additional make-up torque from sealing interface Tse can be relatively accurately calculated by the torque at the initial contact on sealing interface TB and that on shoulder TC.Additional make-up torque from sealing interface T se can be relatively accurately calculated by the torque at the initial contact on sealing interface T B and that on shoulder T C .
To acquire the elastic-plastic contact pressure distribution on a sealing interface, a model for additional make-up torque from the sealing interface should be established first.We regard the real sealing interface as a circular cone surface with its generatrix length of approximately 2w s , as in Figure 3b.Obviously, an additional axial pre-tightening force will generate on the sealing interface and lead to additional thread torque T set .Meanwhile, additional friction torque on sealing interface T sef will also be produced.Consequently, the total additional make-up torque from the sealing interface is the sum of the additional thread torque and additional friction torque.
The relevant structure parameters for a premium connection are shown in Figure 1.As cone taper t s and actual contact width 2w s are both very small in practice, sealing radius r s can be regarded as equal, and the additional axial pre-tightening force F se from the sealing interface is as follows: Further, according to the rotary shoulder connection with Farr's formula [23], the additional thread torque from the sealing interface can be calculated by: In Equation ( 5), the equivalent moment arm for thread friction torque is as follows: 2.1.2.Additional Friction Torque from Sealing Interface T sef Additional friction torque from the sealing interface can be easily obtained by the following integration: 2.1.3.Total Additional Make-Up Torque from Sealing Interface T se By combining Equation (3) to Equation ( 7), the total additional make-up torque from the sealing interface can be obtained as follows: In Equation ( 8), we let: and then the additional make-up torque from the sealing interface can be further written as: When the thread parameters, sealing radius, cone taper and friction coefficient for the premium connection are all constant, the coefficient of ζ is also unchangeable.Apparently, it can be seen from Equation ( 9) that T se is very directly relative to the elastic-plastic contact pressure distribution p sN (x).Consequently, when T se is known, p sN (x) can be calculated with Equation (9).According to the Hertz theory of contact mechanics [21], the elastic contact pressure on a sealing interface for a sphere-type premium connection with sphere radius R s and equivalent elastic modulus E* can be expressed as follows:

The Model of
For the plane strain or axisymmetric problems, the equivalent elastic modulus E* can be calculated by the following equation: For contact of sphere with radius R s1 to sphere with radius R s2 , the sphere radius equals equivalent sphere radius R se :

Initial Yield Condition on Sealing Interface
Based on Hertz contact pressure state and the von Mises yield criterion, Green deduced the critical half contact width w c s with initial yield on the sealing interface for a rigid plane loaded by an elastic-plastic cylinder, as follows [24]: In Equation ( 13), C 1 S y denotes yield strength of the sealing contact pair, and the coefficient of C 1 can be calculated with Poisson's ratio ν as follows: Obviously, for contact between different materials, C 1 S y equals to:

Elastic-Plastic Contact Pressure on Sealing Interface
For sphere-type premium connections, after the initial yield occurs at the maximum contact pressure point, the sealing interface turns into an elastic-plastic contact state along with the continued make-up operation.In the plastic contact zone, we assume that the yield width is 2w y and the contact pressure still equals that when initial yield occurs.In another elastic contact zone, the contact pressure equals to the Hertz contact pressure.By combing Equation (10) with Equation ( 13), the elastic-plastic contact pressure on the sealing interface can be described as follows [25]: By submitting Equation ( 16) into Equation ( 9), the additional make-up torque from the sealing interface can be further written as: Now we discuss Equation ( 17) in the following three cases: (a) When w y = 0 and w s < w c s , the contact between the sealing interface is completely elastic.On this occasion, the additional make-up torque from the sealing interface T e se is as follows: (b) When w y = 0 and w s = w c s , initial yield occurs on the sealing interface.On this occasion, the additional make-up torque from the sealing interface T c se is as follows: (c) When w y > 0, the contact between the sealing interface is under an elastic-plastic contact state.The outer boundary condition in the plastic contact zone can be obtained from Equation ( 16) as follows: Then, by submitting Equation (20) into Equation ( 17), we can get the additional make-up torque from sealing interface T ep se under this condition: It can be seen from Equation ( 21) that when relevant parameters are constant, T ep se can be expressed as a function of w y .Consequently, if T ep se is known, w y can be easily obtained by a trial method.Then, the half-contact width on the sealing interface w s can also be further calculated according to Equation (20).It can be concluded comprehensively that when the actual additional make-up torque from the sealing interface after make-up operation T se is known, if T se < T c se , then w y = 0; if T se = T c se , initial yield occurs; and if T se > T c se , w y can be determined by the following implicit function: By combining Equations ( 18)-( 22), the half-contact width on the sealing interface w s can be expressed as follows:

Average Contact Pressure on Sealing Interface
The average contact pressure on sealing interface p ave can be calculated from Equation ( 16): 2.4.Gas Sealing Capacity for Sphere-Type Premium Connection Base on Sealing Contact Energy Theory The sealing contact energy theory holds that the integral of macro-sealing stress multiplied by sealing length can represent the sealability of a metal-to-metal seal, which is called the gas sealing index.According to the research from Murtagian's paper [20], the gas sealing index W a for a premium connection in HPHT gas wells can be expressed as: Murtagian et al. also proposed that when the gas pressure needing to be sealed p g and the atmospheric pressure p a are both known, a premium connection has reliable gas sealability only when W a in Equation ( 25) exceeds a critical value W ac in Equation ( 26): However, this evaluation criterion is too conservative because the gas leakage rate considered for reliable sealing in Murtagian's experiment was 0.025 cm 3 /15 min, while the permitted limit gas leakage rate for a casing and tubing connection is 0.9 cm 3 /15 min in ISO 13679.Considering this fact and the existing failure cases of tubing premium connections in HPHT gas wells, Xie et al. further proposed a practical critical value of W ac [26]: When the sealing contact pressure distribution is known, by combining Equation ( 16), Equation (25) and Equation ( 27), the gas sealing capacity of a sphere-type premium connection in HPHT gas wells can be obtained as:

Results and Discussion
According to the previous theoretical analysis, the contact pressure distribution on the sealing interface for a sphere-type premium connection of a certain specification is directly related to the additional make-up torque from the sealing interface.Consequently, to provide theoretical guidance on make-up torque control and sealing parameter design to ensure gas sealability for a sphere-type premium connection in tubing and casing strings, by example analysis, the effects of additional make-up torque from the sealing interface on the contact pressure distribution and key sealing parameters, including contact width, yield width, average contact pressure and gas sealing capacity, were analyzed and also compared for three kinds of sphere-type premium connections.This paper takes P110 steel, 127 mm outer diameter casing with API buttress threads and different sphere-type premium connections as an example for analysis.The basic calculation parameters are shown in Table 1.

Contact Pressure Distribution on Sealing Interface
According to the previous theoretical analysis, the contact pressure distribution on the sealing interface for a sphere-type premium connection of a certain specification is directly related to the additional make-up torque from the sealing interface.Sphere-to-sphere contact can be regarded as a sphere-of-equivalent-radius-to-cone contact, and sphere-tocylinder contact can be also taken as the sphere-to-cone with zero taper.Consequently, we firstly took the sphere-to-cone seal with sphere radius of 100 mm and cone taper on diameter of 0.1 as an example to investigate the effects of additional make-up torque from the sealing interface on the contact pressure distribution on the sealing interface.According to Equation (19), we firstly calculated the additional make-up torque from the sealing interface when initial yield T c se was 1644 N•m.Then, we analyzed the contact pressure distribution when the actual additional make-up torque from the sealing interface T se was 1000 N•m, 1644 N•m and 3000 N•m, respectively.The results are shown in Figure 5. Figure 5 shows that the contact pressure on the sealing interface was symmetrically distributed with respect to the normal direction of the initial contact point and in a shape of a downward parabola or a parabolic cup.If Tse < T x-coordinate of sealing interface for sphere−type premium connection (mm) Figure 5 shows that the contact pressure on the sealing interface was symmetrically distributed with respect to the normal direction of the initial contact point and in a shape of a downward parabola or a parabolic cup.If T se < T c se = 1644 N•m, the sealing interface was under a complete elastic contact state; if T se = T c se = 1644 N•m, initial yield occurred on the sealing interface and the yield point was located at the initial contact point; if T se > T c se = 1644 N•m, the sealing interface was under an elastic-plastic contact state and yield width was 1.11 mm.The average contact pressure was, respectively, 278.5 MPa, 357.0 MPa and 379.8 MPa when T se was 1000 N•m, 1644 N•m and 3000 N•m.In general, with an increase of T se , the contact width, yield width and average contact pressure increased gradually.Therefore, sufficient T se should be attained after the make-up operation, ensuring excellent gas sealability for a premium connection.

Sealability Parameters for Sphere-Type Premium Connection
In this section, we also take the sphere-to-cone seal with sphere radius of 100 mm and cone taper on diameter of 0.1 as an example to investigate the effects of the additional make-up torque from the sealing interface on the contact width, yield width, average contact pressure on sealing interface and gas sealing capacity of a premium connection.

Contact Width and Yield Width on Sealing Interface
Contact width and yield width are both important parameters that reflect the gas sealability for a premium connection, and contact width should be as large as possible to increase the leakage resistance.For a sphere-type premium connection, the usual higher contact pressure may easily cause the sealing interface to yield and block off leak passage quickly, thus giving enough gas sealing capacity.However, the yield of the sealing surface also easily leads to sticking and stress relaxation, so the yield width of a spheretype connection should be controlled to be less than 1-2 mm in general.In this paper, contact width 2w s and yield width 2w y were both calculated and compared under different additional make-up torque from sealing interface T se , and the results are shown in Figure 6.It can be seen from Figure 6 that if T se < T c se = 1644 N•m, the sealing interface was under a complete elastic contact state and contact width showed parabolic increases with the increase of T se while the yield width was zero.If T se = T c se = 1644 N•m, initial yield occurred and the critical contact width was 2.45 mm.If T se > T c se = 1644 N•m, the sealing interface was under an elastic-plastic contact state.With the increase of T se , the contact width increased slower and slower, and it finally increased approximately linearly, while yield contact showed a parabolic increase and it gradually approached the contact width.In the example, to make the yield width less than 1 mm, the additional make-up torque from sealing interface T se should be controlled within 2821 N•m.

Average Contact Pressure on Sealing Interface and Gas Sealing Capacity of Premium Connection
The general sealing criterion for a metal-to-metal seal is that the sealing contact pressure should exceed the fluid pressure inside the string, which has also been taken as the early design criterion for tubing and casing connections.However, this criterion has been established based on the assumption of a complete smooth sealing surface and it does not conform to the actual rough contact and sealing.The sealing contact energy theory adopts the gas sealing index W a to reflect the leakage resistance and takes a critical gas sealing index W ac from laboratory testing to denote reliable sealing, which builds a bridge between macro-contact pressure and micro-leakage on the sealing interface.Consequently, the average contact pressure and gas sealing capacity were both calculated and compared under different additional make-up torque from sealing interface T se , and the results are shown in Figure 7.It can be seen from Figure 7 that in the complete elastic contact zone, the average contact pressure on sealing interface pave and gas sealing capacity for premium connection pgm both showed a parabolic increase with the increase of additional make-up torque from sealing interface Tse, although pgm was smaller than pave but gradually approached it and nearly equal to pave at the initial yield point.Later, the sealing interface turned into an elastic-plastic contact state.With the increase of Tse, pgm increased quickly and gradually exceeded pave.This was because the plastic deformation occurred and it blocked off the leakage passage quickly, which led to nearly invariable pave and larger and larger pgm.Consequently, the adoption of pave will overestimate the sealability of a sphere-type premium connection in a complete elastic contact zone while underestimating it in an elastic-plastic contact zone.Therefore, we recommend the sealing contact energy theory.

Sealability Parameters' Comparison for Three Kinds of Sphere-Type Premium Connections
To provide theoretical guidance on sealing parameters' design and make-up torque It can be seen from Figure 7 that in the complete elastic contact zone, the average contact pressure on sealing interface p ave and gas sealing capacity for premium connection p gm both showed a parabolic increase with the increase of additional make-up torque from sealing interface T se , although p gm was smaller than p ave but gradually approached it and nearly equal to p ave at the initial yield point.Later, the sealing interface turned into an elastic-plastic contact state.With the increase of T se , p gm increased quickly and gradually exceeded p ave .This was because the plastic deformation occurred and it blocked off the leakage passage quickly, which led to nearly invariable p ave and larger and larger p gm .Consequently, the adoption of p ave will overestimate the sealability of a sphere-type premium connection in a complete elastic contact zone while underestimating it in an elastic-plastic contact zone.Therefore, we recommend the sealing contact energy theory.

Sealability Parameters' Comparison for Three Kinds of Sphere-Type Premium Connections
To provide theoretical guidance on sealing parameters' design and make-up torque control for sphere-type premium connections, a comparison analysis for three types of premium connections were also conducted.Considering the different additional make-up torque from the sealing interface, the contact width, yield width, average contact pressure and gas sealing capacity for three types of premium connections, including sphere-to-cone seal, sphere-to-cylinder seal and sphere-to-sphere seal, were calculated, and the results are shown in Figures 8-11, respectively.The additional make-up torque from sealing interface T se (N• m) sphere to cylinder, R s =100 mm, t s =0 mm/mm sphere to cone, R s =100 mm, t s =0.1 mm/mm sphere to sphere, R s1 =R s2 =100 mm, t s =0.1 mm/mm The additional make-up torque from sealing interface T se (N•m) sphere to cylinder, R s =100 mm, t s =0 mm/mm sphere to cone, R s =100 mm, t s =0.1 mm/mm sphere to sphere, R s1 =R s2 =100 mm, t s =0.1 mm/mm  The additional make-up torque from sealing interface T se (N•m) sphere to cylinder, R s =100 mm, t s =0 mm/mm sphere to cone, R s =100 mm, t s =0.1 mm/mm sphere to sphere, R s1 =R s2 =100 mm, t s =0.1 mm/mm   It can be comprehensively seen from Figures 8-11 that the sealing capacity for a sphere-to-cylinder premium connection was slightly larger than that for a sphere-to-cone premium connection under the same additional make-up torque from sealing interface Tse.This was because a fraction of Tse was adopted by the thread for the sphere-to-cone premium connection.The value of Tse for a sphere-to-sphere premium connection when initial yield occurred was obviously lower than that of sphere-to-cone and sphere-tocylinder premium connections (see Figure 9).In the complete elastic zone, the gas sealing capacity of each sphere-type premium connection was nearly equal, and this was because The additional make-up torque from sealing interface T se (N• m)   It can be comprehensively seen from Figures 8-11 that the sealing capacity for a sphere-to-cylinder premium connection was slightly larger than that for a sphere-to-cone premium connection under the same additional make-up torque from sealing interface Tse.This was because a fraction of Tse was adopted by the thread for the sphere-to-cone premium connection.The value of Tse for a sphere-to-sphere premium connection when initial yield occurred was obviously lower than that of sphere-to-cone and sphere-tocylinder premium connections (see Figure 9).In the complete elastic zone, the gas sealing capacity of each sphere-type premium connection was nearly equal, and this was because The additional make-up torque from sealing interface T se (N• m) It can be comprehensively seen from Figures 8-11 that the sealing capacity for a sphereto-cylinder premium connection was slightly larger than that for a sphere-to-cone premium connection under the same additional make-up torque from sealing interface T se .This was because a fraction of T se was adopted by the thread for the sphere-to-cone premium connection.The value of T se for a sphere-to-sphere premium connection when initial yield occurred was obviously lower than that of sphere-to-cone and sphere-to-cylinder premium connections (see Figure 9).In the complete elastic zone, the gas sealing capacity of each sphere-type premium connection was nearly equal, and this was because the sphere-tocone and sphere-to-cylinder premium connections had larger contact width and smaller average contact pressure while the sphere-to-sphere premium connection had smaller contact width and larger average contact pressure (see Figures 8 and 10), which produced fair sealing contact energy for the three types of premium connections.In the elastic-plastic zone, although the average contact pressures for three types of premium connections were relatively close to each other, the contact width for sphere-to-sphere was obviously smaller than that of sphere-to-cone and sphere-to-cylinder premium connections, thus leading to obvious lower gas sealing capacity for the sphere-to-sphere premium connection.Consequently, a proper larger sphere radius should be selected for a sphere-type premium connection, which can reduce the risk of yield sticking on the sealing interface and also improve the gas sealability of the connection.

Conclusions
(1) Differently from the widely used FEM method with testing, a theoretical model for evaluating the sealability of a sphere-type premium connection was proposed based on the additional make-up torque from the sealing interface, which combined Hertz contact pressure and the von Mises yield criterion for calculating elastic-plastic contact pressure distribution on the sealing interface and adopted the gas sealing criterion obtained from Murtagian's experimental results for deducing gas sealing capacity.(2) The sensitivity analysis on sealability parameters indicated that the additional makeup torque from sealing interface T se closely influenced sealability parameters' variation and gas sealing capacity:

•
The contact pressure on the sealing interface was symmetrically distributed with respect to the normal direction of the initial contact point and in a shape of a downward parabola or a parabolic cup.The corresponding T c se when initial yield occurred divided the contact state into complete elastic contact and elastic-plastic contact.

•
Under a complete elastic contact state, the contact width showed parabolic increases with increase of T se while the yield width was zero; under an elasticplastic contact state, with the increase of T se , the contact width increased slower and slower and it finally increased approximately linearly, while the yield contact showed a parabolic increase and it gradually approached the contact width.

•
Under a complete elastic contact state, average contact pressure on sealing interface and gas sealing capacity both showed a parabolic increase with the increase of T se , although the latter was smaller but gradually approaching the former and nearly equal to it at the initial yield point; under an elastic-plastic contact state, with the increase of T se , the gas sealing capacity increased quickly and exceeded average contact pressure on the sealing interface.

•
The gas sealing capacity of each sphere-type premium connection was nearly equal in the complete elastic zone, but that of sphere-to-sphere was obviously smaller than that of sphere-to-cone and sphere-to-cylinder premium connections in the elastic-plastic zone.
(3) For ensuring the gas sealability of a sphere-type premium connection for tubing and casing strings, the gas sealing index based on the sealing contact energy theory should be recommended for sealability evaluation alongside average contact pressure on the sealing interface.Make-up torque is also suggested to be controlled accurately to ensure enough average contact pressure and contact width but a proper yield width, and a lager sphere radius should be selected to reduce the risk of yield sticking.(4) This paper mainly adopted macroscopic contact mechanics combined with previous experimental results to establish the model, which could not accurately descript the actual microcosmic contact state between sealing interfaces and may influence the accuracy of the model to some extent, so the model should be improved by considering microcosmic contact mechanics on the sealing interface, and corresponding full-scale experimental testing should be carried out in later work.equivalent elastic modulus for sealing contact pair, MPa R s sphere radius, mm R se equivalent sphere radius for sphere-to-sphere contact, mm R s1 , R s2 two sphere radii, respectively, for sphere-to-sphere contact, mm E p , E c elastic modulus for sphere pin and cone coupling, respectively, MPa ν p , ν c Poisson's ratio for sphere pin and cone coupling, respectively, dimensionless ν Poisson's ratio of material, dimensionless C 1 yield strength coefficient of the softer material for sealing contact pair, dimensionless S y yield strength of the softer material for sealing contact pair, MPa C 1p , C 1c Yield strength coefficient of sphere pin and cone coupling, respectively, dimensionless S yp , S yc yield strength of sphere pin and cone coupling, respectively, MPa T e se additional make-up torque from sealing interface with complete elastic contact, N•m T c se additional make-up torque from sealing interface when initial yield occurs on the sealing interface, N•m T ep se additional make-up torque from sealing interface when sealing interface is under elastic-plastic contact state, N•m p ave average contact pressure on sealing interface, MPa W a gas sealing index for premium connection, mm•MPa 1.4 W ac critical gas sealing index, mm•MPa 1.4 p a atmospheric pressure, MPa p g gas pressure needing to be sealed, MPa p gm gas sealing capacity of sphere-type premium connection, MPa

Figure 1 .
Figure 1.The schematic diagram of a premium connection.

Figure 1 .
Figure 1.The schematic diagram of a premium connection.

Figure 1 .
Figure 1.The schematic diagram of a premium connection.

Figure 3 .
Figure 3. Schematic of sealing formation process for a sphere-to-cone contact premium conn initial sealing contact is for (a) and final sealing contact is for (b).

Figure 3 .
Figure 3. Schematic of sealing formation process for a sphere-to-cone contact premium connection, initial sealing contact is for (a) and final sealing contact is for (b).

Figure 4 .
Figure 4.The typical curve between make-up torque and turns for a premium connection.

Figure 4 .
Figure 4.The typical curve between make-up torque and turns for a premium connection.

Processes 2023, 11 , 256 11 of 19 Figure 5 .
Figure 5. Contact pressure distribution on sealing interfere for sphere-type premium connection under different additional make-up torque from sealing interface.
c se = 1644 N•m, the sealing interface was under a complete elastic contact state; if Tse = T c se = 1644 N•m, initial yield occurred on the sealing interface and the yield point was located at the initial contact point; if Tse > T c se -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1

Figure 5 .
Figure 5. Contact pressure distribution on sealing interfere for sphere-type premium connection under different additional make-up torque from sealing interface.

Processes 2023, 11 , 256 12 of 19 Figure 6 .Figure 6 .
Figure 6.Effects of additional make-up torque from sealing interface on contact width and yield width on sealing interface for sphere-type premium connection.It can be seen from Figure 6 that if Tse < T c se = 1644 N•m, the sealing interface was under a complete elastic contact state and contact width showed parabolic increases with the increase of Tse while the yield width was zero.If Tse = T c se = 1644 N•m, initial yield occurred and the critical contact width was 2.45 mm.If Tse > T c se = 1644 N•m, the sealing interface was under an elastic-plastic contact state.With the increase of Tse, the contact width increased slower and slower, and it finally increased approximately linearly, while yield contact showed a parabolic increase and it gradually approached the contact width.In the

Processes 2023, 11 , 256 13 of 19 Figure 7 .
Figure 7. Effects of additional make-up torque from sealing interface on average contact pressure on sealing interface and gas sealing capacity for sphere-type premium connection.

Figure 7 .
Figure 7. Effects of additional make-up torque from sealing interface on average contact pressure on sealing interface and gas sealing capacity for sphere-type premium connection.

Processes 2023, 11 , 256 14 of 19 Figure 8 .
Figure 8. Effects of additional make-up torque from sealing interface on contact width on sealing interface for three kinds of sphere-type premium connections.

Figure 9 .
Figure9.Effects of additional make-up torque from sealing interface on yield width on sealing interface for three kinds of sphere-type premium connections.

Figure 8 . 19 Figure 8 .
Figure 8. Effects of additional make-up torque from sealing interface on contact width on sealing interface for three kinds of sphere-type premium connections.

Figure 9 .
Figure9.Effects of additional make-up torque from sealing interface on yield width on sealing interface for three kinds of sphere-type premium connections.

Figure 9 .
Figure9.Effects of additional make-up torque from sealing interface on yield width on sealing interface for three kinds of sphere-type premium connections.

Figure 10 .
Figure10.Effects of additional make-up torque from sealing interface on average contact pressure on sealing interface for three kinds of sphere-type premium connections.

Figure 11 .
Figure11.Effects of additional make-up torque from sealing interface on gas sealing capacity for three kinds of sphere-type premium connections.

Figure 10 .
Figure10.Effects of additional make-up torque from sealing interface on average contact pressure on sealing interface for three kinds of sphere-type premium connections.

Figure 11 .
Figure11.Effects of additional make-up torque from sealing interface on gas sealing capacity for three kinds of sphere-type premium connections.
Elastic-Plastic Contact Pressure on Sealing Interface for Sphere-Type Premium Connection 2.2.1.Hertz Elastic Contact Pressure

Table 1 .
The basic parameters for sphere-type premium connection.
Effects of additional make-up torque from sealing interface on average contact pressure on sealing interface for three kinds of sphere-type premium connections.
Effects of additional make-up torque from sealing interface on gas sealing capacity for three kinds of sphere-type premium connections.
T A leading torque, N•m T B make-up torque value at the initial contact on sealing interface, N•m T C make-up torque value at the initial contact on shoulder interface, N•m T D final make-up torque for premium connection, N•m T ti thread interference torque, N•m T se additional make-up torque from sealing interface, N•m T sh additional make-up torque from shoulder, N•m T set additional thread torque from sealing interface, N•m T sef additional friction torque from sealing interface, N•m p sN normal contact pressure on sealing interface, MPa F se additional axial pre-tightening force from sealing interface, N