Study on dynamic characteristics of hydraulic pumping unit on offshore platform

A new technology of offshore oil rod pumping production is developed for offshore heavy oil recovery. A new type of miniature hydraulic pumping unit with long-stroke, low pumping speed and compact structure is designed based on the spatial characteristics of offshore platforms. By combining the strengths of sinusoidal velocity curve and trapezoidal velocity curve, a kinematical model of the acceleration, the velocity and displacement of the pumping unit’s hanging point is established. The results show that the pumping unit has good kinematic characteristics of smooth motion and small dynamic load. The multi-degree-of-freedom dynamic model of the single-well pumping unit is established. The first and second order natural frequencies of the sucker rod string subsystem and the pumping unit subsystem are studied. The results show that the first and the second order natural frequencies among the pumping rod string, pumping unit-platform subsystem and the dynamic excitation have differences over 5 times from each other, indicating that resonance phenomenon will not appear during the operation and the dynamic requirements for field use are met in the system.


Introduction
25% of the proved reserves and 45% of the ultimate recoverable reserves on earth are ascertained from the ocean. The future focus of the world's oil production will gradually shift from onshore to offshore (Terdre, 2011;Shariatinia et al., 2013;Dev et al., 2016). The main producing areas of Chinese offshore oilfield have transited from the northern oil fields of the South China Sea in the last century to the Bohai oilfield nowadays (Guo et al., 2010). Currently, proved oil reserves with viscosity above 350 MPa·s in the Bohai Bay region are over 7.41×10 8 t and un-producing proved reserves reach up to 6.5×10 8 t, which accounts for 87.3% of the total . Therefore, research on the high-efficient development of offshore heavy oil occupies a pivotal position in the reserve discovery and recovery, productivity construction and oilfield production in the Bohai Bay region.
Steam stimulation, of easy operation, quick results, wide application to various types of heavy oil reservoirs, is the current main technology for heavy oil production. Owing to the limitation of the platform space, electric submersible pump is the main lifting equipment of offshore oil recovery (Dong et al., 2008). The pump units and the cables have low temperature resistance, which will influence the implantation temperature and dryness of the steam, thereby affecting the stimulation effect. Thus, a new method of artificial lifting is urgently needed to meet the requirement of high-efficient development of offshore heavy oil . The structure and performance of the platform pumping unit are the key points in the offshore rod pumping production system. The structure of the pumping unit is mainly described in this paper. The kinematic and dynamic performances of the pumping unit are emphatically studied.
2 Design of the hydraulic pumping unit for the offshore platform The target offshore platform is located in LD27-2 oilfield in the eastern Bohai Sea. The platform is divided into five decks. The height of the bottom deck, middle deck and top deck is 5 m, and the distance between the platform wellheads is 1.8 m×2.0 m. The work area of the offshore pumping unit is shown in Fig. 1.
Because of the wellhead space constraints and work over operations' requirement, the new hydraulic pumping unit on the offshore platform must be of compact structure and small occupied area. The current design of the small land-use hydraulic pumping unit is shown in Fig. 2a. The use of the two-stage telescopic hydraulic cylinder can reduce the height of the pumping unit with a single-stage hydraulic cylinder by half and the whole mass of the unit is around 800 kg. The unit can meet the need of 5-meter stroke on the offshore platform. The overall structure is shown in Fig. 2b.

Research on kinematics of hydraulic pumping unit on offshore platform
The speed curve of the pumping unit can be designed as a sinusoidal curve or trapezoidal curve. The trapezoidal velocity curve is divided into three stages: accelerating stage, uniform stage and decelerating stage. The accelerating and decelerating stages respectively account for 1/10 of the whole stroke, while the uniform stage takes the other 8/10 (Guan et al., 2012).
Set the sinusoidal harmonic motion equation in Fig. 3 as: v = Asin(θt).
(1) v max = 1.308 m/s v max = 0.926 m/s With Eq. (1), the maximal pumping unit's frequency is 5 min -1 and the stroke is 5 m, we can obtain A =1.308 and θ=π/6. Then, the maximal velocity of the hanging point in the sinusoidal velocity curve is . For the trapezoidal velocity curve , 71% of the former. To reduce the flow rate of the hydraulic system and the installed power, the hanging point speed should be designated to follow the trapezoidal curve mode. However, the acceleration curve corresponding to the trapezoidal velocity curve is discontinuous, resulting in cyclical impact load on the pumping unit during reversing, then vibration, shortening the operating life of the pumping unit.

Velocity and acceleration analysis of the hanging point
Combining the strengths of the sinusoidal and trapezoidal curve, we set the hanging point speed curve of the pumping unit as shown in Fig. 4.
Analytical expression of the curve: where, T is the period of the pumping unit (s); n is natural number; is the velocity coefficient.

Solution of the hanging point kinematic characteristics
With the pumping unit's frequency of 5 min -1 (i.e. T=12 s), and stroke L=5 m, we have: (3) With the simultaneous Eqs. (2) and (3), the solution: , . The acceleration equation of the pumping unit is: The hanging point of the pumping unit has a continuous acceleration curve, which effectively reduces the impact of the load on the pumping unit and improves its stability. The acceleration period is 6 s and the maximum acceleration is 1.603 m/s 2 . The acceleration value is small, indicating that the pumping unit runs smoothly with small dynamic load. The pumping speed period is 12 s and the maximum velocity is 0.962 m/s, which is 73.5% of the maximum velocity in sinusoidal curve as shown in Fig. 3.

Dynamics study on offshore platform hydraulic pumping unit
The Structure of offshore platform-hydraulic pumping unit-sucker rod string system is shown in Fig. 5.
The oil recovery system shown in Fig. 5 can be divided into two subsystems: the sucker rod string system and the pumping unit-platform support system.

Dynamic analysis of sucker rod string system
In the pumping unit prediction models, the method of solving the wave equation of the sucker rod string is mostly used, which has a huge work load. The mechanical system of mass-spring-damping as the dynamic model to describe the longitudinal vibration of the sucker rod is the trend (Li, X.Y. et al., 2016;Zhang, 2007). The structure of the sucker rod string is shown in Fig. 6.
The oil sucker rod strings are separated into n mass-concentrated and spring components, and the corresponding strings can form n mass-spring-damping sub-systems.

Dynamic model of sucker rod system
Target well LD27-2-A22H is with the vertical depth of 1180 m, primary sucker rod diameter of 19 mm, vertical depth of 420 m; secondary sucker rod diameter of 22 mm, vertical depth of 760 m. The simplest kinetic model for the sucker rod string can apply the two-degree-of-freedom system simulation; and the entire rod string system can be analyzed with three-degree-of-freedom. The dynamic model is shown in Fig. 7.
In Fig. 7, CHANG Zong-yu et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 693-699 displacement vector of the mass of each subsystem; F o (t) is the output of the system. According to the force of m 1 , m 2 , m 3 , using Newton's second law, we obtain the differential equations of motion: Compared with the mass of the sucker rod string and li-quid column, the mass of the pump is relatively small. The fixed tubing pump is used in the system, and the plunger can be considered as a part of the bottom sucker rod string.
The system in Fig. 7 can be further simplified into a two-degree-of-freedom system. Finishing Eq. (5) to obtain the dynamic equation of sucker rod string as: The concentrated mass of the rod string is: where D r is the diameter of the sucker rod (m); D o is the inner diameter of tubing (m); l i is the equivalent length of the sucker rod (m); ρ r is the density of the sucker rod (kg/m 3 ); ρ o is the density of the oil (kg/m 3 ).     Equivalent stiffness: where, E is the elastic modulus of the rod material, 210 GPa; A i is the sucker rod cross-sectional area (m 2 ). Equivalent damping (Zhan and Wang, 1984): μ where is the average viscosity of the liquid (MPa·s); ρ r is the density of sucker rod (kg/m 3 ); A r is the cross-sectional area of sucker rod, (m 2 ); a is the propagation velocity of stress wave (m/s); ω is the angular velocity converted into the crank (rad/s); L is the length of sucker rod (m).

Natural frequency of the rod string system
The characteristic matrix equation of the sucker rod string system of Eq. (6) is: = 0 (10) By solving Eq. (10), the first-and second-order natural frequencies of the system are: The concentrated mass m 1 =6.691×10 3 kg, the equivalent stiffness k 1 =1.050×10 5 N/m; the concentrated mass m 2 =3.407×10 3 kg, and the equivalent stiffness k 2 =1.418×10 5 N/m. From Eq. (11) respectively, we have the first-order natural frequency , and the second-order frequency .
4.2 Dynamic analysis of the single unit-partial platform system of pumping unit In consideration of the influence of a single pumping unit on the platform, the platform support is reduced to a mass beam. The physical model of the pumping unit and the partial platform can be simplified as shown in Fig. 8.
The platform support is simplified to the combination of the concentrated mass and massless beam. The corresponding kinetic model is shown in Fig. 9.
In Fig. 9, F(t) is the input load of hanging point of the pumping unit (N); l is the length of supporting beam for the platform (m); a is the installment position of the pumping unit (m); m 1 and m 2 are the mass of the pumping unit and the supporting beam, respectively (kg); k 1 and k 2 are the equivalent stiffness of the pumping unit and the support beam of the platform, respectively (N/m); y 1 and y 2 are the mass center displacement of the pumping unit and the supporting beam (static equilibrium point as the origin, m), respectively.

Dynamic equations of single unit-partial platform system of pumping unit
The system of Fig. 9 belongs to the two-degree-of-freedom undamped forced vibration system. According to the force of the mass point m 1 and m 2 , Newton's second law is adopted, then: where δ 21 is the dynamic displacement of the concentrated mass converted from Position 2 to Position 1. Using structural mechanics theory (Qu, 2010): (13) Differential equations of motion for the system can be expressed as: To solve the natural frequency of the system is the same as finding the solution for homogeneous Eq. (14), the characteristic matrix equation is:  CHANG Zong-yu et al. China Ocean Eng., 2017, Vol. 31, No. 6, P. 693-699 Therefore, the natural frequency of the system is: 4.2.2 Natural frequency of the single unit-partial platform system of the pumping unit The supporting structure of the pumping unit is simplified as the series spring system shown in Fig. 10 to solve the equivalent stiffness of the pumping unit support. The force segment of the support is divided into Segments 1, 2 and 3. Set the stiffness respectively as k z1 , k z2 , k z3 , and the base stiffness as k z4 .
Based on Fig. 10, the equivalent stiffness of the support of the pumping unit satisfies the following formula: The equivalent stiffness is: Under the load of W=10 t, the static deflections of each equivalent section are respectively δ z1 = 0.979 -3 m; δ z5 = The equivalent stiffness of each segment is:  (18), then the equivalent stiffness of the pumping unit support . The piston rod dimensions of the two-stage telescopic cylinder of the pumping unit are shown in Fig. 11.
Stiffness of each section for the piston: where E is the piston rod material elastic modulus, 210 GPa; A i is the piston rod cross-sectional area (m 2 ); l i is the length of piston rod (m).
Therefore, the equivalent spring stiffness of the piston rod for the pumping unit is: According to the structural parameters of Fig. 11, the equivalent stiffness of the piston rod for the pumping unit is The supporting system and piston rod system in series constitute the entire spring system of the pumping unit, whose equivalent stiffness is: Two installation positions of the single-well pumping unit for the target platform are shown in Fig. 12.
Under the load W=10 t, the maximum static deflection of the support beam is when the pumping unit is installed at Position 1 or Position 4 of the target platform; the maximum static deflection is δ p2 = 10 -3 m for Positions 2 or 3.
With Fig. 9 and Fig. 12, the equivalent stiffness of the platform support is: Based on Eq. (23), the equivalent stiffness of the platform beam is k 2 =1.703 N/m when the unit is installed at Position 1 or Position 4; The effective stiffness is k 2 = 0.974 N/m for Position 2 or Position 3.
With the total unit mass m 1 =1.266 kg, equivalent stiffness k 1 =8.526 N/m, the mass of the simplified beam m 2 =3.248 kg, the equivalent stiffness of the beam k 2 =1.703 N/m (Position 1 or 4), k 2 =0.974 N/m (Position 2 or 3). Based on Eq. (16), for Position 1 or Position 4, the first order natural frequency of the single pumping unit is 55.306 s -1 , the second order natural frequency is 107.443 s -1 ; for Positions 2 and 3, the first order natural frequency of the single pumping unit is 44.024 s -1 and the second order natural frequency is 102.078 s -1 .

Study of the hydraulic cylinder excitation impact on the system
The pumping unit is connected with the sucker rod string through the hydraulic cylinder. The working force of the hydraulic cylinder is the excitation source of the above two subsystems. In accordance with the requirements of the produced fluids of the oil well, the pumping speed is 2 to 5 times per min, i.e. the excitation frequency is 0.033 to 0.083 s -1 . The relationship between the subsystems and excitation frequency is shown in Table 1.
From Table 1, the first and second order natural frequencies of the two subsystems and the excitation frequency differ by more than 5 times, which means that there will be no resonance phenomenon in the rod production system of the offshore platform during the working process.

Conclusion
A new technology of offshore oil rod pumping production is developed for the thermal recovery of offshore heavy oil, and a small, two-stage telescopic hydraulic pumping unit for the offshore platform is studied. The kinetical model of the hanging point for the pumping unit is established. The maximal acceleration of the pumping unit is 5 min -1 , the stroke is 5 m, the maximum velocity of suspension is 0.962 m/s, the maximum acceleration is 1.603 m/s 2 , and the acceleration curve is continuous with small extreme value, indicating that the pumping unit movement is smooth, of small dynamic load. The multi-body dynamics model of the sucker rod string, the pumping unit and platform is established to study the first-and second-order natural frequencies of the two subsystems of the pumping unit-platform and sucker rod string. The results show that the first and the second order natural frequencies for the sucker rod string, pumping unit-platform subsystem and the dynamic excitation frequency have differences over 5 times from each other, indicating no resonance phenomenon in the rod pumping and little dynamic impact on the platform from the unit.