The RDD-ESPP lifting system consists of a downhole linear motor, an electrical submersible plunger pump, a surface control unit, and a power transmission cable, as shown in Fig. (1). The frequency of the alternating current is firstly adjusted by the surface control unit. Then, current is transferred to the downhole linear motor. The downhole linear motor makes a reciprocating movement due to the effect of electromagnetic induction, which drives the plunger directly connected to the mover, moving it up and down. Then, the formation fluid is continuously lifted to the surface after being pressurized by the pump [11, 12].

2.1.3. Surface Control Unit

Controlling the operating state of the downhole linear motor is the key technology that must be mastered for the smooth operation of the reciprocating movement of an electric submersible plunger pump. Such operation is realized by surface variable-frequency control technology. The three-phase alternating current is firstly rectified into single-phase direct current, which is then inverted into square-wave alternating current at the required frequency. The intelligent control technology and high-performance electronic components are integrated to realize different control functions, such as constant-speed stepping, intermittent power supply, repeated startup and shutdown operations, repeated direction changes, automatic restart after power cut-off, automatic alarm and protection.

Fig. (1). Schematic diagram of the reciprocating direct-drive electric submersible plunger pump lifting system.

Fig. (2). Structure of the downhole linear motor.

Fig. (3). Differences between a conventional plunger pump and an electric submersible plunger pump.

At present, the stroke of the downhole linear motor is usually fixed at 1.23 m and the pumping speed can be adjusted from 0.1 rpm to 8 rpm. Due to the special design of the downhole linear motor, the maximum pump depth and displacement are limited. The recommended practices as defined by the manufacturer are listed in Table 1.

When compared to conventional sucker rod pumping systems and electrical submersible pumping systems, there are some significant differences in the RDD-ESPP lifting system. Therefore, the mathematical models for the lifting design are also different. 1) Compared with conventional sucker rod pumping systems, the RDD-ESPP lifting system does not need a rod string to transfer power. Therefore, the lifting load model and pump efficiency model do not need to consider the effects of rod weight and its elastic stretching and shrinking. 2) Compared with conventional electrical submersible pumping systems, the temperature model only needs to consider the ther mal effects of the motor and cable. 3) The RDD-ESPP pumping system works using an intermittent power supply, meaning that the input power depends on the power supplied during the time required for the upstroke and downstroke.

To acquire the temperature distribution in the wellbore, the following assumptions are made. 1) Radial heat losses between the wellbore and the formation are considered and axial heat losses along the wellbore are not included. 2) A steady-state heat transfer system is in existence in the downhole environment. 3) Heat capacity changes in the flowing fluid along the wellbore are very small and this value is taken as a constant. 4) The heat generated by the motor and cable are entirely contribute to heating the fluid. 5) The production rate is assumed to be constant.

According to the configuration of the RDD-ESPP lifting system, the wellbore temperature distributions are divided into three parts, namely, from the bottomhole to the motor, the motor and pump body, and from the pump outlet to the wellhead [13, 14].

3.2.1. Bottomhole-to-Motor Section

Using the energy balance equation, the fluid temperature in the bottomhole-to-motor section can be calculated using Eqn. (1):

(1)

where, T(h) is the temperature at any location in the bottomhole-to-motor section, °C; t ₀ is the surface temperature, °C; α is the geothermal gradient, °C/m; H_w is the reservoir depth, m; h is the depth at any location in the bottomhole-to-motor section, m; K_t is the heat transfer coefficient between the fluid and the formation in the bottomhole-to-motor section, W/(m²·°C); C_H is the fluid heat capacity, J/(kg·°C); and G is the fluid mass flow rate, kg/s.

3.2.2. Motor and Pump Body (Pump Outlet)

The pump outlet temperature is a comprehensive result of motor heating, cable heat dissipation, and heat exchange between the fluid and the formation. It can be calculated using Eqn. (2):

(2)

Where T_p is the fluid temperature at the pump outlet depth, °C; T(H_L) is the fluid temperature resulting from heat exchange between the fluid and the formation at the pump outlet depth and can be calculated by Eqn. (1), °C; H_L is the pump outlet depth, m; N_m is the downhole linear motor input power, kW; η_m is the downhole linear motor efficiency, %; L_s is the length of the flat cable attached on the outside of the downhole linear motor, m; I is the working current of the downhole linear motor, A; and R is the resistance per unit length of the flat cable attached on the outside of the downhole linear motor, Ω/m.

3.2.3. Pump Outlet-to-Wellhead Section

In this section, the heat transfer includes the heat exchanged between the fluid and the formation, and cable heat dissipation. Similarly, the wellbore temperature at any point can be calculated via the energy balance equation, as expressed in Eqn. (3) and Eqn. (4):

(3)

(4)

where T(h_D) is the temperature at any location in the pump outlet-to-wellhead section, °C; h_D is the depth at any location in the pump outlet-to-wellhead section, m; K_D is the heat transfer coefficient between the fluid and the formation in the pump outlet-to-wellhead section, W/(m²·°C); and g is gravitational acceleration, which is defined as 9.81 m/s².

According to the working principle of the RDD-ESPP, the standing valve opens and the traveling valve closes during the upstroke. At this condition, the liquid column in the tubing above the pump acts on the plunger. Considering the inertial effect of the liquid column and the friction between the plunger and the working barrel, the load acting on the downhole linear motor can be calculated using Eqn. (5) [15, 16]:

(5)

Where F is the load acting on the downhole linear motor, N; A_p is the cross sectional area of the plunger, m²; A_rd is the cross sectional area of the push rod connecting the downhole linear motor and the plunger, m²; p_d is the pressure at the pump outlet depth or the discharge pressure, pa; p_s is the pressure at the pump inlet depth or the submergence pressure, pa; F_f is the friction force between the plunger and the working barrel, N; and F_inertia is the inertial force of the liquid column, N.

The friction force between the plunger and the working barrel can be calculated using Eqn. (6):

(6)

Where L_p is the plunger length, m; D is the plunger diameter, m; Δp is the pressure difference across the pump plunger, Pa; δ is the radial clearance between the pump plunger and the working barrel, m; μ is the fluid viscosity, Pa.s; v_p is the plunger moving velocity, m/s; ε is the eccentricity ratio, which is defined as ε = e/δ and is dimensionless; and e is the eccentricity between the pump plunger and the pump inner diameter in m.

The inertial force of the liquid column is calculated by an empirical equation as follows:

(7)

Where a is the acceleration of the pump plunger, m/s², and F_liquid is the static load of the liquid column, kN.

During the downstroke, the standing valve closes and traveling valve opens. This causes the submergence pressure to act on the pump plunger and the liquid column to act on the tubing. At this time, the load exerted on the downhole linear motor is calculated using Eqn. (8):

(8)

The plunger is directly driven by the downhole linear motor. The pump efficiency for the RDD-ESPP lifting system is mainly affected by the gas volume in the pump, the pump leakage rate and the volume change of the liquid in the pump. Therefore, the pump efficiency for the RDD-ESPP lifting system is defined as follows:

(9)

(10)

(11)

(12)

(13)

(14)

(15)

Where η is the pump efficiency, dimensionless; β is the pump admission coefficient, dimensionless; η_l is the pump leakage coefficient, dimensionless; η_B is the liquid volume effect coefficient, dimensionless; Q is the real production rate, m³/d; Q_t is the theoretical production rate, m³/d; B_L is the liquid volume coefficient, dimensionless; q_leak is the pump leakage, m³/d; f_p is the cross sectional area of the plunger; m₂; S is the stroke, m; N is the pumping speed, rpm; K is the pump clearance ratio, dimensionless; R is the gas-to-liquid ratio in the pump, m³/m³; R_P is the produced gas-to-oil ratio at the surface, m³/m³; R_S is the dissolved gas-to-oil ratio in the pump, m³/m³; f_w is the water cut, dimensionless; P₀ is the absolute pressure at standard conditions and is defined as 10⁵Pa; T₀ is the absolute temperature at standard conditions and is defined as 293K; T_in is the absolute temperature at the pump inlet; and Z is the gas compression factor, dimensionless.

Due to no speed-reducing and reversing mechanisms and by adopting an intermittent power supply, the RDD-ESPP lifting system will save more energy than a conventional sucker rod pumping system. When the working frequencies for the upstroke and downstroke are known, the daily power supply time can be calculated with the pumping speed. Then, the daily power consumption can be calculated using Eqn. (16) [17]:

(16)

(17)

Where W_M is the daily power consumption, kWhU_S is the surface voltage, V; I_U is the upstroke current, A; I_d is the downstroke current, A; T_u is the daily running time of the upstroke, h; T_d is the daily running time of the downstroke, h; cosφ is the power factor, dimensionless; and N_M is the surface input power, W.

Then, the system efficiency of an RDD-ESPP lifting system can be defined by Eqn. (18):

(18)

Where η_p is the system efficiency, dimensionless; HP_H is the hydraulic power, W; Q_t is the daily production rate, defined as t/d; ρ is the liquid density, kg/m³; and L is the effective lifting head, m.

When the RDD-ESPP lifting system is in operation, the maximum tubing tensile force appears at the wellhead. To ensure operational reliability and safety, the tubing strength should be evaluated. The weights of the tubing string, liquid column in the tubing, cable, pump and motor and vibration load are included in the tubing safety validation:

(19)

(20)

Where F_t is the total load acting on the wellhead tubing hanger, t; W_tube is the tubing string weight, t; W_liquid is the liquid column weight, t; W_cable is the downhole cable weight, t; W_pump is the pump weight, t; W_motor is the motor weight t; W_vib is the vibration load, t; σ_t is the tensile strength of the tubing thread buckle, t; and [S] is the safety coefficient, dimensionless.

Usually, it is very difficult to calculate the vibration load, which is affected by the motor’s vibration frequency, the tubing string’s natural frequency, the liquid damping coefficient and other factors. During lifting design, one way to tackle this problem is to use a larger safety coefficient to ensure proper tubing strength.

The Beggs-Brill method is used to calculate the pressure distribution in the wellbore. Based on the aforementioned developed mathematical models, nodal analysis method is adopted to select the optimal working parameters for an RDD-ESPP lifting system. Fig. (4) shows the detailed calculation and optimal working parameter selection processes.

The Ma 2 well block is located in the northern region of Manas Lake in the Junggar basin. The oil-bearing layer includes the Triassic Baikouquan Formation and the Permian Wuerhe Formation. The formation is a lithologic reservoir without bottom water and edge water. The Wuerhe formation is a very poor reservoir with secondary pores, a fine throat, low porosity and ultralow permeability. The porosity changes from 6.0% to 14.23% and the average porosity is 8.14%. The permeability ranges from 0.36×10^-3 μm² to 1763.4×10^-3 μm² and the average permeability is 6.42×10^-3 μm² [18, 19].

The X-I well is a producer that was completed at 3621 m in the Wuerhe Formation. It was taken into production as a flowing well after a hydraulic fracturing treatment in July 2013. The reservoir pressure decreased quickly due to an insufficient supply of formation energy. The current reservoir pressure is 31.43 MPa, and the bubble pressure is 5.81 MPa. Table 3 lists its production data when operated with a conventional sucker rod pumping system. The tubing, casing and motor data are shown in Table 2.

Under these working conditions, the pump efficiency is 38.9% and the power consumption associated with lifting one ton of liquid is 134.75 kWh. The system efficiency is less than 1%. The reasons for this are significant stroke loss and an unsuitable pump depth. In December 2017, a RDD-ESPP lifting system was installed in this well. The developed models and optimal working parameter selection method are used to optimize the most appropriate working parameters for this well. The downhole linear motor used for this well is a WFQYDB 114-1140-(30~50). Table 4 shows the design results for an allocated production rate of 3 t/d and a stroke of 1.23 m.

The pump efficiency and system efficiency can be improved largely by adopting this new RDD-ESPP lifting system and changing the pump depth. For the same stroke, speed and pump diameter, we observed that a large increase in the pump depth could not significantly improve the lifting efficiency. Considering that increasing the pump depth will increase the tubing and cable cost, a pump depth of 1950 m, a pump diameter of 38 mm, a stroke of 1.23 m and a speed of 2 rpm were adopted. The currents and frequencies for the upstroke and downstroke were 25 A and 12 A, and 8 Hz and 20 Hz, respectively. Using these optimized working parameters, a daily production rate of 2.8 t/d, a power consumption of 86.39 kWh per ton of liquid and a system efficiency of 4.97% were achieved in the X-I well after the RDD-ESPP lifting system was installed. The lower system efficiency is due to the installation of an unsuitable motor due to limitations of the supply of motors in the field. The daily power consumption decreased from 363.8 kWh to 241.9 kWh after optimization and a power savings rate of 33.5% was realized. The pump efficiency increased from 38.9% for the sucker rod pumping system to 80.5% for the RDD-ESPP lifting system. Compared with other field applications by changing the sucker rod pumping system to an RDD-ESPP lifting system, average pump efficiency improvements such as 44.3% in the Jinlin oilfield (Zhang, 2015) and 60.6% in Zhengting oilfield (Wang, 2012), 22% in the Changqing oilfield (Zheng et al., 2013) and 45.64% in the Daqing oilfield (Wang et al., 2007) were achieved. The application of the RDD-ESPP lifting system in the X-I well is demonstrated to have a remarkable energy-savings effect. Additionally, errors between the calculated results and the actual measurements of the production rate, pump efficiency, system efficiency and power consumption are 7.14%, 7.55%, 9.46% and -5.98%, respectively. These results show that the proposed method can be used to select optimal operating parameters for a RDD-ESPP lifting system and analyze its working state.