Numerical investigation of contact stress between rotor and stator in a two-lead progressing cavity pump

Author’s Accepted Manuscript Numerical investigation of contact stress between rotor and stator in a two-lead progressing cavity pump Long Pan, Jinzhu Tan www.elsevier.com/locate/petrol

PII: DOI: Reference:

S0920-4105(15)30073-5 http://dx.doi.org/10.1016/j.petrol.2015.07.026 PETROL3139

To appear in: Journal of Petroleum Science and Engineering Received date: 4 March 2015 Revised date: 24 July 2015 Accepted date: 29 July 2015 Cite this article as: Long Pan and Jinzhu Tan, Numerical investigation of contact stress between rotor and stator in a two-lead progressing cavity pump, Journal of Petroleum Science and Engineering, http://dx.doi.org/10.1016/j.petrol.2015.07.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Numerical investigation of contact stress between rotor and stator in a two-lead progressing cavity pump Long Pan, Jinzhu Tan* School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China *Corresponding author. Tel./fax: +86 25 58139954. E-mail address: [email protected](J.Z. Tan).

Abstract: The contact stress between rotor and stator in a progressing cavity pump (PCP) is critical to seal performance and service lifetime of the PCP. In this paper, an approach to predict the contact stress and friction behavior in PCPs is proposed. The approach is based on the results from the finite element analysis (FEA) and Stribeck-type curve. A three-dimensional finite element model for the contact stress between the rotor and the stator within a two-lead PCP is successfully established. The fluid pressure, the eccentric force and the laden torque are applied to the model. And the finite element analysis is conducted to investigate the contact stress between the rotor and the stator in the two-lead PCP. The simulated results show that the contact stress in the spiral seal lines (SPSLs) is always high compared to that in warping seal lines (WSLs) at the identical relative position, and the contact stress increases along seals from the first seal line to the last one. The maximum contact stress of the PCP appearing in every seal line is off-center. It is found that the serious wear and/or damage for the progressing cavity pump during operating always occurs on the paths which are off-center of the spiral seal lines. The results are validated against the actual phenomenon which appeared in oilfields from open literatures.

Keywords: Progressing cavity pump, finite element analysis, Stribeck-type curve, contact stress

1

1. Introduction A single-lobe progressing cavity pump (PCP) mainly consists of a single-lobe screw (rotor) and a double-lobe nut (stator). The rotor turns inside an elastomer-lined stator. PCPs are positive-displacement pumps, which are becoming more widespread in a variety of applications, such as pumping oil, food and drink, coal slurry, viscous chemical and so on (Nelik and Brennan, 2005). This is primarily due to their ability to pump high viscosity fluid, to tolerate free gas and impurities, and to have high efficiency, low capital and operating costs (Revard, 1995; Cholet, 1997). Based on the pumping operation principle discovered by Moineau (1930), PCP was invented and found able to pump fluids of different viscosities. Although PCP was considered as one of the most effective artificial lifting devices (Beauquin et al., 2007; Liu et al., 2005; Ramos et al., 2007; Wu and Li, 2010), PCP has met some difficulties and challenges particularly in predicting wear and damage since the 1970s when PCP was applied to oil artificial lifts in low to medium depth oil wells. The wear and damage led to the decrease of pumping performance and lifetime in some oilfields. There are many articles in the open literatures concerning the degradation of the progressing cavity pump. For example, Sathyamoorthy et al. (2013) found that there were wear marks on surfaces of both rotor and stator after a short period of operation and evidence of debonding along the whole elastomer insert and the steel tube. Liang et al. (2011) presented some attempts of anti-scaling techniques for the PCPs in the Daqing oilfield with the intention of avoiding the unusual friction torque and wear so that service lives of the PCPs were improved. In order to improve the service life of PCPs, experiments were performed by Wang et al. (2013) and Lv et al. (2013) to investigate friction and wear behavior of the PCP stator rubbers in various chemical environments. In addition, finite element 2

method has been used to study various behaviors of PCPs for several years. For example, Liu et al (2010) studied the relation between interference and laden torque on the basis of finite element method. Zhou et al. (2013) found that the laden torque of the PCP with the stator of the even thickness elastomer was smaller than that of the conventional PCPs using finite element analysis (FEA). Chen et al (2013) found that the interference and the stator thickness were the two main factors influencing the volumetric efficiency by establishing the new finite element model of the PCP which consisted of the stator, the rotor, the lifted fluid and the fluid-solid interaction. The Stribeck curve is an overall view of friction variation over the entire spectrum of lubrication, including the boundary, mixed, and full-film hydrodynamic or elastohydrodynamic lubrication regimes. In the beginning of the last century, its concept was initialized by Stribeck (1902). The Stribeck curve can give the functional relationship between the coefficient of friction and the product of sliding speed and viscosity divided by the normal load. Many researchers used the Stribeck curve to study the lubricant regimes in different conditions. For example, Stribeck curves were used by Vijaykumar et al. (2015) to reveal the lubrication regimes with the CuO nano-particles in the solution. And Stribeck curves were investigated by Zhu et al. (2015) in a wide range of speed and lubricant film thickness based on the simulation results with various types of contact geometry using machined rough surfaces of different orientations. Although several works related to the optimization of PCPs were published, there are few reports regarding the contact stress distribution and the friction behavior which are the main factors influencing the performance of PCPs. In this paper, an approach to predict contact stress and friction behavior between rotor and stator in PCP was proposed. This approach combined a finite element model (FEM) with 3

Stribeck-type curve, and was applied to a two-lead PCP. Firstly, the full 3D transient contact stress distributions were obtained by FEM, considering the fluid pressure distribution, the eccentric motion of rotor and the laden torque within the PCP. And then a Stribeck-type curve was used to study the friction behaviors between the rotor and stator under various contact stress and relative sliding velocity. Finally, coupling the FEA model with the Stribeck-type curve, the friction behavior was studied in this work.

2. Finite element analysis 2.1. The geometrical structure and material properties The PCP studied in this work mainly consisted of a single-lobe screw (rotor) and a double-lobe nut (stator) with interference between the rotor and stator. The outside diameter of the rotor, DSR, was larger than the inside diameter of the stator, DST (i.e. DSR > DST). For the PCP, DSR is the diameter of the circle of the rotor in cross section, and DST is the inside diameter of the semi-circle of the stator in cross section. The rotor was made of steel with the surface chromium plated to prevent medium corrosion and to improve wearability. In order to ensure the sealing efficiency, the stator was mainly made of the rubber material to tolerate the repeated deformation due to the rotor motion. The stator rubber elastomer with the outside diameter of 100mm was glued to the steel tube with the thickness of 2mm as shown in Fig.1. The internal space of the PCP was separated into consecutive cavities (i.e. cavity1, cavity2, cavity3, cavity4 and cavity5) by seal lines which generated from interference between the rotor and the stator. Based on the eccentric motion of the single rotor, the cavities are moved upwards in the pump. And the movement of the cavities makes a pressure increase over the pump. In this work, the geometrical characteristic parameters for the 4

PCP are listed in Table 1. In order to study the mechanical properties of the stator material for the two-lead progressing cavity pump, the nitrile butadiene rubber (NBR) material was chosen. The elastomeric material was fabricated in accordance with our previous work (Pan et al., 2015). The formula of the elastomeric material for the stator is listed in Table 2. The samples from the fabricated elastomer block were prepared for the tensile and compressive tests, respectively. The tensile test sample is a dumbbell-style slice of work length 25mm, thickness 2 mm and width 6mm according to ISO 37-2011. The compressive test sample is a cylindrical disc of diameter 13 mm and thickness 6.3mm according to ISO 7743-2011. The tensile and compressive tests were performed using the universal instrument (Model: MZ-4000D) with a precision of force measurement (±0.5%) and a precision of displacement measurement (±0.5%). The forces and displacements can be controlled and recorded automatically by computer. All tests

℃

were conducted at room temperature (about 25 ) and relative humidity of 45%. All tensile tests were performed at the rate of 500 mm/min and the compressive tests were performed at the rate of 10 mm/min under displacement-controlled mode. Fig.2 shows the tensile test results for the elastomeric rubber material. The compressive test results for the elastomeric rubber material are shown in Fig.3. In order to simulate contact stress between the rotor and the stator in the two-lead progressing cavity pump, it is assumed that the elastomeric material for the stator is nearly incompressible in this work. For the elastomeric material, the Mooney-Rivlin function as shown in Eq. (1) was used to model the hyperelastic mechanical behavior.

(

)

(

)

(

)(

)

W = c10 I1 − 3 + c01 I 2 − 3 + c11 I1 − 3 I 2 − 3 +0.5k ( J − 1)

2

(1)

where W is strain energy function. c  , c and c

are Rivlin coefficient, the

values of the c10, c01 and c11 are -2.838, 4.162 and 0.669, respectively. ̅ = J − 5

(2/3) ·  (p=1,2) is deviation strain invariant; k is initial volume modulus; J is volume ratio. For the rotor material, i.e. 45# steel in Chinese grade, Poisson ratio is 0.3, and the elastic modulus is 210 GPa. Although the elastic modulus of the rotor material is much larger than that of the stator rubber material, the rotor with a structure similar to a spring cannot be assumed as a rigid body in the finite element analysis because of the deformation under various forces during the operation. Therefore, the rotor was assumed as an elastomer to obtain more accurate simulation results in this work. In addition, in order to consider the effect of the rotor stretch on the simulated results, the length of the rotor is larger than that of the stator as shown in Fig.1. 2.2 seal lines Contact stress between the rotor and the stator within the PCP, as an essential quantitative parameter, is critical to the seal performance of the PCP, the friction behavior and the wear behavior. However, it is difficult to experimentally observe inner contact status and to obtain the contact stress distribution between the rotor and the stator because of adiaphanous stator. The seal line represents the line in which the contact stress between the rotor and the stator in the PCP appeared in this work. The seal line reflects the contact condition between the rotor and the stator for the PCP. The three-dimensional seal lines for the two-lead PCP are showed in Fig. 4. The seal lines are divided into three categories and named as: (1) semicircle seal lines (SSLs), (2) spiral seal lines (SPSLs) and (3) warping seal lines (WSLs). The SSLs form when the rotor surface completely contacts with the semicircle segment of the stator in cross section, and move spirally with the eccentric motion of the rotor along the surfaces which generate from outline of semicircle regions of the stator inner wall. The SPSLs and WSLs appear when the 6

rotor surface contact with the straight line segment of the stator inner wall in cross section, and move spirally with the eccentric motion of the rotor along the surfaces which generate from outline of straight line regions of the stator inner wall. 2.3. Assumptions of the FEA model In order to study the contact stress between the rotor and the stator for the PCP, the finite element model of the PCP for oil artificial lifts was simplified by the following assumptions: (1) the inner fluid pressure distribution was assumed uniform in every cavity. For the PCP, the inner fluid pressure actually had a gradient particularly in the clearance between the adjacent cavities, but the fluid pressure was nearly identical in each cavity, that can be quoted (Paladino et al., 2008; Paladino et al., 2009; Paladino et al., 2011; Andrade et al., 2011); (2) the rotation speed of the PCP was assumed as a constant; (3) the effects of fluid temperature, chemical corrosion, abrasion were not considered in this work. 2.4. Numerical simulation strategy Fig. 5. represents the simulation process for finite element analysis of the contact stress distribution between the rotor and the stator in the PCP over time. It can be seen from the Fig. 5 that the finite element models for the PCP was simplified to a series of static FEA models in an increment of time for continuous motion. The three types of loads, i.e. the hydraulic pressure, the eccentric force of rotor and the laden torque, were applied to the model for each static simulation for the PCP. The each transient simulation included two steps. First step, contact stresses were calculated at the given interference between the rotor and stator. In this step, the regions of seal lines and cavities in the curved surfaces of the rotor and the stator were preliminarily distinguished, and then different fluid pressures were applied to the cavities within the 7

PCP. Second step, the inner fluid pressure, the eccentric force load and the laden torque were applied to the model automatically using ANSYS parameter design language (APDL), respectively.

2.4.1. Mesh and boundary conditions In this study, the augmented Lagrangian method was used to analyze the contact stress distribution between the rotor and the stator in the PCP. A three-dimensional FEA model of the PCP with the rotor and the stator was established and shown in Fig. 6. The mesh refinement was performed in the inner surface of the stator where the size of mesh is as one fourth as the size of mesh in other regions. The local mesh refinement could increase the accuracy of the simulation results but not take a large increase in the computing time. Outside surface of the stator and the top surface of the rotor were fixed, as the boundary conditions for the FEA. As stated above (see Fig.1), the stator of the PCP consisted of the steel tube and the elastomer lined inside the steel tube in this study. The steel tube was assumed as a rigid body and ignored for the FEA in the work. Therefore, the outside of the elastomer was considered not to be deformed. In addition, the top surface of the rotor connected with the transmission shaft, which led to change of the location of the top surface over time. Therefore, fixing the top surface, as a reference to distinguish these static models in different time, was performed. 2.4.2. Inner fluid pressure load Many researchers (Paladino et al., 2008; Paladino et al., 2009; Paladino et al., 2011; Andrade et al., 2011; Gamboa et al., 2002; Gamboa et al., 2003) studied the fluid pressure distributions in the pumps. The inner fluid pressure distributions for the pumps changed with the operating conditions, such as viscosity and density of the 8

fluid, rotate speed, lifting pressure and so on. For an elastomeric stator pump, the contact stress of seal lines increases with the increase of fluid pressure in the two adjacent cavities. The seals are broken when the fluid pressures in the adjacent cavities are larger than the contact stress. And then the fluid pressure in the cavity with lower fluid pressure increases until the contact stress reaches the seal requirement. Although the actual pressure distribution may be nonlinear, the increase of the fluid pressure in each cavity is similar and the deformation of the stator is more or less constant. Therefore, it is assumed in this study that the fluid pressure distribution is linear for the FEA. In order to simulate the contact stress for the PCP, it is also assumed that the fluid pressure is constant within the cavities and increase along the seals. The fluid pressure in the Nth pressurized cavity is given by Equation (2): P =

∆∗ 

+ P

(2)

where ∆P represents the differential fluid pressure between the suction cavities and the discharge cavities, N is the number of pressurizing, P represents the fluid

pressure of the suction cavities. In this work, the fluid pressure of the suction cavities is zero and the fluid pressure of the discharge cavities is 1.2 MPa. The PCP consists of two half-formed suction cavities, two half-formed discharge cavities and (2NPST-2) integrated cavities, that leads to the fluid pressure pressurizes (2NPST-2) times in the integrated cavities and the last pressure increase occurs in the discharge cavities, so at that time the number of pressurizing is N = 2N

!

− 1. However, as shown in Fig.

1, at the special moment when one of the suction cavities has closed but the last integrated cavity has not opened, the quantity of the integrated cavities increases and the number of pressurizing is N = 2N !. Eq. (2) is only applied to the Newtonian fluids. 9

For the operating pump, the clearances appear at the seal lines where the fluid has a pressure gradient (reduction). However, the scope of the pressure gradient is small. Therefore, the fluid pressures at the clearances were not applied in this model. In that case, some of the contact stress results obtained from this model should subtract the fluid pressure when the contact stress cannot meet the seal requirement.

2.4.3. Inertial force load The dynamics load as body load applied to the nodes of rotor for the static FEA models was implemented in this work. The inertial force generally act through the center of mass which is difficultly found, but according to the finite element method the inertial force discretely applied in all notes of the rotor is adopt using ANSYS parameter design language (APDL). The motion of the PCP rotor is shown in Fig. 7. In the figure, the point O represents the central axis of the stator. The circle around point O as the center, with a radius of 2E, consists of the centers belonging to the semicircle segment of the stator, which gives the boundary of the centers of the cross section of the rotor. The point O1 represents the central axis of the rotor, and the distance from point O to point O1 is the eccentricity E. The circle around point O1 as the center, with a radius of E, consists of the centers of the cross section of the rotor (i.e. the locus of the center of the rotor helix). The planetary motion of the rotor could be seemed as the center circle of the rotor rolls along the inner locus of the minor center circle of the stator. On one side, the center circle of the rotor rotates around point O1 with an angular velocity # %%%%%%%&, $ that is called the rotation. On the other side, the center circle of the rotor rotate along the dashed circle with an angular velocity # %& = −# %%%%%%%&, $ that is called the revolution of

the pump. The angle '($ and θ as the rotation angle and the revolution angle, are 10

references that specify the positions of the rotor. In this work, the location, velocity and acceleration at every point in/on the rotor can be calculated for any angle θ (i.e. for any instant of time t) by the following equations: θ = θ + ωt

(3)

cos θ ∗ x6 + sin θ ∗ y6 + cos θ ∗ E x r = .y2 = 3−sin θ ∗ x6 + cos θ ∗ y6 + sin θ ∗ E: z z6

− sin θ ∗ ωx6 + cos θ ∗ ωy6 − sin θ ∗ ωE x; r; = 3y; : = 3− cos θ ∗ ωx6 − sin θ ∗ ωy6 + cos θ ∗ ωE: 0 z;

x − cos θ ∗ ω= x6 − sin θ ∗ ω= y6 − cos θ ∗ ω= E x< = y r< = 3y< : = −ω . 2 = 3 sin θ ∗ ω= x6 − cos θ ∗ ω= y6 − sin θ ∗ ω= E : z z< 0

(4)

(5)

(6)

Where the initial position of the rotor is represented when the angle θ equals zero, as

shown in Fig. 6. The coordinate of the point in/on the rotor relating to the stator coordinate system is assumed as (x, y, z), and the coordinate (x6 , y6 , z6 ) represents the coordinate of the point in/on the rotor relating to the body-fixed coordinate system. In this study, the rotate speed of the PCP is 100 r/min. Based on D'Alembert principle, the accelerating rotor can be transformed into an equivalent static system by adding the so-called "inertial force" and the external forces. The inertial force for each node can be calculated by the following equation F@ = −mB r<

： (7)

where mB represents the mass of the node of the rotor after discretization, the acceleration r< is obtained from locations of the nodes multiplying by −ω=. The

mass for each node was established after each element mass was averagely distributed to the nodes belonging to the element.

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2.4.4. Laden torque load The total torque over the rotor is given by the integral over the rotor surface of the stress tensor (including pressure and friction) multiplied by the local radius and the area as follows: %%& × EF C

G H(IJ(KLKM (N MOL M(MPQ M(BRSL

=

T

B(M. YSBN

U̿ ∙ X& × Z[ \]

(8)

This total torque is equal to the shaft driving torque and can be further separated in the torques due to fluid stress and due to contact stress as follows: %%& × EF C

G H(IJ(KLKM (N MOL M(MPQ M(BRSL

=

+

T

B(M. YSBN NQS_` H(KMPHM

T

^ ∙ X& × Z[ \] +

B(M. YSBN YMPM(B H(KMPHM

U̿

T

B(M. YSBN NQS_` H(KMPHM

H(KMPHM YMBLYY K(BIPQ /MPKbLKM_PQ

U̿a_YH(SY ∙ X& × Z[ \]

∙ X& × Z[ \]

(9)

The total torque for an operating PCP consists of two main parts, i.e. the hydraulic torque and irreversible torque. The first integral term in Eq. (9) is the hydraulic torque and exerted only by the "conservative forces" (pressure), as it does not consider any losses. The second integral term is the irreversible torque due to viscous loss of fluid, but it was ignored in this FEA model because the fluid in each cavity was stable. The last integral term is the irreversible torque due to contact stress and consists of the normal component and the tangential component. The friction torque is equal to the tangential component, which is one of the main irreversible torques in this study. The other important part of the shaft power is consumed in the normal deformation of the elastomer. The normal deformation process consists of compression stage and resilience stage. For the PCP, the two stages appear in the same time, the energy is absorbed in the compression stage while the energy is released in 12

the resilience stage. For the resilience stage, the released energy may compensate part of the compress energy and transmit to the fluid for fluid pressure increase and viscous loss. And the hysteresis loss energy is the main energy loss due to the normal deformation. Because the value was not large enough compared to the friction torque, the irreversible torque due to the normal deformation of the elastomer was ignored in the model. The hydraulic torque herein can be calculated by dividing the hydraulic power by the angular velocity. And the hydraulic torque for pumping the fluid can be given as: q = AP ! = (4ED

M=

j∆ =k

=

!

− πδ D

!

lmnop qkrs nop tkrsu =k

+ πδ = )P !

P ! ∆P

(10) (11)

where q is the volume displaced in a rotation, A represents the flow area of the cross section and M represents the hydraulic torque. The derivation process of the flow area of the cross section was given in detail by Nguyen et al. (2014). The volumetric capacity in Eq. (10) is theoretical capacity and not the actual capacity. However, the theoretical capacity is generally close to the actual capacity for the elastomeric PCPs when the differential pressure is not large enough to cause serious leakage, and the hydraulic loss and transmission efficiency could be ignored. When the differential pressure makes the seal broken seriously, the engine torque must be obtained from the experiment but not from above equations. Therefore, the model herein only gives the results when the whole seal is slightly broken. But the model can be extended for the large differential pressure to predict the place easy to wear on the stator inner surface based on the actual engine torque from experiments. The friction torque as the main irreversible torque can be calculated by the following equation:

13

MN =

T

B(M. YSBN YMPM(B H(KMPHM

v ∗ wH ∗

xyz \] =

(12)

Where f represents the friction coefficient between the rotor and stator and Pc is contact stress. The coefficient of friction was determined based on the contact stress from the FEA results and Stribeck-type curve. The procedure of the Stribeck-tyoe curve used with FEM is given as follows. Firstly, the contact stress between the rotor and the stator from the FEA model, only considering the effect of interference, was obtained. Secondly, the Stribeck-type curve was used to determine the coefficient of friction (predicted value) depending on the FEA results only considering the effect of interference. Thirdly, the predicted friction torque can be calculated and applied in the FEA model. And then the FEA results considering all loads were obtained. Finally, the coefficient of friction and the friction torque were refreshed depending on the Stribeck-type curve and the new FEA results. This procedure cycles until the value of friction torque changes little. The total torque was calculated through Eqs. (11) and (12) in this study. In order to simulate the contact stress between the rotor and the stator under the total torque, the way of “rigid beam” in ANSYS was selected. And then the torque was applied to the model for the FEA. 3. Results and discussion In order to study the effect of the mesh density on the accuracy of the results and to optimize the computational time, several different finite element meshes were used in the simulation of the PCP at the beginning time of zero using the commercial ANSYS code. Fig.8 shows the calculated results of the contact stress for different finite element meshes. The maximum contact stresses in the second SPSL of the PCP for the different finite element meshes are shown in Fig. 9. It can be seen from Fig. 8 14

and Fig. 9 that the results are sufficiently accurate from mesh size with 2003784 nodes, which corresponds to element size of about 1 mm. The element size was then used throughout the study. Fig.10 shows the contact stress distribution in all seal lines under the fluid pressure, the eccentric force and the laden torque. The results were obtained from the computational mesh size with 1849077 elements in which the contact elements had 356216. It can be found that the contact stress in the SPSLs was always higher than that in WSLs at the identical relative position, and the contact stress increased successively from the first seal line to the last one (i.e. the fourth seal line). This is mainly because the angular variation of rotor form line is twice that of stator form line. It leads to the large width of WSL compared with the width of SPSL in actual model. In addition, the contact stress distribution for each kind of seal lines is similar. The contact stress near center was higher than that at ends, and the highest contact stress appeared off-center. It is found from Fig.10 that the highest contact stress near center of the seal lines appeared in the SPSL and was off-center about 5mm. However, it can be seen from Fig. 10(c) that the contact stresses increased sharply on one end of the first SSL which was near point of intersection in the inlet cross section and on the other end of the fifth SSL which is near point of intersection in the outlet cross section, where the value of contact stress is close to the maximum value of contact stress in SPSL. This is mainly because the rotor distortion makes the contact positions changed obviously. Fig. 11 shows contact stress contours considering the operating load effects at different times. It can be seen from Fig. 11 that the high contact stress also appeared in the SPSL of the PCP. This result indicates that the place easy to wear could be in the SPSL of the PCP. 15

The coefficients of friction between the rotor and the stator under oilfield are different depending on different temperatures and fluid types (Livescu et al., 2014; Livescu and Craig, 2015). The temperature of the contact surfaces may increase due to the mechanical friction, the normal deformation and the downhole temperature. For different oilfields, the pumping fluids are various, and the temperature also affects the viscosity of the fluids. Therefore, for different operating conditions, the Stribeck-type curve used to determine the actual coefficient of friction should be obtained according to the actual operating conditions. In order to study the friction behavior, the Stribeck curve from experiments by Wang et al. (2013) was used here for an example. The results from Wang et al. (2013) indicated that the solid-to-solid contact (i.e. boundary lubrication regime) appeared as the value of ην/Ρ (η is the dynamic viscosity, ν is the sliding velocity, P is the

mean contact stress) was less than 1.75 × 10q . And the coefficient of friction almost kept constant, when the seal lines are in the boundary lubrication regime. Fig. 12 shows the local value of ην/Ρ plotted along the contact lines. For the PCP in this study, η is the dynamic viscosity of water, ν is the sliding velocity, P is the mean

contact stress obtained from the FEA results. It can be found from Fig.12 that the value of ην/Ρ changed in the different contact regions of the PCP for each cross section. Based on the Stribeck curve from experiments by Wang et al. (2013), the friction coefficient can be obtained according to the value of ην/Ρ along the seal lines. It can be found from Fig.12 that most of the values of ην/Ρ were less than 1.75 × 10q . And the coefficient of friction almost kept the constant. Therefore, the

friction coefficient between the rotor and the stator in the PCP could be assumed as a constant (0.22). Note that if the mean contact stress for each cross section becomes lower and there are obvious regions out of the boundary lubrication regime, the 16

coefficient of friction used in the FEA model could not be assumed as a constant and should strictly follow the Stribeck-type curve obtained from the experiment. In order to validate the simulated results using the FEM, the simulation results were compared with the results from Sathyamoorthy et al. (2013). The results from Sathyamoorthy et al. (2013) indicated that the wear and damage of the PCP appeared in the areas which were off-center a distance in the SPSL in the inner surface of the PCP. The simulated results in the study are in agreement with the experimental results from Sathyamoorthy et al.(2013). The same phenomenon appearing in the PCP used in the Daqing oilfield was reported by Liang et al. (2011), as shown in Fig.13 and Fig.14. Fig.13 shows the wear in the inner surface of the PCP stator used in Daqing oilfield. In the figure, the wear (near the middle of the line region in cross section) appeared in the area which was off-center a small distance in the SPSL in the inner surface of the PCP. Fig.14 shows the wear on the surface of the PCP rotor used in Daqing oilfield. In the figure, the phenomenon which appeared in the surface of the PCP rotor was also observed. And the wear in the rotor appeared more and more serious from one end to the other end, which could well explain that the contact stress increased with seal line from the first seal line to the last one. As expected, the numerical simulated results in the study are in good agreement with the actual wear and damage results from Sathyamoorthy et al. (2013) and Liang et al. (2011). It can be also found that the wear was mainly relevant to the contact stress and increased with the increase of contact stress. The FEA results indicate that the serious wear and/or damage during operating for the progressing cavity pump may occur on the paths which are off-center of the spiral seal lines (SPSL) and near the points of intersection in the inlet and outlet cross sections.

17

4. Conclusions An approach to predict the contact stress and friction behavior in PCPs was proposed. This approach was based on the results from the finite element analysis (FEA) and Stribeck-type curve. A detailed 3D finite element model for the contact stress between the rotor and stator within a two-lead PCP was successfully established. The fluid pressure, the eccentric force and the laden torque were applied to the model. And then the finite element analysis was conducted to investigate the contact stress between the rotor and the stator in the two-lead PCP. The simulated results show that the contact stress in the spiral seal line (SPSL) was always high compared to that in warping seal line (WSL) at the identical relative position, and the contact stress increased successively with the seals from the first seal line to the last one. The maximum contact stress of the PCP appeared in the SPSL and was off-center about 5mm. In addition, it is found that the serious wear and/or damage during operating for the progressing cavity pump always occurred on the paths which is off-center of the SPSL and near the points of intersection in the inlet and outlet cross sections. The approach can well explain the actual wear and damage phenomenon for the PCPs used in oilfield from literatures (Liang et al., 2011; Sathyamoorthy et al., 2013). The FEA model can be extended to provide detailed information about the contact status for different geometrical parameters and operating conditions. The tangential stress and friction torque, which are the significant design parameters, can be estimated using the results obtained from the FEA model and Stribeck-type curve from wear and damage experiment between various rotor and stator materials under various operating conditions. Studies on the clearances at the seal lines and the pressure gradient are under investigation and will be reported later.

18

Acknowledgements This work is sponsored by the Graduate Student Scientific Innovative Project of Jiangsu Province (No. CXZZ13_0429), China. This work is also supported by the Natural Science Foundation of China (51175241) and the “Six Talent Peaks” of Jiangsu Province in China. References

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Andrade, S.F.A., Val rio, J.V., Carvalho, M.S., 2011. Asymptotic model of the 3D flow in a progressing-cavity pump. SPE paper 142294. Society of Petroleum Engineers. 16 (2), 451-462. Beauquin, J., Ndinemenu, F.O., Chalier, G., et al. 2007. World’s first metal PCP SAGD field test shows promising artificial-lift technology for heavy-oil hot production: Joslyn Field case. SPE paper 110479. In: 2007 SPE Annual Technical Conference and Exhibition, November 11-14, Anaheim, CA, USA. Chen, J., Liu, H., Wang, F.S., et al. 2013. Numerical prediction on volumetric efficiency of progressive cavity pump with fluid-solid interaction model. Journal of Petroleum Science and Engineering. 109, 12-17. Cholet, H., 1997. Progressing Cavity Pumps. Editions Technip, Paris, France. Gamboa, J., Olivet. A., Espin, S., 2003. New approach for modeling progressive cavity pumps performance. SPE Paper 84137. In: the SPE Annual Technical Conference and Exhibition, October 5-8, Denver, USA. Gamboa, J., Olivet. A., Gonzales, P., et al. 2002. Understanding the performance of a progressive cavity pump with a metallic stator. In: 23rd International Pump User Symposium, March 5-8, Houston. Liang, Y.N., Cao, G., Shi, G.C., et al. 2011. Progressing cavity pump anti-scaling techniques in alkaline-surfactant-polymer flooding in the Daqing Oilfield. 19

Petroleum Exploration and Development. 38(4), 483-490. Liu, H., Cao, G., Wang, D.M., et al. 2005. The successful application of 2000 PCP wells in Daqing oilfield. SPE paper 10032. In: Proceedings of 2005 International Petroleum Technology Conference, November 21-23, Doha, Qatar. Liu, H., He, X.D., Wu, H.A., et al. 2010. Three dimensional FEM simulation and parameter study on laden torque of interference friction of PCP. SPE paper 135808. In: Proceedings of 2010 SPE Progressing Cavity Pumps conference, September 12-14, Alberta, Canada. Livescu S., Craig S.H., Watkins T., 2014. Challenging the Industry''s Understanding of the Mechanical Friction Reduction for Coiled Tubing Operations. SPE paper 170635. In: 2014 SPE Annual Technical Conference and Exhibition, October 27-29, Amsterdam, the Netherlands. Livescu S., Craig S., 2015. Increasing lubricity of downhole fluids for coiled-tubing operations. SPE Journal. 20(2), 396-404. Lv, X.R., Huo, X.Y., Qu, G.Z., et al. 2013. Research on friction and wear behavior of rubber in the mixture of crude oil with water. Applied Mechanics and Materials. 300-301, 1254-1258.Moineau, R., 1930. Le Noveau Capsulism. Ph D. Thesis, University of Paris, France. Nelik, L., Brennan, J., 2005. Progressing Cavity Pumps, Downhole Pumps and Mudmotors. Gulf Publishing Company, Houston, Texas. Nguyen, T., Al-Safran, E., Saasen, A., et al. 2014. Modeling the design and performance of progressing cavity pump using 3-D vector approach. Journal of Petroleum Science and Engineering. 122, 180-186. Paladino, E.E., Lima, J.A., Almeida, R.F.C., et al. 2008. Computational modeling of the three-dimensional flow in a metallic stator progressing cavity pump. SPE 20

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Tables

Table 1 Geometrical parameters of the PCP. Geometrical parameter Value Rotor diameter DSR

39.6 mm

Stator diameter DST

39 mm

Interference δ =( DSR - DST)/2

NBR3965

Carbon

100

62

0.3 mm

Eccentricity E

8.5 mm

Stator pitch PST

128 mm

Number of stator pitches NPST

2

Table 2 The formula of NBR material for the PCP stator. Carbon-white ZnO SA S TMTD DOP 18

5

1

0.5

3

Figures

Fig. 1. Cutaway view of the two-lead PCP.

23

15

4010NA

Total

2

206.5

Fig. 2. Tensile stress-strain curve of the NBR material sample.

Fig. 3. Compressive stress-strain curve of the NBR material sample.

24

Fig. 4. Seal lines of the two-lead PCP.

Fig. 5. Flowchart of simulating the contact stress distribution over time.

25

Fig. 6. Finite element model of the PCP .

26

Fig. 7. Diagram of the motion for the PCP.

27

Fig. 8. The contact stresses along the z-axis of the second SPSL only considering the interference with different mesh number of nodes at time of zero.

Fig. 9. The maximum contact stresses of the second SPSL only considering the interference with different mesh number of nodes at time of zero.

28

29

Fig. 10. The contact stress distribution in (a)SPSLs, (b)WSLs and (c)SSLs under the fluid pressure, the eccentric force and the laden torque.

30

Fig. 11. Contact stress contours considering effect of the fluid pressure, the eccentric force and the laden torque at the time of (a) 0s, (b) 0.15s, (c) 0.3s and (d) 0.45s.

31

Fig. 12. The values of ην/Ρ along (a)SPSLs, (b)WSLs and (c)SSLs.

Fig. 13. Wear marks in the inner surface of the PCP stator used in Daqing oil field from Liang et al. (2011). 32

Fig. 14. Wear marks in the surface of the PCP rotor used in Daqing oil field from Liang et al. (2011).

Highlights: 1.

A three-dimensional finite element model for a two-lead PCP is established.

2.

The contact stress distribution in seal lines for the operating PCP is presented.

3.

An approach based on Stribeck-type curve to predict friction behavior in PCPs is proposed.

4.

The actual wear and damage phenomena of PCPs appearing in oilfields are explained.

33