SPE 107899Integrated Analysis for PCP Systems Oscar Becerra Moreno, Petrobras Energía S.A., and Maria Eugenia Mena Romero, Universidad Simon Bolivar Copyright 2007, Society of Petroleum Engineers This paper was prepared for presentation at the 2007 SPE Latin American and Caribbean Petroleum Engineering Conference held in Buenos Aires, Argentina, 15–18 April 2007. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435. Abstract One of the concerns that artificial lift engineers have to face when they are working with PCP systems is the lack of information about effective procedures for PCP diagnosis and optimization. Most of the information available is limited to explaining the importance of keeping the fluid level over the pump to avoid dry run. On the opposite side, there are some studies that explain the hydrodynamics of the pumps, and how some variables, like intake void fraction and fluid viscosity affect the pump performance. However, there is no information available that may help field engineers understand the relationship between operational conditions and internal performance of the pump to know how each condition in the well could affect the pump, and how to determine and understand which condition is best for the system. This paper introduces a new procedure to be used for understanding PCP’s wells performance and therefore determine the best condition for system operation. Based on a set of simple mathematical models and patterns recognition techniques, this procedure is a useful tool to reduce the timing to analyze PCP systems and to automatically determine the best operational conditions in terms of production and performance. Introduction PCP system is a fast growing artificial lift application, during the last years many operator have been introducing this method to lift light oil and high WOR wells, thus the application window of PCP system is on expansion and the method is been used to lift lighter, gassy and low viscosity fluid a better understanding of the system performance is required. Unfortunate the information that is available to understand the downhole pumps performance is limited to some experimental studies and simplified models that don’t establish a relationship between the standard system variables and pump performance. Keeping in mind that pump is the critical element of and PCP artificial lift system. This work is dedicated to explain how to predict the pump performance under a wide range of downhole condition and how to recognize the optimum operational point or downhole pump adverse condition. The set of models that are presented in this study are based on Vetter [8] and Gamboa [11] developments, whom presented simplify models to predict and analyze the hydrodynamics phenomena inside of twin screw pump and Progressive cavity pump respectively. Internal Forces in a PCP When a PCP is operated some forces are generated due to the differential pressure, the eccentricity and rotational speed, some of these forces could affect the PCP duration and installation integrity. Figure 1, is used to visualize the direction of the forces that are generated along of a half stage by differential pressure between two cavities applied on the center of mass of rotor and stator. Tracking the forces acting on the center of the rotor through out a half cavity as figure 2 shows, a forces balances for X and Y direction can be expressed as differential forces. The following equation represents the differential force in X direction. dFx = 2 ⋅ ΔP ⋅ d r ⋅ cos β ⋅ dl ……………………eq. (1) Where, β= 2 ⋅π ⋅ l …………………………………….…eq. (2) PLSt Then to calculate a differential of length (dl), dl = PLSt ⋅ dβ ………………………………….…eq. (3) 2 ⋅π Following the same procedure the forces balance in both directions X and Y is found. ∑ Fx = π ΔP ⋅ d r ⋅ PLSt ⋅ 2 ⋅ ∫ cos β ⋅ dβ …..….eq. (4) 0 2 ⋅π ∑ Fy = 2 ⋅ ΔP ⋅ d r ⋅ PLSt π ⋅ ∫ sin β ⋅ dβ ………eq. (5) 0 2 ⋅π Since this original condition is transitory. [9]. the rotor is forced to be reclined to one side of the stator cavity. (9) For position of the rotor illustrated in figure 2.…eq.2 SPE 107899 Solving equation 4 is found that forces in X direction are equilibrated and therefore the resulting force is cero. the centrifugal force is orientated to the positive side of X axe. these models have been formulated for pure liquid under steady state operational conditions to be validated with experimental data. (12) 2 ⋅π To get the Hydraulic Torque for a complete stage length equation 8 has to be integrated from π to 0 as is shown in equation 13. A centrifugal force caused by the rotation of the rotor around stator is also acting on the center of mass of the rotor. the mass change as a function of time for a single cavity is: dm n = Q sl (( n − 2 ) − n ) ⋅ ρ n − 2 + Q st (( n −1 ) − n ) ⋅ ρ n −1 dt − Q sl ( n − ( n − 2 )) ⋅ ρ n − Q st ( n − ( n −1 )) ⋅ ρ n …………. Using figure 3 representations. an expression for differential torque is represented by equation 8. [8].. (7) 60 ⎠ ⎝ 2 Solving equation 13 a final expression for hydraulic torque is deduced. (8) Where the mass of the rotor (Mr). Using Gamboa [9]. (−π + θ ) ⋅ PLSt …………………………. The following lineal formulation was proposed for Gamboa [11] to describe the .…………eq. [11]. (13) π 2 ⋅π Internal Slip in a PCP The internal slip is the phenomenon that allows the PCP to increase the pressure from one stage to the next one and defines the pump performance in terms of volumetric efficiency and lifting capacity.eq. therefore at any section of the pump.eq. has to be higher enough to defeat the seal lines and create a gap that allows the fluid flow. Elastomeric PCPs are manufactured with interference between stator and rotor. a mass balance for a single stage can be proposed taking into account that fluid flow through the seal lines can be described as HagenPoiseuille Flow. generates a reactive force on the stator. τ hyd = 4 ⋅ e ⋅ d r ⋅ PLst ⋅ ΔP ……………….PLSt .eq.. represents the volumetric capacity of the pump for a single rotation The internal forces analysis indicates that two important forces are acting on the rotor during the pump operation.…………………………………. For Y direction the total force is expressed by equation 6.. which means that differential pressure between stages.…eq. so that if the differential pressure changes through the pump this force will be concentrated as well as hydraulic torque in these stages where the differential pressure is higher. Some authors [7]. [11] approach. could be calculated using the following simplified equation. τ hyd = − 0 ΔP ⋅ d r ⋅ e ⋅ PLSt ⋅ 2 ⋅ ∫ sin θ ⋅ dθ ……eq. (15) A lineal pressure distribution is assumed to start a trial and error process to calculate slippage flow rates between stages. [11] have developed simplify models to determine the pressure and slip distribution.dr.. (6) 2 ⋅π This result indicates that a contact force between rotor and stator is generated by differential pressure. Then.…eq. P dl = − LSt ⋅ dθ …………………………….……. 2 Fc = M r ⋅ AC …………………………….e.. ∑ Fy = 4 ⋅ d r ⋅ PLSt ⋅ ΔP ………………………eq. the centrifugal Force (Fc) will be distributed along of the pump because it is a consequence of the rotational speed and the lineal mass of the rotor. Field and laboratory studies have demonstrated that slip and differential pressure between stages could change along the pump [3]. (10) Where.. Fy is a consequence of the differential pressure between two stages. Nevertheless. (11) l=− 2 ⋅π Then.. ⎛ RPM ⋅ 2π ⎞ Ac = ω ⋅ r = ⎜ ⎟ ⋅ e ………………eq. the value of this force is equivalent to the centrifugal acceleration times the mass of the rotor. An additional analysis can be done to infer the hydraulic torque. depending on fluid properties and operational conditions. Both resulting forces from equations 6 and 9.. Although. dτ hyd = 2 ⋅ ΔP ⋅ d r ⋅ e ⋅ sin θ ⋅ dl ……………. there is not an available model to predict how an actual pump could operate in terms of pressure and slip distribution through out the pump considering two phases flow. n ⎛ π ⋅ dr 2 ⎞ ⎛ ⎟⎟ ⋅ ⎜⎜ 4πe ⋅ nstages + PLSt ⋅ stages M r = ⎜⎜ 2 ⎝ 4 ⎠ ⎝ ⎞ ⎟⎟ ⋅ ρ Steel ⎠ ……………………………………………. the process has to be calculated for sufficient time steps to get the definitive slippage and pressure distribution. (14) 2 ⋅π Where 4. Using figure 2. rotor minor diameter=0. w = ao + a1 ⋅ ΔP ………………………………. (17) 2 If the stator is manufactured using a rigid material. this force excites the vibration in the system and grows in quadratic form versus speed (Figure 4).86..04 M. Pressure distribution inside the pump was calculated for a total differential pressure of 14000 kPa and 68. PCP System Analysis To explain the utility of the last set of equation predicting and understanding a PCP system performance. Relationship between operational torque and downhole conditions Anyone that has been in contact with PCPs operations knows that torque and rotational speed (RPM) are the most important variables to be observed to regulate and protect the system. specific gravity of the gas 1.1656 M. where discharge pressure is composed by wellhead pressure plus hydrostatic pressure and flow pressure losses. This vibration is a common cause of looseness and fatigue because it acts at the edge of a pendulous system if it is not anchored. viscosity of liquid 16 cp. QL = V p ⋅ RPM ⋅ ε VOL …………………………….eq. Fy is also a contact force between rotor and stator that acts over a small area along of the sealing lines. Some fluids properties considered for this simulation were: specific gravity of liquid 0. Considering that. a constant clearance has to be supposed. Centrifugal force is relatively low because the rotational speed of a typical PCP is low. w= d r − de ……………………………………….eq. the hydraulic torque can represent the greatest part of the operational torque if viscous and friction losses are lower. hydraulic torque expression can be simplified as: τ= ΔP ⋅ V p ………………………………………. the force Fy generated inside the pump is over 6 Tons (Figure 5). equations from 18 to 21 are combined. (16) Where. Operating at total lifting capacity.eq. a constant differential pressure between stages was generated as a result of the simulation.SPE 107899 3 interference or clearance change as a function of differential pressure in case of elastomeric stator. this contact force will be higher in this cavities causing higher deformation and stress on the elastomer of the stator. Some of the parameters of the pump are: Eccentricity= 0. Pressure distribution inside the pump To calculate the pressure distribution inside the pump a simple program was created. According to equation 12. The operational torque that is measured at surface must be higher because the viscous and friction torque inside the pump and at the rod string. some operational condition for a typical installation are calculated. τ hyd 1 ⎞ ⎛ 1 ⎞ ⎜ ⎛⎛ ⎞2 ⎟ ⎜⎜ ⎛ V p ⎟ ⎟ Vp ⎜ ⎞n ⎟ 2 = ⋅ ⎜ Pwh + Phyd + Pf − ⎜ ⎜1 − ⎜⎜ ⋅ RPM ⋅ ε vol ⎟⎟ ⎟ ⋅ Pe ⎟ ⎟ 2 ⋅π ⎜ ⎜⎜ ⎜ ⎝ C1 ⎟⎟ ⎟ ⎠ ⎟ ⎠ ⎝⎝ ⎠ ⎟ ⎜ ⎝ ⎠ ………………………………………………………. (18) 2 ⋅π Fetkovich equation is used to describe the inflow performance ratio for an oil well. To take into account the transitory condition of the process changes in boundary conditions due to the opening and closing of cavities at the pump intake and discharge during the operation were included in the mathematical model. The initial interference between rotor and stator (a0) was assumed as 0.eq. the very close relationship between theses two variables is not as well known. This program was also used to determine the time when the pressure distribution along the pump could be considered under steady state condition (Figure 6). and downhole flowing pressure is represented by in taking pressure at the pump.0024 mm and the proportional constant (a1) of 0. ΔP = Pwh + Phyd + Pf − Pint ……………………. Therefore. Nevertheless.eq. This contact force is transfered to the stator generating a torque that tries to unscrew the pump from the tubing.eq. volumetric efficiency and pump capacity as equation 18 describes. The first example was performed for an elastomeric stator PCP. changing the void fraction at pump intake for different rotational speeds.….94 kPa of intaking pressure. For a pure liquid intake condition. n ⎛ Pint 2 ⎞ QL = C1 ⎜⎜1 − 2 ⎟⎟ ………………………………. in some cases the differential pressure could be concentrated only in some cavities. Pitch length=0. (19) Pe ⎠ ⎝ The amount of liquid that a PCP can handle is a function of the rotational speed. That is why a good practice for PCP installation consists on the use of anchors and centralizers. Internal forces during a simulated operation All the examples will be conducted for a 24 cavities PCP and 40 M3/Day at 100 RPM volumetric capacity.eq.01 M. (20) The total differential pressure for an actual PCP installation could be simplified as the difference between discharge pressure and intake pressure. (21) To get an expression for hydraulic torque as a function of RPM. Nevertheless.004377 mm/Kpa. (22) The last equation is representing only hydraulic torque. These distributions don’t have any dependency with rotational speed because the fluid is considered incompressible . an average reduction of volumetric efficiency of 10% is required.4 (Figure 7). when the system is getting close to the maximum rate due to the higher void fraction entering to the pump. For pure liquid operation simulation. where the operational torque reduces the dependence with rotational speed. It is easy to recognize a slope change in both curves. A second example was performed considering a void fraction of 0. That’s why in some cases a good practice consist on keeping some submergence in order to reduce the void fraction entering to the pump improving the volumetric efficiency. the volumetric efficiency will be reduced while the lack of liquid at the pump intake will be occupied by gas from the annulus. When the intaking void fraction is increased (figures 8 and 9) the pressure distribution developes a dependence with rotational speed as a consequence of the change in the amount of mass that can slip during a pump’s rotation. Some of the parameters of the simulated system are: Pump Depth= 1219. In the same way as PCPs with elastomeric stator. For 250 and 500 RPM there is not any differential pressure between stages close to the pump’s intake. a slightly quadratic pressure distribution inside the pump was negated. no flow caused by parted rods or stator failure. improving the heat transfer from PCP’s components to the fluid. The previous exercise shows that higher slip helps the PCP to establish a more equilibrate pressure distribution when void fraction and rotational speed are increased. If the void fraction is increased a higher change in pressure distribution with rotational speed is observed (Figures 11 and 12). [8]. In the same way. clearance between cavities depends on differential pressure. it means that a constant clearance between stator and rotor was assumed. plugging caused by solids or paraffin deposition. the pressure distribution when the void fraction is increased change with rotational speed (Figure 11 and 12).5 M3/day). how operational torque under steady state conditions change with rotational speed.5 Then. SPE 107899 Recognizing operational conditions To illustrate the operational conditions recognition using equation 22. All the fluid that is slipping front the discharge is kept inside the pump to compress the fluid at last stages. allowing slippage if any differential pressure exists between cavities.5 M Pe= 8272 kPa C1= 79. viscous torque or friction torque. This difference is cause by the combined effect of higher slip and constant clearance for PCP with rigid stator. The second example was performed for a rigid stator’s PCP. If one cavity is losing pressure the downsteam clearance is reduced and the upstems clearance is increased to keep more mass inside of the cavity.2 at the pump intake under atmospheric pressure. Figure 13 shows. so that at higher speeds the pressure gradient in the last part of the pump has to be higher to allow more slip compressing the fluid to reach the discharge Pressure. But the slope’s change for toque versus RPM is smooth as long as the void fraction and pump’s slip is increased. In case of rigid stator the clearance is constant and the pressure of the cavity is controlled only by the pressure drop caused by the fluid flow trough the seal lines that have a constant geometry.4 at pump intake for atmospheric pressure. this dependence is very weak due to the higher slip rate (5. . abnormal or dangerous conditions in the system are normally related to: sticking caused by elastomer swelling. the reduced differential pressure between cavities close to the pump intake indicates the absent or reduced slip between cavities. Nevertheless. a typical system performance was simulated using the same PCP that was used for previous examples. Since. To get the last 8% of the maximum flow rate. If total differential pressure is concentrated in some cavities the stress on stator and rotor material due to contact forces will be also concentrated. Nevertheless.53 M3/Day. it means that there is not any slip between these stages too. the pressure distribution inside the pump is better distributed when the slip rate is higher. The first case was performed for a void fraction of 0. Higher slip was also allowed in order to observe the effect of this variable on pressure distribution. low flow caused by worn or leaks. If the rotational speed is increased beyond of the point for maximum flow rate. Patterns recognition Operational torque is affected by any irregularity that changes the differential pressure. This distribution doesn’t have any dependence with rotational speed (Figure 10). This condition allows the clearance to work as flow regulator of the cavity. show that the point where the system is lifting the maximum flow rate is again recognizable on torque curves. Spite of the different slip rate for both simulated conditions the volumetric efficiency is the same at higher speeds because the slip rate trends to cero at higher rotational speeds as was previously explained. The fluid compressibility also plays an important role to generate this dependence because it increases the amount of flow that is required to boost the pressure of one cavity. Results for both cases indicate the maximum flow rate is reached at low volumetric efficiency (Figures 14 and 16).[3]. The amount of mass that a cavity receive during a rotation is proportional to the rotation duration. this change is lower compared with the same condition for elastomeric stator. two different total slip rates at maximum differential pressure at the pump (48 M3/Day and 96 M3/Day) were supposed to compare the system performance. Results in figure 15. For elastic stator. This change indicates the point where the system has reached the maximum allowable liquid rate as is shown in figure 14. which means that internal forces inside the PCP are better distributed and the average void fraction inside the pump is lower. The volumetric efficiency of the pump drops. Nevertheless. The total slip calculated for these conditions was 1.6 M3/Day n= 0. Nevertheless a simplified analysis described in this work could be performed in order to determine the best operational conditions. 2. tubing or rod flaws can be recognized. A better ways to compare different pumps consist on a single stage performance representation for each one. a simple example was simulated as is shown in figure 17. A lineal pressure distribution inside the PCP represents its best operational condition.M) θ : Angle that is formed between force at the rotor and eccentricity (M) ε VOL : Volumetric efficiency Q L : Liquid Flow Rate (M3/S) Pwh : Wellhead Pressure (kPa) Phyd : Hydrostatic pressure (kPa) Pf : Pressure losses due to friction (kPa) Pint : Intake pressure (kPa) Pe : Static Pressure (kPa) dmn : Change in mass of a cavity as a function of time dt Qsl : Slippage rate between two cavities (M3/S) w : Diametric Interferential or Clearance (M) ao : Initial interference between rotor and stator (M) a1 : Proportional constant for interference reduction versus pressure (M/kPa) . The downhole performance prediction for PCP is a very complex process. An equilibrate pressure distribution improves the contact forces distribution between rotor and stator and increases the average liquid fraction inside the pump which means less stress concentration and better stator refrigeration. because many variables are playing an important role at the same time. Figure 17 is only an example for one particular problem but many different problems like inflow changes. Once the model is matched. In this way. This condition is a sign of an irregular condition like: elastomer swelling or discharge plugging.A for all the support during this work preparation and publication. After day 100 the operational torque started to increase. Nevertheless. then each of these conditions could be related with a particular torque versus RPM or time pattern. which took more than 20 days. 4. Acknowledgements The Authors wish to acknowledge Petrobras Energia S. 6. Then the speed is kept stable to get the maximum flow rate. all of them affecting differential pressure or friction forces. Nomenclature Fx : Force X direction (Nw) Fy : Force Y direction (Nw) ΔP : Differential Pressure between two consecutive cavities (KPa) d r : Rotor Diameter (M) β : Angle force of the rotor (Rad) PLSt : Stator Pitch Length (M) Ac : Centrifugal Acceleration (M/S2) ω : Angular velocity (Rad/S) e : Eccentricity (M) r : Radius (M) RPM : Revolutions per minute Fc : Centrifugal Force (Nw) M r : Mass of Rotor (kG) n stages : Total Stages number ρ Steel : Density of Steel (kG/M3) τ hyd : Hydraulic Torque (Nw. Operational torque versus RPM plots represents a useful tool to determine the optimum operational condition for a PCP system. these curves are not directly comparable for different pumps if they don’t have the same amount of stages. understanding the operational torque pattern that these flaws or changes would generate. 5. The first part of the graphic represents the optimization process when the speed of the system in increased to get the optimum flow rate. the best test bench condition for any particular PCP application could be established. The hydrodynamic model for PCP pressure distribution and slip that is proposed in this work can be use to mach test bench data for PCP’s by changing interference or clearance constants. It could be done dividing the pump head by the stages number. downhole condition could be estimated.SPE 107899 etc. Nevertheless. while the representation of operational torque as a function of time is suitable for failures recognition and can be used to identify changes 5 in the system performance that in some cases could require an operational regulation to avoid failures. the increase of pressure trends to be concentrated at the last stages of the pump if void fraction and rotational speed are increased. Conclusions 1. 3. The way to equilibrate the pressure distribution consists on increasing the slip by reducing the initial interference between rotor and stator. The PCPs performance curves are normally provided for the entire pump. To explain how one of the common abnormal conditions could looks like using a torque versus time graphic. Olivet. The University of Paris.. 16-17 September. 5-8 October. S. Gd Solutions C... 21-23 April.6 de : Stator cavity minor diameter (M) Q L : Liquid Rate (M3/S) C1 : Maximum flow rate (M3/Day) n : Cavity number References [1] Gamboa J and Mendez J (2002) Experimental Study of the effect of Fluid Temperature and Running Time on Performance of a Progressive Cavity Pump. [9] Olivet. Paper presented at the SPE Annual Technical Conference and Exhibition. A.. (1998). J (2002) Understanding the performance of progressive Cavity Pump with a Metallic Stator. Paper prepared for representation at the 2002 Progressing Cavity Pump Workshop. 16-17 September. Gamboa J. [4] Gamboa. Paper presented at the SPE Annual Technical Conference and Exhibition. SPE77730. W. and Paluchowski. and Kenyery F..A. Modeling of NPSHR for Progressing Cavity Pumps. Caracas. Rene. (1997). A. J. 29 September-2 October. Texas US.. Colorado.. G. D. [8] Vetter.. and Pregler. A New Capsulism. [3] Martin. [10] Gamboa. (2002) New Approach for understanding the Behavior of Progressive Cavity Pumps. Paper presented at the 1998 SPE Permian Basin Oil and Gas Recovery Conference. . SPE 107899 SPE 84137. (2000). [2] International Standard for Downhole Progressing Cavity Pump (ISO 15136-1) 2000. SPE 53967. Multiphase Pumping with Twin-Screw PumpsUnderstand and Model Hydrodynamics and Hydroabrasive Wear. 2003. Espin. Gonzalez P and Iglesias.S. Universidad Simon Bolivar (2003) New Approach for Modeling Progressive Cavity Pumps Performance. J. Proceedings of the 17th International Pump Users Symposium. S. A. Wirth. ASME Fluids Engineering Division Summer Meeting. [5] Moineau. A. San Antonio. Midland.A.Paper presented at the 1999 SPE Latin American and Caribbean Petroleum Engineering Conference. J. [7] Robello. Calgary. SPE 39786. G and Saveth K. Canada. Venezuela. Texas. (2002) Experimental Study of Two-Phase Pumping in a Progressive Cavity Pump Metal to Metal. (1999). Progressing Cavity Pump (PCP): New Performance Equations for optimal Design.Intenational Standard Organization. Kenyery F. U. Experimental Study of Two Phase Pumping in Progressive Cavity Pumps. [11] Gamboa.. and Tremante A. [6] Vetter. G. Canada. Paper prepared for presentation at the 2002 Progressing Cavity Pump Workshop. Calgary. Held in Denver. (1930). Doctoral Thesis. Proceedings of the 20th International Pump Users Symposium. Olivet. Forces acting on rotor and stator originated by differential pressure along of half stage. Simplified Longitudinal Representation of a PCP Showing Cross section slip and longitudinal slip in a PCP . Position and direction of forces acting at the center of the rotor through out a half cavity Figure 3.SPE 107899 7 Figure 1. Figure 2. 8 SPE 107899 Void Fraction=0 1600 25 1000 20 800 15 600 14000 12000 10 400 200 0 0 100 200 300 400 Pressure(kPa) 1200 Ac(M/S2) 1400 Fc (Nw) 16000 30 RPM 100 RPM 250 10000 RPM 500 8000 6000 5 4000 0 2000 500 0 RPM CAVITY Figure 7. Pressure evolution in two stages during PCP operation simulation 0 1 6 11 16 21 26 CAVITY Figure 9.6 void fraction and elastomeric stator . Force in Y direction 1000 Void Fraction=0.6 900 16000 800 14000 RPM100 700 12000 RPM 250 600 Pressure (kPa) Pressure 11 500 400 300 200 100 RPM 500 10000 8000 6000 Cavity 4 Serie1 4000 Cavity 10 Serie2 2000 0 0 5 10 15 20 25 Rotations Figure 6. Pressure distribution along of PCP for 0. Pressure distribution inside of a PCP for pure liquid and elastomeric stator Figure 4. Centrifugal acceleration and centrifugal force for a simulated example 16000 60000 14000 50000 12000 Pressure(kPa) Fy (Nw) Void Fraction = 0. Pressure distribution along of PCP for 0.4 void fraction and elastomeric stator Figure 5.4 70000 40000 30000 20000 RPM100 RPM 250 10000 RPM 500 8000 6000 4000 10000 2000 0 0 5000 10000 15000 0 1 ΔP (kPa) 6 16 21 26 CAVITY Figure 8. 2 Void Fraction=0 700 16000 600 14000 Troque (Nw-M) Pressure (kPa) 500 RPM 100 12000 RPM 250 10000 RPM 500 8000 6000 400 Torque Slip 96 300 Torque Slip 48 200 4000 100 2000 0 0 0 1 6 11 16 21 100 200 26 RPM 300 400 500 CAVITY Figure 13. Flow Rate versus RPM for void fraction =0.4 Void Fraction =0.7 60 0.4 void fraction and rigid stator Void Fraction 0.6 void fraction and rigid stator Figure 15. Pressure distribution along the Pump for 0.2 10 0.6 700 16000 600 14000 RPM100 500 RPM 250 Torque (Nw-M) Pressure(kPa) 12000 RPM 500 10000 8000 6000 400 Torque Slip 96 300 Torque Slip 48 200 100 4000 0 2000 0 0 100 200 300 400 500 RPM 1 6 11 16 21 26 CAVITY Figure 12.6 50 0.5 40 0.SPE 107899 9 Void Fraction 0.4 .1 0 4000 Volumetric Eficiency Void Fraction 0. Operational Torque versus RPM for void fraction =0. Pressure distribution along the Pump for pure liquid and rigid stator Void Fraction =0.3 20 0.2 Figure 11.9 70 0.2 90 0 100 200 2000 300 0 500 400 RPM 0 1 6 11 16 21 26 Flow Rate Slip 96 Flow Rate Slip 48 Volumetric efficiency Slip 96 Volumetric Effiency Slip 48 CAVITY Figure 14.4 30 0.4 Flow Rate (M3/Day) 16000 14000 RPM100 12000 Pressure (kPa) RPM 250 10000 RPM 500 8000 6000 1 80 0. Pressure distribution for along the Pump for 0.8 0. Operational Torque versus RPM for void fraction =0.2 Figure 10. Operational torque versus time 140 .8 0.7 60 0.1 0 500 0 0 100 200 300 Volumetric Efficiency 1 80 400 RPM Flow Rate Slip 96 Flow Rate Slip 48 Volumetric efficiency Slip 96 Volumetric efficiency Slip 48 Figure 16.4 90 500 400 Torque 300 200 100 0 0 20 40 60 80 100 120 Day Figure 17. Flow Rate versus RPM for void fraction =0.6 50 0.9 70 0.3 20 0.4 Historical Torque 800 700 600 Torque (Nw-M) Flow Rate ((M3/Day) Void Fraction 0.5 40 0.10 SPE 107899 0.2 10 0.4 30 0.