REVIEW AND APPLICATION OF THE TULSA LIQUID JET PUMP MODEL

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REVIEW AND APPLICATION OF THE TULSA LIQUID JET PUMP MODEL Pål Jåtun Pedersen Trondheim December 2006 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Preface This report is a mandatory project assignment in the 9th semester of the petroleum production engineering studies at NTNU It was written at the institute for petroleum technology and applied geophysics, fall 2006 The assignment consists of 71 pages, and was delivered the 19th of December 2006 I would like to thank Professor Jón Steinar Guðmundsson for good help and advice throughout the project Also, I am very grateful for all the help I have got from the people at Petroleum Experts Ltd., regarding the version update of PROSPER I Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Summary As a water drive reservoir is depleted, production will fall and inflow decrease As a consequence of this, the well will either stop flowing or produce only a limited amount of oil and gas In these cases, an artificial lift system can be installed to increase the production and save the well One of these lift systems is the Jet Pump The Jet Pump operates on the principle of the venturi tube, converting pressure into velocity head by injecting power fluid through a nozzle This creates a suction effect which drives the production fluid through the pump At the diffuser the velocity head is converted into pressure, allowing the mix of power and production fluid to flow to the surface through the return conduit There have been made several theoretical models for the Jet Pump Among these are the one reviewed in this project: “Performance model for Hydraulic Jet Pumping of two-phase fluids” by Baohua Jiao from 1988 The model is an approach to calculate pump performance while pumping a compressible fluid Important elements in the model are the nozzle and throatdiffuser friction factors The nozzle friction factor is estimated by optimisation based on high pressure data, while the equation for the throat-diffuser friction factor is developed using regression analysis The dimensionless pressure recovery, N, and the efficiency, is very dependent on these values Especially is it dependent on the throat-diffuser friction factor, which again depends on the gas-oil ratio Calculations were performed on a North Sea well with a gas-oil ratio on 95 Sm Sm , using both the Tulsa model and models based on incompressible flow The calculated efficiency was, as expected, higher for the models based on incompressible flow The well performance program PROSPER was used for pressure drop calculations Also, the Jet Pump function in the program gave about similar results as the Tulsa model Perhaps is the Tulsa model used as the Jet Pump function in this program Anyhow, the similarity in results between the Tulsa model and PROSPER indicates that the calculations performed in this project is reasonable and that the model is applicable to a field situation as presented here II Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Table of contents Introduction Jet Pump Literature Survey 2.1 When is artificial lift required 2.2 Jet Pump compared to other artificial lift methods 2.4 Jet Pump principles Review of the Tulsa Jet Pump Model 3.1 Development of the model 3.2 Presentation of the model and its main principles Tulsa Jet Pump performance 4.1 Main factors to control pump performance 4.2 Sizing of the pump 10 Application of the Tulsa Jet Pump Model 13 5.1 Sizing and performance calculations 13 5.2 Evaluation of results 17 Application of the Tulsa model on a North Sea well 18 6.1 Case description 18 6.2 Model calculations 19 6.3 Evaluation of results 22 Application of Other Models on a North Sea Well 23 7.1 JSG model calculations 23 7.2 NTNU project calculations 24 Discussion 28 Conclusion 30 10 References 31 11 Tables 32 12 Figures 34 13 Appendixes 46 III Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Introduction In the course of a field’s life, reservoir pressure will fall and inflow decline As a water drive reservoir is depleted, water cut will rise and production decrease This can cause wells either to stop flowing or to produce only limited amounts of oil In these cases, different artificial lift systems can be installed to save the well and increase production A wide range of artificial lift systems are available The choice of lift system is dependent on well characteristics, well location and costs considerations One of these lift systems is the one reviewed in this paper: the Hydraulic Jet Pump Several different Jet Pump models have been developed, varying in accuracy and complexity However, few models for predicting the behaviour of compressible flow are developed Among these few models is the “Performance model for Hydraulic Jet Pumping of two-phase fluids” by Baohua Jiao, published in a thesis at Tulsa University in 1988 (in this project referred to as the Tulsa model) The project assignment is to review the Tulsa model, convert the basic equations to SI and perform calculations for a production well in the North Sea, using PROSPER for pressure drop calculations Then, look at previous NTNU projects/thesis and perform calculations on the North Sea well with a few other models Finally, compare the results with the Tulsa model The project starts with a literature survey of the Jet Pump, giving a brief introduction to the Jet Pump principles, different Jet Pump models and comparison between Jet Pump and other artificial Jet Pump models In chapter 3, and the Tulsa Jet Pump model is introduced and documented, and in chapter the model is used for calculations on a North Sea well Chapter contains calculations using other models, for comparison Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Jet Pump Literature Survey 2.1 When is artificial lift required The objective of any artificial lift system is to add energy to the produced fluids, either to accelerate or to enable production Some wells may simply flow more efficiently on artificial lift, others require artificial lift to get started and will then proceed to flow on natural lift, others yet may not flow at all on natural flow In any of these cases, the cost of the artificial lift system must be compared to the gained production and increased income In clear cut cases, such as on-shore stripper wells where the bulk of the operating costs are the lifting costs, the problem is usually not present In more complex situations, which are common in the North Sea, designing and optimising an artificial lift system can be a comprehensive and difficult exercise This requires the involvement of a number of parties, from sub-surface engineering to production operations The requirement for artificial lift systems are usually presented later in a field’s life, when reservoir pressure decline and well productivity drop If a situation is anticipated where artificial lift will be required or will be cost effective later in a field’s life, it may be advantageous to install the artificial lift equipment up front and use it to accelerate production throughout the field’s life All reservoirs contain energy in the form of pressure, in the compressed fluid itself and in the rock, due to the overburden Pressure can be artificially maintained or enhanced by injecting gas or water into the reservoir This is commonly known as pressure maintenance Artificial lift systems distinguish themselves from pressure maintenance by adding energy to the produced fluids in the well; the energy is not transferred to the reservoir (Jahn, Cook & Graham, 1998)2 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 2.2 Jet Pump compared to other artificial lift methods The Jet Pump has many advantages towards other artificial lift systems There are no moving parts, the pump is tolerant not only of corrosive and abrasive well fluids, but also of various power fluids Maintenance and repair are infrequent and inexpensive, the pump can be replaced without pulling the tubing (casing type installation) and it consists of few parts The pumps are suitable for deep wells, directional wells, crooked wells, subsea production wells, wells with high viscosity, high paraffin, high sand content, and particularly for wells with GOR up to 180 Sm Also, the pureness of the power fluid can be relatively low compared to Sm the quality of for instance the hydraulic piston pump power fluid Other great advantages of the jet pumps are that water can be used as power fluid and that the power source can be remotely located and can handle high volume rates Hydraulic Jet Pumps are adaptable to all existing hydraulic pump bottomhole assemblies, can handle free gas and are applicable offshore However, using a Jet Pump as the artificial lift solution will also bring disadvantages First and foremost, it’s a relatively inefficient lift method As seen in Figure 4, the hydraulic efficiency of the Jet Pump is very low compared to for instance the Progressive cavity pump (PC) or the Beam Pump (BP) It also requires at least 20% submergence to approach best lift efficiency and is very sensitive to changes in backpressure Also, the pump requires high surface power fluid pressure The casing type installation is the most common solution, using the casing-tubing annulus as the return conduit and the tubing as the power fluid string For this type of installation, the production of free gas through the pump causes reduction in the ability to handle liquids The advantage is, as mentioned above, that the Jet Pump can be replaced without pulling the tubing (Brown, 1982, Jiao, 1988)1,3 Figure shows a comparison for the different artificial lift methods Review and Application of the Tulsa Liquid Jet Pump Model December 2006 2.4 Jet Pump principles Jet Pumps operate on the principle of the venturi tube A high-pressure driving fluid (“power fluid”) is ejected through a nozzle, where pressure is converted to velocity head The high velocity – low pressure jet flow draws the production fluid into the pump throat where both fluid mix A diffuser then converts the kinetic energy of the mixture into pressure, allowing the mixed fluids to flow to the surface through the return conduit (Jiao, 1988)1 Figure illustrates the principle Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Review of the Tulsa Jet Pump Model 3.1 Development of the model The Tulsa Jet Pump model is presented in the thesis “Performance Model for Hydraulic Jet Pumping of Two-Phase Fluids” by Baohua Jiao from 1988 The model is based on experimental studies conducted at Tulsa University, and is a further development of the model presented in his master thesis “Behaviour of Hydraulic Jet Pumps When Handling a GasLiquid Mixture” from 1985 Experimental studies were performed using a mixture of water and air as the production fluid and water as the power fluid The operating pressures were set to typical values found in the field, with power fluid, for example, reaching 3000 psig (20 MPa) and production intake fluid exceeding 1200 psig (8.3 MPa) The performance data acquired were the power fluid pressure, the pressures at the intake and discharge, the flow rates of the power fluid, the two phases of the production fluid, and the appropriate temperature so that the air-liquid ratio could be computed For further description of the experimental facility and test data it is referred to the thesis The analysis of the data followed the model of Petrie, Wilson and Smart (PWS) This model is based on conservation of mass and energy, and is widely familiar to production engineers The PWS model and the Tulsa model differ only in the treatment of the two empirical, dimensionless parameters, K n and K td , which are the loss parameters for the nozzle and the throat-diffuser, respectively The objective of both models is to predict a dimensionless pressure recovery ratio, N, as a function of a dimensionless mass flow ratio, M (Jiao, 1988)1 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 3.2 Presentation of the model and its main principles The model is originally derived in oilfield-units, but is presented here in SI-units Conversion from field to SI-units is a task specifically mentioned in the project-description, and is conducted on both the “Derivation of the Jet Pump Model” (Appendix A) and the “Pump Sizing Procedure” (Appendix B) Following is a presentation of the main principles of the Tulsa model For the model derivation in its entirety it is referred to Appendix A The terminology used in the model is detailed in the Nomenclature (Appendix A, page 61-62) and shown in Figure The brackets on the right side of the mathematical expressions contain the equation number in the derivation As mentioned earlier, the purpose of the model is to predict pressure recovery, N, as a function of dimensionless mass flow ratio, M The dimensionless pressure recovery is the pressure increase over the pump divided by the pressure difference between the drive fluid and the pump discharge Mathematically it’s defined as follows: N≡ Pd − Pi Pp − Pd ….(19) The dimensionless mass flow ratio between the suction (producing) fluid and the power fluid is defined as: M ≡ mint ake ρQi Q = = i , mnozzle ρQ p Q p for one phase flow, assuming equal density for the two fluids Extended to include gas, the mass flow ratio can be expressed as: M = Qi + Qia × 1.227 Qp ….(37) Review and Application of the Tulsa Liquid Jet Pump Model December 2006 hence, Mscf ⎡ lbm ⎤ , where Qia (flow rate of air through pump) is in Qiam = Qia × 1000 × 0.0763, ⎢ D ⎣ D ⎥ ⎦ For standard conditions in SI, we have: ρ air = PM 101325 * 28.97 g kg = = 1225 = 1.225 RT 8.314472 * 288 m m ⎡ kg ⎤ Qiam = Qia × 1000 × ρ air = Qia × 1000 × 1.225, ⎢ ⎥ ⎣ s ⎦ Where Qia is in ….(31) Mm s The mass flow can be expressed as: M = ρ water Qi + ρ air Qint ake ,air ρ Q ρ water Qi =( ) + ( air int ake ,air ) ρ water Q powerfluid ρ water Q powerfluid ρ water Q powerfluid ⎡ kg ⎤ ⎡ kg ⎤ where ρ water = 999 ⎢ ⎥ and ρ air = 1.225⎢ ⎥ ⎣m ⎦ ⎣m ⎦ The second term in the above equation simplifies to kg (Qia × 1000 × 1.225) ρ air Qint ake ,air s = Qia × 1.227 )= ( kg ρ water Q powerfluid Q powerfluid (999 × Q powerfluid ) s The first term simplifies to ( ρ waterQi ρ waterQ powerfluid )= ( Qi Q powerfluid ) 57 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Hence, the equation for mass flow becomes: M = Qint ake + Qint ake ,air × 1.227 Q powerfluid ….(32) The consumed input hydraulic power becomes: E C = ( Pp Q p − Pd Q p ) = ( Pp − Pd )Q p , ….(33) while the transferred useful hydraulic power is expressed: Et = Pd (Qi + Qiaw ) − Pi (Qi − Qiaw ) = ( Pd − Pi )(Qi + Qiaw ) ….(34) Substituting equation (32) into equation (34): Et = ( Pd − Pi )(Qi + 1.227 × Qia ) ….(35) Hence, the pump efficiency is: transferred useful power = consumed input power E t ( Pd − Pi )(Qi + Qiaw ) ( Pd − Pi )(Qi + 1.227 × Qia ) = = = EC ( Pp − Pd )Q p ( Pp − Pd )Q p η= ….(36) ⎛ Q + 1.227 × Qia ⎞ ⎟ N⎜ i ⎟ ⎜ Qp ⎠ ⎝ This efficiency restates the efficiency computed by equation (28), using: M = Qi + Qia × 1.227 Qp ….(37) 58 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 From equation (36) we see that N and M gives us the efficiency: η = Efficiency = N × M ….(38) Jiao’s Jet Pump Model uses a functional form for N = f (M ) that is based on work by Cunningham(3), who developed this function on mass energy conservation principles Simplifying the typing of this function, we define two component elements: [ B = R + (1 − R )( M R ) /(1 − R ) ] C = R (1 + M ) ….(39) ….(40) (where R is the ratio of the nozzle to throat area) Using equation (39) and (40), equation (21) becomes: N= B − (1 + K td )C (1 + K n ) − B + (1 + K td )C ….(41) Where K n is the effective friction factor at nozzle, and K td is the friction factor at throat and diffuser K n is in this jet pump model set to 0.04 This value was selected by Jiao from optimization based on high pressure data K td is given by: K td = 0.1 + (3.67 * 10−3 )( R p ) −2.33 ( AWR) 0,63 R 0.33 Where Rp = ….(42) Pd Pp 59 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Equation 42 is developed by Jiao, using regression analysis The analysis was done by a computer program, performing a multiple linear least squares regression on the logarithms of the variables R , R p and AWR (Air-Water-Ratio) For single-phase flow, the right side of the equation simplifies to the constant 0.1, as AWR=0 Converting the equation to SI-units: K td = 0.1 + (3.67 * 10 −3 )( R p ) −2.33 ( AWR) 0.63 ( 0.15898 0.63 0.33 ) R 0.30483 Rearranging: K td = 0.1 + (10.88 * 10 −3 )( R p ) −2.33 ( AWR) 0.63 R 0.33 ….(43) 60 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Nomenclature Ai Area of flow path at pump suction, sq m An Area of flow path at nozzle exit, sq m At Area of flow path at throat, sq m Asm Minimum flow area throat annulus to avoid cavitation, sq m As Throat annulus area, sq m AWR Air water ratio, Sm^3/Sm^3 B Parameter used in computing N, dimensionless C Parameter used in computing N, dimensionless Dp Depth to the pump, m Et Transferred useful hydraulic power, Watt Ec Consumed input hydraulic power, Watt Gi Gradient of pump intake fluid, Pa/m Go Gradient of oil, Pa/m Gp Gradient of pump power fluid, Pa/m Gw Gradient of water, Pa/m Kt Friction factor at throat, dimensionless Kd Friction factor at diffuser, dimensionless K td Friction factor at throat and diffuser, dimensionless K 'n Friction factor at nozzle (theoretical), dimensionless Kn Friction factor at nozzle (effective), dimensionless 61 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 M Dimensionless mass flow ratio mi Mass flow rate through pump suction, kg/s mn Mass flow rate through nozzle exit, kg/s mt Mass flow rate through throat, kg/s N Dimensionless pressure recovery factor Pd Pump discharge pressure, Pa Pe Pressure at throat entrance, Pa Pi Pump intake pressure, Pa Ps Static pressure, Pa Pp Power fluid pressure at nozzle entrance, Pa Qd Pump discharge flow rate, Sm^3/s Qg In situ gas flow rate, Sm^3/s Qi Flow rate at pump intake, Sm^3/s Qiam Mass intake flow rate of air at standard conditions, Mm^3/s Qp Flow rate of power fluid through pump, Sm^3/s Qr Flow rate of returning fluid, Sm^3/s Qsc Maximum noncavitating flow rate, Sm^3/s R Dimensionless ratio of nozzle area to throat area Rp Ratio of discharge pressure to power fluid pressure, Pd / Pp Vi Fluid velocity through pump intake, m/s Vn Fluid velocity through nozzle, m/s Vt Fluid velocity through throat, m/s 62 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Appendix B Sizing of Hydraulic Jet Pumps Step by step procedure The following text is a step by step procedure for sizing downhole liquid Jet Pumps Most of the equations used are derived from basic fluid mechanical and momentum equations in Appendix A, “Derivation of the Jet Pump Model” Select the type of hydraulic jet pump installation: If a casing type open system is selected, the annulus will be the return conduit Compute the pump intake fluid gradient ( Gi ), ⎡ Pa ⎤ Gi = (Gw × WC ) + (1 − WC )Go , ⎢ ⎥ ⎣m⎦ Where WC is the water cut, and G w and Go are the water and oil gradients in Pa/m Compute Asm , the minimum throat annulus area to avoid cavitation Using the equation: ⎡ G GOR ⎤ )⎥ , m Asm = Qi ⎢0.28 i + (1 − Wc )(550)( Pi Pi ⎦ ⎣ [ ] Where Qi is the pump-inlet flow rate, Pi is the pump-inlet pressure, and Gi the fluid gradient at the inlet in Pa/m GOR is the gas-oil ration in Sm3/Sm3 63 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Using the manufacturers part list, select a value of At − An that exceeds Asm , to avoid cavitation: ( At − An ) > Asm Where At is the area of flow path at throat and An is the area of flow path at nozzle exit Both values are in square meters Manufacturer tables are found as figure 5 Compute the dimensionless ratio of nozzle to throat area, R R ≡ An / At Select a reasonable value for a surface operating pressure, Ps The choice depends upon the available pressure of the surface pump, the jet pump setting depth, and other properties of the well, the fluid and the inflow performance Estimate Nozzle pressure and flow rate, Pp and Q p using the following equations: Pp = Ps + (G p × D p ) − ( Pfp ) estimated , [Pa ] Q p = 4.3 An ( Pp − Pi ) Gp [ , Sm / s ] Where Pfp is the friction loss in the power tubing given in Pa Pfp depends on oil viscosity, water cut, tubing length, production etc When estimating this value, it is usually possible to assume that Pfp is approximately 1% of Pp 64 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Compute the frictional loss in power tubing, Pfp Velocity of fluid through tubing: v= Qp Q → v = 1.27 p , [m / sec] d d5 π ( )2 Where d is the tubing inside diameter in meters Computing the Reynolds number: Re = ρvd , μ where, N R < 2100 , laminar flow N R > 2100 , turbulent flow For laminar flow we have: 64 Re , [Pa ] f L Pfp = ρu d f = For Turbulent flow we have: Pfp = f L ρu d where f is calculated from Haaland’s equation: ⎡ ⎛ ⎞ ⎛ k ⎞ 1 =− log ⎢⎜ ⎟ +⎜ ⎟ f n ⎢⎝ Re ⎠ ⎝ 3.75d ⎠ ⎣ n 1.11n ⎤ ⎥ ⎥ ⎦ alternatively the moody diagram can be used 65 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Compute the nozzle pressure, Pp Using the same equation as in point 7, swapping the estimated pressure loss with the calculated pressure loss: Pp = Ps + (G p × D p ) − Pfp , [Pa ] where G p is the fluid gradient in Pa/m, D p is the pump setting deep in meters, and Pfp is the frictional loss in Pa for the power fluid 10 Compute the volumetric flow rate of the power fluid, Q p Using equation (23) from Appendix A: Q p = 4.3 An ( Pp − Pi ) Gp [ , Sm / s ] 11 Compute the flow rate of the returning fluid, Qr Using the relation: Qr = Qi + Q p , [Sm^3/Sm^3] The returning fluid flow equals the power fluid flow + the intake fluid flow 12 Calculate the gradient of the return fluid, Gr Units in Pa/m: Gr = (Gi × Qi ) + (G p × Q p ) Qr 66 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 13 Calculate the return water cut, Wcd = (WC ) Qi Qr 14 Compute the gas-liquid ratio, GLR = Qi (1 − WC )GOR / Qr Gas production / Total return-liquid prod = GLR in return line 15 Compute the frictional loss in the returning fluid, Pfr , using the appropriate single phase or two-phase model, for GLR ≅ and GLR > respectively For the two-phase case, it is necessary to select an appropriate two-phase correlation, such as Hagedorn & Brown, Aziz, Govier and Fogarasi, Beggs and Brills, or alternatively one might use gas-lift charts For the single-phase we as follows: The single phase flow is divided into two cases: laminar and turbulent flow To distinguish between the two cases we use the Reynolds equation: NR = ρvd h μ where d h is the hydraulic diameter equal to d1 − d N R < 2100 , laminar flow N R > 2100 , turbulent flow 67 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 For laminar flow with GLR ≅ , the total friction loss through the casing-tubing annulus is calculated using the following equations: Fluid velocity through casing-tubing annulus: v = 1.27 * Qr ( d − d ) −1 , [m / s ] 2 where d1 is the casing inner diameter and d is the outer diameter of the tubing The total friction loss through the casing-annulus for the laminar flow case: f = 64 Re Pfr = f L ρu d For turbulent flow at GLR ≅ , we use the following equation: Pfr = f L ρu d where f is calculated from Haaland’s equation: ⎡⎛ 6.9 ⎞ n ⎛ k ⎞1.11n ⎤ 1 =− log ⎢⎜ ⎟ +⎜ ⎟ ⎥ f n ⎢⎝ Re ⎠ ⎝ 3.75d ⎠ ⎥ ⎣ ⎦ alternatively the moody diagram can be used 68 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 16 Calculating a reasonable pump discharge pressure, Pd using the equation: Pd = (G r D p ) + Pfr + Pwh This pressure equals the sum of the return fluid pressure gradient, the pressure friction factor and the flowing wellhead pressure 17 Set the dimensionless nozzle loss coefficient, K n equal to 0.04 Based on the empirical value from the Tulsa thesis 18 Compute K td , the dimensionless loss parameter for the throat and diffuser using equation (43) from Appendix A: K td = 0.1 + (10.88 * 10 −3 )( R p ) −2.33 ( AWR) 0.63 R 0.33 AWR equals GLR, which is calculated in point 14 R is calculated in point nr 5, and R p is given from the relation: Rp = Pd , the ratio of the discharge pressure to the power fluid pressure Pp 19 Compute M, the dimensionless mass flow rate, Explained and calculated in Appendix A (equation 37): M = Qi + Qia × 1.227 Qp where Qi is the volumetric production fluid flow rate in Sm^3/s, Qia is the volumetric gas flow rate at standard temperature and pressure in Mm , and Q p is the power fluid s flow rate in Sm^3/s 69 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 20 Compute the two dimensionless parameters B and C, used to simplify the model Equation 39 and 40 from Appendix A: [ B = R + (1 − R )( M R ) /(1 − R ) ] and C = R (1 + M ) 21 Now, calculate the dimensionless pressure recovery, N Equation 41 from Appendix A: N= B − (1 + K td )C (1 + K n ) − B + (1 + K td )C 22 Recompute the nozzle pressure, Pp from Rearranging equation 19 from Appendix A: Pp = (Pd − Pi ) + Pd N 23 Recompute the pump intake pressure, Pi Rearranging equation 22: Pi = Pd − N ( Pp − Pd ) 24 Recompute surface pump operating pressure, Ps Rearranging equation from point 9: Ps = Pp − G p D p + Pfp 70 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 25 Calculate the non cavitation flow rate, Qsc , in Sm3/s Qsc = Qi ( At − An ) Asm 26 Compute pump efficiency, η , from equation (38) in Appendix A: η = Efficiency = N × M 27 Determine the power requirement of the surface pump Power = Q p × Ps Assuming that the typical efficiency of triplex pumps is 90%, we have: Power = Q p × Ps 0.9 = 1.1 × Q p × Ps , [W] In horsepower: HP = 740W HP = 1.1 × Q p × Ps 740 = Q p × Ps 672 28 Repeat steps to 27 for a different set of throats and nozzles Compare results of successive iterations to obtain the optimum combination: the highest efficiency and lowest horsepower 71 ... Table The example-well from the Tulsa thesis is hereafter called Well A 14 Review and Application of the Tulsa Liquid Jet Pump Model December 2006 15 Review and Application of the Tulsa Liquid Jet. .. of the Jet Pump, giving a brief introduction to the Jet Pump principles, different Jet Pump models and comparison between Jet Pump and other artificial Jet Pump models In chapter 3, and the Tulsa. .. and Application of the Tulsa Liquid Jet Pump Model December 2006 13 Appendixes Appendix A DERIVATION OF THE JET PUMP MODEL IN SI-UNITS The Derivation of the Pump Model performed in the Tulsa thesis