Pedersen 2006

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


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 throat-

diffuser 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 33 SmSm , 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


Table of contents

1. Introduction ............................................................................................................................ 1

2. Jet Pump Literature Survey .................................................................................................... 2

2.1 When is artificial lift required .......................................................................................... 2

2.2 Jet Pump compared to other artificial lift methods .......................................................... 3

2.4 Jet Pump principles .......................................................................................................... 4

3. Review of the Tulsa Jet Pump Model .................................................................................... 5

3.1 Development of the model ............................................................................................... 5

3.2 Presentation of the model and its main principles............................................................ 6

4. Tulsa Jet Pump performance .................................................................................................. 9

4.1 Main factors to control pump performance...................................................................... 9

4.2 Sizing of the pump ......................................................................................................... 10

5. Application of the Tulsa Jet Pump Model............................................................................ 13

5.1 Sizing and performance calculations.............................................................................. 13

5.2 Evaluation of results....................................................................................................... 17

6. 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

7. Application of Other Models on a North Sea Well .............................................................. 23

7.1 JSG model calculations .................................................................................................. 23

7.2 NTNU project calculations............................................................................................. 24

8. Discussion ............................................................................................................................ 28

9. Conclusion............................................................................................................................ 30

10. References .......................................................................................................................... 31

11. Tables ................................................................................................................................. 32

12. Figures................................................................................................................................ 34

13. Appendixes......................................................................................................................... 46

III


1. 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, 4 and 5 the Tulsa Jet Pump model is introduced and

documented, and in chapter 6 the model is used for calculations on a North Sea well. Chapter

7 contains calculations using other models, for comparison.

1


2. 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

2


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 3

3

SmSm . Also, the pureness of the power fluid can be relatively low compared to

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 4 shows a comparison for the different artificial lift methods.

3


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 2 illustrates the principle.

4


3. 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 Gas-

Liquid 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, and , 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)

nK tdK

1

5


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 1. 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:

dp

id

PPPPN

−−

≡ ….(19)

The dimensionless mass flow ratio between the suction (producing) fluid and the power fluid

is defined as:

p

i

p

i

nozzle

ake

QQ

QQ

mmM ==≡

ρρint ,

for one phase flow, assuming equal density for the two fluids.

Extended to include gas, the mass flow ratio can be expressed as:

p

iai

QQQM 227.1×+

= ….(37)

6


As shown in the derivation of the pump model (Appendix A).The numerator in equation (37)

describes the total producing fluid mass flow. This includes both liquid and gas, where the

term represents the gas mass flow (derivation page 56-58). iaQ×227.1

In the Tulsa thesis, it is assumed equal density for the power fluid and the produced liquid

phase. For an oil production case with high water cut, it could be argued that in equation

(37) should be adjusted for difference in oil and water density.

iQ

The product of the two parameters N and M is the ratio of the transferred useful power to

consumed input power. Explained mathematically:

MNEfficiency ×==η ….(38)

The model use a functional form of )(MfN = that is based on work by Cunningham4, who

developed this function on mass energy conservation principles. Simplifying the typing of this

function, two component elements are defined:

[ ]222 )1/())(21(2 RRMRRB −−+= ….(39)

22 )1( MRC += ….(40)

where R is the ratio of the nozzle to throat area. As shown in the derivation (Appendix A) N

can be written:

CKBKCKBN

tdn

td

)1()1()1(++−+

+−= ….(41)

where and are the dimensionless loss parameters for the nozzle and throat-diffuser,

respectively.

nK tdK

7


In the above expression of N, the importance of the loss parameters is obvious. The nozzle

loss parameter, , is in this model set to 0.04. This value was estimated in the Tulsa thesis

from optimization based on high pressure data.

nK

tdK is a combination of the loss parameter for the throat and the diffuser , respectively.

The equation for was developed using regression analysis. The analysis was done by a

computer program, performing a multiple linear least squares regression on the logarithms of

the variables and AWR (Air-Water-Ratio). For single-phase flow, the right side of the

equation simplifies to the constant 0.1, as AWR=0. The expression is presented as:

tK dK

tdK

pRR,

33.063.033.23 )())(10*88.10(1.0 RAWRRK ptd

−−+= ….(43)

where R is the ratio of the nozzle to throat area, is the ratio of the discharge pressure to the

power fluid pressure and AWR is the air-water ratio, equivalent to GOR in a gas-oil system.

pR

8


4. Tulsa Jet Pump Performance

4.1 Main factors to control pump performance

The performance of the jet pump can be expressed by comparing different important elements

of the Jet Pump model. Figures 7-16 are based on data from the thesis-experiment described

in Chapter 3.1 “Development of the Model”. They illustrate jet pump performance under

varying conditions.

Figure 7 is a plot of the throat-nozzle loss parameter versus air-water-ratio, for three fixed

values of (ratio of discharge pressure to power fluid pressure). The trend shows that an

increasing air-water ratio results in an increasing throat-nozzle friction factor. Figure 8 are the

same plot as Figure 7, but with AWR in field units. is also expressed in figure 9. Here the

throat-nozzle friction factor is plotted against for five different values of AWR. Clearly it

shows that decrease as increase. Hence, referring to equation (19) in chapter 3.2, the

higher pressure recovery ratio the lower the friction loss in the throat and diffuser. Following,

as N increase and M remains the same, the efficiency will increase. Also, the horsepower

needed to drive the power fluid will decrease as the horsepower requirement varies with N

(Appendix B, step 27, 24, 22).

pR

tdK

pR

tdK pR

Figure 10 shows vs. R with five different values of AWR. As R increase, increase,

the only exception is for AWR=0 where remains constantly equal to 0.1. A common

trend for the plots mentioned is that the throat-nozzle friction factor increase with increasing

air-water-ratio.

tdK tdK

tdK

The Figures 11 and 13 describe the dimensionless pressure recovery ratio vs. dimensional

mass flow rate. As explained in chapter 3, for optimal performance of the pump it is important

to find the values of N and M that together result in the highest efficiency and lowest power

demand for the power fluid. As seen on Figure 11 and 12, increasing nozzle/throat area ratio

9


results in decreasing flow of the production fluid to the power fluid and overall lower pump

efficiency. The nozzle/throat relation is described in more detail in chapter 4.2.

Figure 13 restates the influence of air (gas) in the system. The higher AWR, the lower the

total efficiency(N*M) and production flow rate to fluid flow rate. This can also be seen on

figure 14. The last two figures, 15 and 16 describes N vs. M and efficiency vs. M for different

values of . influence which again influence N. Decreasing leads to decreasing

efficiency and decreasing production fluid flow to power fluid flow.

pR pR tdK pR

4.2 Sizing of the pump Dimensioning a jet pump is an important part of a jet pump installation process. The

nozzle/throat combination defines the degree of pump optimization and performance, another

consideration is that a minimum area of throat annulus is required to avoid cavitation.

Following is a description of these two important elements of Jet Pump sizing:

The nozzle/throat relation Jet Pump performance is well specific and careful selection of the nozzle/throat combination

is therefore necessary to ensure optimum well performance. Due to this fact, manufacturers of

Jet Pumps have made a wide range of nozzles and throats available (Figure 5), where the

optimum combination represents a compromise between maximum oil production and

minimum power fluid rates.

In general, the areas of nozzles and throats increase in geometric progression. Because of this,

fixed area ratios between nozzles and throats, R, can be established. The different

configurations of the nozzle/throat relation are given in Figure 5. A given nozzle (N) matched

to the same number throat (N) will always give the same area ratio, R. This is referred to as an

A ratio. For a given nozzle(N): B, C, D….ratios represent throats with number N+1, N+2 and

N+3 respectively. It is possible to match a given nozzle with a throat which is one size

smaller; this is a A combination (by some manufacturers also referred to as an X

combination). Because of geometric considerations, application of successively smaller

throats is not suitable.

10


A specific nozzle/throat combination is defined by a number, which refers to the nozzle size,

followed by a character which defines the throat size. For example a 10A combination refers

to a 10/10 nozzle/throat combination, a 12B a 12/13 combination and so on (Figure 3).

The A (X)-ratio is for high lift and low production rates compared with the power fluid rate,

while for instance the C ratio is for low lift and high relative production rates. This is

explained in the paper “Jet Pumping Oil Wells” by Petrie, Wilson and Smart:

“Physical nozzle and throat sizes determine flow rates while the ratio of their flow areas

determines the trade off between produced head and flow rate. For example, if a throat is

selected such that the area of the nozzle is 60% of the throat area, a relatively high head, low

flow pump will result. There is a comparatively small area around the jet for well fluids to

enter, leading to low production rates compared to the power fluid rate, and with the energy of

the nozzle being transferred to a small amount of production, high heads will be developed.

Such a pump is suited to deep wells with high lifts.

Conversely, if a throat is selected such that the area of the nozzle is only 20% of the throat

area, more production flow is possible. But since the nozzle energy is being transferred to a

large amount of production compared to the power fluid rate, lower heads will be developed.

Shallow wells with low lifts are candidates for such a pump“ (Petrie, Wilson Smart, 1983,

Allan, Moore, Adair, 1989)5,6

Cavitation and sizing of throat entrance area When sizing a hydraulic Jet Pump for multiphase flow, one of the most important factors is to

avoid cavitation.

Cavitation can damage the Jet Pump, and the throat in particular. When oil reaches the bubble

point, it is saturated with gas, so any lowering of pressure means that more gas will come out

of the solution. The cavitation phenomenon is caused by the collapse of these gas bubbles on

the throat surface as the pressure increases along the jet pump axis (Figure 6). This collapse of

vapour bubbles may cause erosion known as cavitation damage and will decrease the jet

pump performance.

11


Within the throat, pressure must remain above liquid-vapour pressure to prevent throat

cavitation damage. Note that pressure drops below pump-intake pressure as produced fluids

accelerate into the throat mixing zone. If pressure drops below the liquid-vapour pressure,

vapour bubbles will form. The throat entrance pressure is controlled by the velocity of the

produced fluid passing through it. From fluid mechanics we have the Bernoulli equation that

states that as the fluid velocity increase, the fluid pressure will decrease and vica verca.

In order to maintain the throat entrance pressure above the liquid-vapour pressure, the nozzle

and throat combination must be carefully selected. The nozzle and throat flow areas define an

annular flow passage at the throat entrance. This area decides the velocity of the fluid, and

therefore the fluid pressure. The smaller flow area, the higher velocity of the fluid. The static

pressure of the fluid drops as the square of the velocity increase and will reach the vapour

pressure of the fluid at high velocities. This low pressure can cause cavitation. Thus, for a

given production flow rate and a given pump intake pressure, there will be a minimum

annular flow area required to avoid cavitation. (Grupping, Coppes, 1988, Christ, Petrie,

1989,Petrie, Wilson, Smart, 1983)8,7,5

A step-by-step guide for sizing hydraulic Jet Pumps is enclosed in Appendix B. The

procedure was first presented in the Tulsa thesis, and is in this project converted to SI-units.

12


5. Application of the Tulsa Jet Pump Model

5.1 Sizing and performance calculations Following is a calculation example for the Jet Pump Model. This example is originally

presented in the Tulsa thesis, and is included in this project to illustrate the application of the

model. Input data has been converted to SI-units, and calculations have been carried out

following the “step by step”-procedure enclosed in Appendix B.

The two set of calculations (in this project, 5.1 and in the example in the Tulsa thesis) only

differs in the computing of the Reynolds number. In this project, a considerable higher

Reynolds-number was computed, which results in turbulent flow in the power fluid tubing. In

the Tulsa thesis, laminar flow was calculated in the power tubing. In this project the relation

μρud

=Re [SI] is used, while in the Tulsa thesis, the relation μρudN 124Re = [field] is used.

In both cases, the following criteria for flow-regime determination are used: Reynolds

numbers above 2100 implies turbulent flow (transient between 2100 and about 4000) and

below 2100 implies laminar flow. This results in different flow regimes for the two cases, as

the Reynolds number is different.

In the Tulsa thesis, the following calculations are made:

Velocity in power fluid tubing = 4.152 sec/ft

Density of the power fluid = 52.93 3/ ftlbm

Diameter of the tubing = 1.995 inches = 0.16625 ft

Viscosity of the power fluid = 5 cp

1.9065

16625.0152.493.52124124Re =××

==μρudN

The constant 124 is a conversion factor between cP and sec2 ×

×ft

inchlbm

906.1 is the value used in the Tulsa thesis.

13


The above equation assumes that the diameter is in inches, not in feet. This is stated on page

61 of the Tulsa thesis. As seen in the above calculations, feet are used as the diameter unit.

This results in an incorrect Reynolds number.

Using inches as the tubing inside diameter unit, we get:

108735

995.1152.493.52124124Re =××

==μρudN

This is about the same number calculated in this project, therefore it seems like the

calculations using the standard relation μρud

=Re are correct.

A conversion of the input data, from field to SI units, are given in Table 1

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 Review and Application of the Tulsa Liquid Jet Pump Model December 2006

15

15


16

16


5.2 Evaluation of results With the given nozzle/throat configuration the model gives an efficiency of 23% and a power

requirement for the surface pump on 88.5 horsepower (65 kW). Comparing the above results

to the Tulsa thesis example results, it is concluded that they are both correct according to the

Tulsa Jet Pump model. The values from the thesis are for the efficiency 0.2263 23% and for

the power requirement 88 HP. For this example, it is obvious that the difference in the

Reynolds-number mentioned earlier has very little influence on the pump efficiency and

power demand.

≅

However, the above calculations are not performed with the optimal nozzle/throat

configuration. This was found to be 6D (National, Figure 5) which gives efficiency of 23.7%

and power requirement on 82.4 HP.

17


6. Application of the Tulsa Model on a North Sea Well

As presented in the introduction, one of the tasks in this project was to apply the Tulsa model

to a North Sea oil well and suggest if it is suitable for use in the given well. Well 34/10-C-16,

located in the “Gullfaks”-field, was chosen as the basis for the North Sea Well. Well data

were found in the “Gullfaks database” on the IPT computer network. For the following case,

the reservoir pressure was adjusted such that the well became a candidate for artificial lift.

The depth of the well was slightly increased and the well was made vertical to simplify the

case. PROSPER, a well performance, design and optimisation program, was used to make the

inflow and outflow curves. The Characteristics of the well are found in Table 2. The well is

hereafter called Well B.

6.1 Case description Well B is located in the southern part of the “Gullfaks” field in the North Sea (Figure 17). No

pressure maintenance solution has been added, the reservoir pressure is falling and the

production declining. At present time the well is producing 585 dSm3 liquid. With a water

cut of 35% , this gives an oil production on 380 dSm3 (Figure 18 and Table 2&3). To

increase oil production and save the well from depletion, Jet Pumping has been selected as the

artificial lift method. The desired liquid flow rate is set to 1000 dSm3 , which gives an oil

flow rate on 650 dSm3 . Gas-oil ratio for the well is 95 33 SmSm .

For power fluid, processed oil with the same characteristics as the producing oil is selected

(for power fluid oil, GOR=0). The pump supplier is “National” (Figure 5). Surface pump

pressure together with pump efficiency for the given well are to be computed. In Chapter 6.2

the Tulsa Jet Pump Model is used for these calculations.

Inflow and tubing performance curves for the well are found in Figure 18. Data for Well B are

found in Table 2 and data for the performance curves are found in Table 3.

18


6.2 Model calculations The following calculations were performed using different nozzle/throat combinations. The

efficiency/power relations for the different combinations are found in Figure 19. Following is

the calculations for the optimal combination found, 14D:

19


20

20


The annular friction loss for the returning fluid was calculated using PROSPER (table 4), by

setting the return conduit to “casing-tubing annulus” and the return water cut to 20% (step

13), PROSPER calculates an annular friction pressure loss of 38 bar for the return fluid flow

equal to 1769 sm3/sm3 (step 11).

21


6.3 Evaluation of results Choosing nozzle/throat combination D will, as shown in the calculations, give relatively high

production flow compared to the input HP and power fluid flow. However, the nozzle energy

is being transferred to a large amount of production compared to the power fluid rate, so

relatively low heads will be developed. For this case, even though the well is 2900m deep,

low heads will work, as the well is producing prior to the jet pumping with a reasonably high

bottomhole pressure. An efficiency of about 30% seems as a realistic value, but the power

requirement might be a little off as it is pretty sensitive to the “selected surface operating

pressure” value that has to be estimated prior to the calculations.

The values for power fluid rate, surface pump operating pressure, nozzle friction factor and

throat-diffuser friction factor from the calculations in 6.2 were inserted in the Jet Pump

function in PROSPER. Based on this input data, the program computed the plot seen in figure

20. PROSPER illustrates the lift by plotting the pump discharge pressure vs. liquid flow rate.

The Jet Pump Model in PROSPER gave a liquid flow rate of 957 dSm3 . Hence, the

difference between the Tulsa Jet Pump Model and the model used by PROSPER is about

4.3%. Probably a large part of the difference can be explained with the fact that the

bottomhole pressure is adjusted from 271 to 261 bar when using the jet-pump feature in

PROSPER. The above calculations uses 271 bar as bottomhole pressure. Setting the

bottomhole pressure manually to 271 bar after applying the Jet Pump in PROSPER, the

program gives production flow rate of about 1000 dSm3 with the given input data.

Probably is the Jet Pump function in PROSPER based on the Tulsa model or a very similar

model. Anyhow, the program indicates that the Tulsa Jet Pump model gives, in this case,

reasonable results.

22


7. Application of Other Models on a North Sea Well

For a basis of comparison, the following chapters include some other models applied to the

case presented in Chapter 6.1

7.1 JSG model calculations This model is presented in the paper “Modell for strålepumpe” by Jón Steinar Guðmundsson

(01.06.06). The JSG model is a simple jet pump model, derived from the basic Bernoulli

equation for steady state incompressible flow:

.21 2 ConstuP

=+ρ

, and the mass flow relation MLG mmm =+

Neglecting the kinetic energy (very small), we have

L

L

G

G

M

M PPPρρρ

+= , for real systems, efficiency must be added:

L

L

G

G

M

M

P

PP

ρ

ρρη−

= , where the subscript G represents the production fluid.

Gρ is the production fluid (65% oil and 35% water): (0,35)1030 + (0,65)843 = 908 kg/Sm3

Lρ is the density of the power fluid, in this case oil with density 843 kg/Sm3.

The Pump discharge pressure ( ) we have from point 16 of the calculations in Chapter 6.2.

This can also be used here, independent of the models. Pump discharge pressure is 318 bar.

MP

For the power fluid pressure, it is referred to point 9 in Chapter 6.2, where this is calculated to

be 503 bar.

23


Calculating:

3

3

3

/880843)56.01()908(56.0)1(

56.0

843908)58.01(58.0

58.0

)1(

58,05.75.10

5.10

24*60*601*769*843

24*60*601*1000*908

Smkg

xx

x

numeratord

SmSmkg

mmmx

LGM

L

G

LG

G

=−+=−+=

=−+

=−+

=

=+

=+

=+

=

ρααρρ

ρρα

efficiency becomes:

%29289.0

843)318503(

908271

880318

==−

−=

−=

L

L

G

G

M

M

P

PP

ρ

ρρη

This seems like a reasonable result compared to the Tulsa model, but applying the model to

well A, it gives about 45% efficiency, which is very high. Probably is this explained with a

considerable difference between production and power fluid flow. This is further commented

in Chapter 8.

7.2 NTNU project calculations In the NTNU project “Ejektorpumpe” (Moxnes, 2005)9, a Jet Pump model is presented. The

model is based on a note by Harald Asheim from 200410, containing some small

modifications. In this model, calculations are performed for each part of the pump, divided

into inflow, ejection, throat (mixing chamber) and outflow.

The Bernoulli based model assumes incompressible flow. Following is a presentation of how

to apply the model:

1) Fluid velocity in each part of the pump is calculated: tubing power fluid velocity,

ejection power fluid velocity, producing fluid velocity, production fluid suction

velocity, mixed fluid velocity in throat and discharge (outflow) velocity of mixed

fluid. Using AQV =

24


2) Calculating power fluid ejection pressure:

)(21 22

prodsuctionprodprode VVPP −−= ρ

3) Pressure at the throat-exit:

)]()([1suctionthroatprodprodthroatepowerpower

throatethroat VVQVVQ

APP −−−+= ρρ

4) Discharge (outflow) pressure is calculated:

)(21

arg22

edischthroatthroatthroatd VVPP −+= ρ

Where throatρ is assumed to be homogenous

5) Pressure at nozzle inflow:

)(21 22

, powerejectionprodetubingpower VVPP −+= ρ

6) Efficiency of the pump is given by:

power

prod

EE

=η

Where

prodprodprodprodedischproddprod QVPVPE )21

21( 2

arg2 ρρ −−+=

and

poweredischpoweredischpowerpowerpowerpower QVPVPE )21

21( arg

2arg

2 ρρ −−+=

7) Power requirement of surface pump:

psurfacepumpower PQP ×= , [W]

To calculate tubing power fluid flow and pressure, the equations from Chapter 6.2, step 7 are

used. Calculations are performed in Excel, using the input values from Table 2.

25


ompared to the efficiency value calculated in Chapter 6.2, the efficiency here is remarkably

he same calculations were performed on well A from Chapter 5.1, using input data from

C

high. This is further discussed in Chapter 8.

T

Table 1:

26


The efficiency for well A is also high compared to the values calculated with the Tulsa model.

This is further discussed in the next Chapter.

27


8. Discussion

ter 6 and 7, calculations using the different models gave deviant results.

he input data were similar for all three models: well data from table 2 and a preset

ssure

and the NTNU project model differ in the way pressure loss over the pump

calculated. The NTNU model uses velocity calculations and acceleration terms while the

quired surface pump pressure on 312 bar. For comparison, the Tulsa model calculated a

n other

exceeds

00 m/s. According to the “Ejektorpumpe”-project (Moxness, 2005)9 velocities exceeding

). ll

clude gas, which the Tulsa model is. The

portance of the gas-oil relation in the Tulsa model can be shown by setting the GOR equal

s

As presented in Chap

T

reasonable pump surface pressure, used to calculate the power fluid pressure at the nozzle

inflow. This value is further used to compute the flow rate in the power fluid tubing.

Considerable changes in the assumed surface pump pressure have high influence on the

results. When comparing the models, it is important to use the same surface pump pre

assumption.

The Tulsa model

is

Tulsa thesis uses the nozzle and throat-diffuser friction factors to calculate pressure loss.

For well B, the NTNU project model calculated a discharge pressure on 332 bar and a

re

required surface pump pressure on 335 bar for a preset discharge pressure on 318 bar. I

words, the pressure loss over the Tulsa Jet Pump is higher than over the pump in the other

model. Calculations performed on well A with both models show the same trend as mentioned

above, the Tulsa model computes lower efficiency and higher power requirement.

In the NTNU project calculations, the velocity of the mixed fluids at the throat exit

1

mach 0.3 (≈100 m/s) results in compressible flow, which violates the assumption for the

Bernoulli equation. The velocity decreases as the pressure increases throughout the throat

(Figure 3.1 This means that if the velocity is about 100 m/s at the throat-exit, the flow wi

be compressible in the entire throat-section.

The NTNU project model is not adjusted to in

im

to 0. This sets the nozzle-throat friction factor equal to 0.1. For well A, which has a GOR

value of nearly 180, it results in a calculated efficiency of 28.1% and a power requirement of

64 HP. If, in addition, the nozzle friction factor is set to 0, the efficiency for well A exceed

30%, which is about the same that was calculated using the NTNU project model.

28


The friction factors in the Tulsa model are based on data from experiments using water and

air as the two fluid phases. The experiments were conducted under typical field conditions,

luid conditions are calculated. The model computes efficiency

r a pump, given the production fluid, power fluid and discharge fluid pressure and density.

Tulsa model seems to give very good results. It

possible that the PROSPER Jet Pump function is based on the Tulsa model presented here,

however it is not certain that these friction factor values are accurate when handling other

fluids. Further development of the model should probably include experiments with other

fluids and fluid properties.

In the JSG model, none of the f

fo

The model does not take the difference in power and production volume fluid flow into

consideration. It should, however, give quite good results for cases with incompressible fluids

and low power/production fluid flow ratio.

Comparing to the results from PROSPER, the

is

or the improved Tulsa model from 199811, anyhow, the similarity in results shows that the

model is applicable in the field as it is presented in this project.

29


9. Conclusion

From the calculations conducted in this project, it is clear that there is a difference between

p models when pumping a two-phase fluid. The Tulsa model is

presented models are assuming incompressible

from PROSPER shows that the calculations done here are reasonable. Based on this, it

s like the model is applicable to a field situation as it is presented in this project. For the

e two phases. Experimenting with other fluids and using unequal power and production

er

the presented Jet Pum

extended to include gas, while the two other

flow.

Although this project does not include comparison to other models for compressible flow, the

results

seem

North Sea well, presented in Chapter 6.1, the calculations performed with the Tulsa model

gave an efficiency of about 30% and a surface pump power requirement of 444 HP.

To improve the Tulsa model, experiments should be conducted, using different production

and power fluids. In the experiments described in the Tulsa thesis, water and air were used as

th

fluid could contribute to further development of the model. An improved version of the Tulsa

model has already been made, and is presented in the SPE Journal in September 199811.

Among the improvements are more precise values for both the nozzle and the throat-diffus

loss factors.

30


10. References

rmance model for hydraulic jet pumping of two phase fluids”, (Tulsa,

2. Jahn, F., Cook, M., Graham, M.: “Hydrocarbon exploration and production”,

3. Overview of Artificial Lift Systems”, (SPE Article, 1982)

Fluid

rld Oil,

6. : “Design and Application of an Integral Jet Pump/Safety valve in

7. , H.L.:”Obtaining Low Bottomhole Pressure in Deep Wells With

8. .: “Fundamentals of Oilwell Jet Pumping”,

9.

11. ”Improved Two-Phase Model for

1. Jiao, B.: “Perfo

August 1988)

(Nederland, 1998)

Brown, K. E.: “

4. Cunningham, R. G.: “Gas Compression with the Liquid Jet Pump”,(Journal of

Engineering, 1974)

5. Petrie, H.L., Wilson P.M., and Smart, E.E.: “Jet Pumping Oil Wells”, (Wo

November 1983)

Allan, Moore, Adair

a North Sea oilfield”,( SPE article, 1989)

Christ, F.C, Petrie

Hydraulic Jet Pumps”, (SPE Production Engineering, August 1989)

Grupping, A.W. Coppes, J.L.R., Groot, J.G

(SPE Production Engineering, February 1988)

Moxness, V.W: “Ejektorpumpe”, (NTNU Project, 2005)

10. Asheim, Harald: “Ejektorpumpe”(Note),( NTNU, 2004)

Noronha, F.A.F., Franca, F.A., Alhanati, F.J.S.:

Hydraulic Jet Pumps”, (SPE article, September 1998)

31


11. Tables

able 1: Data for Well A, Tulsa thesis example (Jiao, 1988)1 T Input Data Unit [SI] Value Unit [Field] ValuePump Setting depth, Dp m 1828 ft 6000Casing OD m 0.1397 inCasing ID, d1 m 0.1242 in

5.5.4.892

ubing OD, d2 m 0.0603 in 2.375Tubing ID, d5 m 0.050673 in 1.995Tubing coupling OD, d4 m 0.0778 in 3.063Length of tubing m 1828 ft 6000Bottomhole temperature, T °C 62.7 F 145Flowing bottomhole pressure, Pi Pa 6895000 psi 1000Wellhead back pressure, Pwh Pa 689500 psi 100Desired flowrate, Qi Sm^3/s 0.0011 bbl/D 600Water Cut, Wc 0.25 0.25Gas-oil ratio, GOR Sm^3/Sm^3 178.1 scf/STB 1000Oil gravity kg/Sm^3 848 API 35Water gradient, Gw Pa/m 10291 psi/ft 0.455Oil viscosity Pas 0.005 cP 5Ol gradient, Go Pa/m 8324 psi/ft 0.368Power fluid gravity kg/Sm^3 848 API 35Power gradient, Gp Pa/m 8324 psi/ft 0.368

T

Table 2: Well B data for the case presented in Chapter 6 (http://gullfaks.ipt.ntnu.no) Input Data Unit [SI] ValuePump Setting depth, Dp m 2900Casing OD m 0.1778Casing ID, d1 m 0.1607Tubing OD, d2 m 0.1143Tubing ID, d5 m 0.0995Tubing coupling OD, d4 m 0.127Length of tubing m 2900Bottomhole temperature, T °C 83Flowing bottomhole pressure, Pi Pa 27100000Wellhead back pressure, Pwh Pa 5588000Desired flowrate, Qi Sm^3/s 0.0115Water Cut, Wc 0.35Gas-oil ratio, GOR Sm^3/Sm^3 95Oil gravity kg/Sm^3 843Water gradient, Gw Pa/m 10300Oil viscosity Pas 0.00135Ol gradient, Go Pa/m 7100Power fluid gravity kg/Sm^3 843Power gradient, Gp Pa/m 7100

32


Table 3: Data table for the performance curves in Figure 18 (PROSPER generated table, using data from the gullfaks database: http://gullfaks.ipt.ntnu.no).

Table 4: Data table for the return annular liquid flow. This is calculated by PROSPER, setting the return conduit to “casing-tubing annulus” and the return water cut to 20 (step 13, Chapter 6.2). Highlighted are the return liquid flow rate (step 11, Chapter 6.2) and the corresponding friction pressure loss.

33


12. Figures

1Figure 1: The Jet Pump with letters and markings used in the Tulsa model (Jiao, 1988)

Figure 2: Jet pump operating principles (Allan, Moore, Adair, 1989)

6

34


igure 3: General Jet Pump Nozzle/Throat combinations (Allan, Moore, Adair, 1989) 6F

Figure 3.1: Pressure profile along Jet Pump cross-section (http://www.weatherford.com/weatherford/groups/public/documents/general/wft007479.pdf)

35


Figure 4: Comparison of different artificial lift methods (Jahn, F., Cook, M., Graham, M., 1998)2

36


Figure 5: Jet Pump nozzle and throat sizes from 3 manufacturers; Kobe, National and Guiberson ( Petrie, H.L., Wilson P.M., and Smart, E.E, 1983)5

37


Figure 6: Pressure history of produced fluid as it enters and travels through the jet pump. Cavitation occurs where the throat pressure is drawn below the produced fluids vapour pressure (Christ, F.C, Petrie, H.L, 1989)7

Ktd vs. AWR, [SI units]

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 50 100 150 200 250 300

AWR [SCM/SCM]

Ktd

Rp=0,5Rp=0,6Rp=0,7

Figure 7: Ktd vs. AWR graph based on experiment-data from thesis by Jiao1

38


Ktd vs. AWR, [Field units]

0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 250 500 750 1000 1250 1500 1750

AWR [Scf/STB]

Ktd

Rp=0,5Rp=0,6Rp=0,7

Figure 8: Ktd vs. AWR graph in field units, based on experiment-data from thesis by Jiao1

Friction factor at throat and diffuser vs. Ratio of discharge pressure to power

fluid pressure

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7

Rp

Ktd

AWR=18 Scm/ScmAWR=36 Scm/ScmAWR=54 Scm/ScmAWR=0 Scm/ScmAWR=180 Scm/Scm

R=0.233

Figure 9: Ktd vs. Rp graph based on experiment-data from thesis by Jiao1

39


Friction factor at throat and diffuser vs. ratio of nozzle area to throat area

0

0,2

0,4

0,6

0,8

1

1,2

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

R

Ktd

AWR=18 Scm/ScmAWR=36 Scm/ScmAWR=54 Scm/ScmAWR=0 Scm/ScmAWR=180 Scm/Scm

Rp=0.

Figure 10: Ktd vs. R graph based on experiment-data from thesis by Jiao1

Deminensionless pressure recovery ratio vs. dimensionless mass flow ratio

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,5 1 1,5 2 2,5

M

N

R=0.2331R=0.2994R=0.3786R=0.3817

AWR=17.8 Scm/ScmRp=0.6Kn=0.04

Figure 11: N vs. M graph based on experiment-data from thesis by Jiao1

40


Efficiency vs. dimensionless mass flow ratio

0

5

10

15

20

25

30

35

0 0,5 1 1,5 2 2,5

M

Effic

ienc

y [%

]

R=0,2331R=0,2994R=0,3786R=0,3817

AWR=17.8 Scm/ScmRp=0.6Kn=0.04

Figure 12: Efficiency vs. M graph based on experiment-data from thesis by Jiao

1

Dimensionless pressure recovery ratio vs. dimensionless mass flow ratio

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 0,5 1 1,5 2 2,5

M

N

AWR=18 Scm/ScmAWR=36 Scm/ScmAWR=180 Scm/ScmAWR=0 Scm/Scm

R=0.2331Kn=0.04Rp=0.7


41



0

5

10

15

20

25

30

35

40

0 0,5 1 1,5 2 2,5

M

Effic

ienc

y, [%

]

AWR=18 Scm/ScmAWR=36 Scm/ScmAWR=1000 Scm/ScmAWR=0

R=0.2331Kn=0.04Rp=0.7

igure 14: Efficiency vs. M graph based on experiment-data from thesis by Jiao1F

Dimensionless pressure recovery ratio vs. Dimensionless mass flow ratio

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0 0,5 1 1,5 2 2,5

M

N

Rp=0,7Rp=0,6Rp=0,5Rp=0,4

R=0.2331Kn=0.04AWR=18 Scm/Scm


42



0

5

10

15

20

25

30

35

0,1 0,3 0,5 0,7 0,9 1,1 1,3 1,5 1,7 1,9 2,1

M

Effic

ienc

y, [%

]

Rp=0,7Rp=0,6Rp=0,5Rp=0,4

R=0.2331Kn=0.04AWR=18 Scm/Scm

Figure 16: Efficiency vs. M graph based on experiment-data from thesis by Jiao1

Figure 17: The Gullfaks field is a part of the Tampen-area in the North Sea. Well B is locatedin “Gullfaks sør”, found on the picture to the right. Gullfaks is operated by Statoil ASA

(www.statoil.com)

43


44

Figure 18: Performance curves for Well B. The two pictures differ in the axis. The

nu.no)

uppermost picture plots pressure vs. liquid rate (water and oil). The lowermost plots pressure vs. oil rate. (PROSPER generated plots, using data from the gullfaks database: http://gullfaks.ipt.nt


Efficiency vs. power requirement

0

5

10

15

20

25

30

35

0 200 400 600 800 1000 1200

Power, [HP]

Effic

ienc

y, [%

]

Nozzle 13Nozzle 14Nozzle 15Nozzle 16Nozzle 17Nozzle 18Nozzle 19Nozzle 20

14D

14

14

13D

13

15C

15

16A

16C 17

17

2019

19

18

18C

Figure 19: Pump performance curves for well B, based on trials with different nozzle/throat combos.

Figure 20: Jet Pump performance curves. PDP = Pump Discharge Pressure. (PROSPER generated)

45


13. Appendixes Appendix A DERIVATION OF THE JET PUMP MODEL IN SI-UNITS The Derivation of the Pump Model performed in the Tulsa thesis is based on the field unit

system, and the fundamental physical constant is used. This constant expresses the

proportionality between force and momentum change. In the field unit

system has been given an experimental value; 32.1739

cg

, cg2sec⋅

⋅lbf

lbmft (Reynolds & Perkins,

1970). In SI-units the gravitational constant is set to = 1 by choice.

The following derivations of the energy-balance-momentum equations are based on the Tulsa

The following assumptions are made for the analysis:

1. Steady State flow

2. Uniform properties of a single-phase fluid

3. Incompressible fluid

4. Pressure constant over the entire section

5. Zero thickness of the nozzle wall at the exit

6. Complete mixing at the exit of the throat

7. No space between the nozzle exit and the throat entrance

cg

thesis.

46


The continuity equation states:

ρ

∇

.0

constVAmV

===⋅

Further, we define the following:

,

,/

p

i

p

i

n

i

tn

QQ

QQ

mmM

AAR

==≡

≡

ρρ

,tin AAA ≡+

….(1)

(3)

Ap

we have:

....(2)

….

plying the extended Bernoulli equation:

lossgzVPVin

ininout

outout −++=++22

22

ρρ

e choose to neglect the gravitation terms in this derivation, as they represent a

significant pressure-drop. The equation simplifies to:

gzP

W

in

nnn VMRMRVV )1( +=+

nnntt

nn

nnii

t

R

AmAAV

VR

MRRA

MRm

MmMmm

m

/

,1)1(

,

=

=

⋅−

=−

⋅

⋅===

=

ρ

ρ

in mm +

Vtnti RAAAA )1()( −− ρρρ

n

ninnint mmAmmmm /+=

+=

tntt AAAA ρρρ

=

47


48

,21 2

2outin KVVPP ∑ ⎞⎛+

Δ=

−

,2

)1(

222

22

nnep

nnneP

VKPP

VKVPP

ρ

ρρ

′+=−

′+=−

2 ii⎟⎠

⎜⎝ρ

….(4)

Applying the equation above to the power-section, we find:

….(5)

Where is the theoretical friction factor at nozzle (dimensionless).

take section:

….(6)

Substituting equation (2) into equation (6) we have for the intake section:

nK ′

For the in

22

22

)1(2)1( niei V

RRMKPP

−+=−

ρ ….(7)

arge section:

….(8)

for the disch

Substituting equation (3) into equation (8) we have for the discharge section:

222

2)1()1( ndtd VMRKPP ρ

+−=− ….(9)

,2

)1(

222

22

iiei

iiiei

VKPP

VKVPP

ρ

ρ ρ

+=−

+=−

2

22

2)1(

22

tdtd

tdttd

VKPP

VKVP ρ ρP

−=−

−=−

ρ


The momentum equation states:

Applying the above equation to the mixing section (throat), we get:

FmVPA −Δ=Δ ….(10)

2

2) tttttiinntet VAKVmVmVmAPP ρ

−−+=− ….(11)

earranging:

(

R

=− et PP 2

2 ttt

tt

t

ii

t

nn VKAVm

AVm

AVm ρ

−−+

Now, using the continuity equation, the equation above becomes:

21

2

2222

2222

ρρρρ

ρρρρ

tttt

ni

t

nn

ttttt AAA

⎞⎛⎞⎛

ttiinnet

VKVAAV

AAV

VKAVAVAVP

−−⎟⎟⎠

⎜⎜⎝−+⎟⎟

⎠⎜⎜⎝

−−+=−

Further, this gives:

P

=

2222

2)1( tttinet VKVRVRVPP ρρρρ −−−+=− ….(12)

Substituting equations (2) and (3) into equation (12), we obtain:

222222

22

2

)1()1( nnet V

RRRVPP −

−−+=− ρρ

22

)1(2

)1( ntn VMRKVMR

R

+−+ρρ

M

49


Therefore,

⎥⎦

)1)2((

MKt ⎤

⎢⎣

⎡++−

−+=− 22

222 (

)122

2R

RRMRVPP net

ρ ….(13)

ince

e get:

S

)()( eiepip PPPPPP −−−=−

W

⎥⎦

⎤⎢⎣

⎡−

+−′+=− 2

22

22

)1()1(1

2 RRMKKVPP nnip

ρ ….(14)

or a well rounded entry , hence 02 ≅KF

2

22

2 )1(1

2RRM

V

PPK

n

ipn −

+−−

=′ ρ

We define:

2

22RMKK nn +≡′)1( R−

Where the effective nozzle loss parameter, , is defined as nK

1

22−

−= ip PP

n

n

VK ρ

50


Using the identities:

….(15)

nd

….(16)

ombining equations (7),(9),(13) and (15), we get

)()()( eiettpip PPPPPPPP −−−+−≡−

a

)()( idipdp PPPPPP −−−≡−

C

,

)1()1()1()2(

122)1(1( 2

2222

⎢⎢ +−

=M

RM ρρ)

221

)1()1()2()1(

22

22)1()1(

2

2222

222

2222

2222

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎣

⎡

−+−++−

−++

−

⋅+−⎥⎦

⎤⎢⎡

++−−

+

⋅++−=−

RRMKMRK

RRMRRK

VVR

KMRKRRMR

VVMRKPP

it

d

nn

it

nndidρ ρ

⎣

and

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+−

+++−−

+=−

2

22

2222

2

)1()1(

)1()1(1

22

2RRMK

MRKKRRMR

VPP

i

dt

nidρ ….(17)

fig nd in practice negligible, we set

.

bining equation (14),(16), and (17) we obtain:

Since the coefficient iK is a theoretical con uration factor a

K 0=i

Com

51


[ ])(12

)()1(

)1(12

2

2 id PP ⎥⎦ ….(18)

222

idnn

inndp

PPKV

RRMKKVPP

−−+

=⎤

⎢⎣

⎡−−

−+−′+=−

ρ

ρ

mensionless recovery ratio, N, is defined as:

A di

dp PPN

−≡ id PP − ….(19)

Substituting equations (17),(18) into (19), we obtain:

NumeratorKRRMKMRKK

RRMR

Nn

idt

−+−

+−+++−−

+=

1)1(

)1()1()1(1

22 2

2222

22

….(20)

Merging coefficients into a single and setting dt KK + tdK 0=iK , we rearrange

e numerator. Equation (20) becomes: th

[ ]NumeratorK

MRKRRMRRNn

td

−+++−−−+

=1

)1()1()1/())(21(2 22222

….(21)

Calculating the jet pump efficiency, we express:

Output power = )( ii PPm

−ρ

d

nd a

)( dpn PPm

−ρ

Input Power =

Hence, we express the efficiency, η , as

52


NMPPmPPm

dpn

idi ⋅=−−

=)()(η ….(22)

Now, the single-phase model defined by equation (21) and (22) must be modified to include

as. Jiao’s approach to do so, is to first define the volumetric flow rate for the gas-liquid

n (21), and regress on the pump energy loss

oefficients, and to predict a two-phase pressure recovery factor, N. The regression is

on pump geo perties of the

mixture.

In the Tulsa derivation, power fluid flow rate (bbl/D) is given by:

g

mixture, then substitute this flow rate into equatio

tdK nKc

metric factors and operating conditions, not fluid physical pro

pipnp GPPAQ )(832 −=

This is an empirical eq so the units us in the equatiouation, ed n do not “add-up” to bbl/D.

onverting this equation to take SI-units:

Steps

C

ftinchft

PsiPsiinch2 1. Units in field unit system: 2=

its in SI: mmm

PaPam 22 = Un

. Unit relations: 2

smDbblmfoot

minch

/10*84013.1/13048.0100064516.01

36

22

−=

==

3. Calculating the relation between the unit systems, using

53


the SI-values above (2):

00516629.010*84013.1*00035618.0

00035618.0)552086.0)(00064516.0(6 =→=

==− xx

mAn

4. Now, we multiply the original equation by x, and change the units to sm3

⎥⎦

⎢⎣

−=

sm

GPPAQ

p

ipnp

3

,)()00516629.0)(832( ⎤⎡

ence,

he equation in SI-units:

H

T

p

ipnp G

PPAQ

)(3.4 −= ….(23)

….(24)

becomes

The dimensionless are ratio is:

tn AAR /≡

The dimensionless mass flow ratio

),() GQ ⎤(p

i

p

giGQ

QM ×⎥⎦⎢⎣⎡ +

= ….(25)

This can be expressed in terms of gas-oil ratio and water cut:

[ ]( ) [ ])/()1()/(8.21 2.1ppiCCii xGQGWWPGORQM ×+−+=

54


Conver ing the relations mentioned earlier and in addition:

ting to SI-units (input), us

STBScf

SmSm 6146.51 3

3

= = GOR

This leads to:

1 Psi = 6894.757 Pa

[ ]( )[ ]=

×+−+=− ))/()(G10*84013.1(

)1()/())757.6894)(6146.5((8.21*)10*54.5(6

2.12.15

ppi

CCii

xGQ

WWPGORQM

[ ]( )[ /()(10*84013.1( 6

i

xQG− ]))

)1()/)(10*9.8(1*)10*54.5( 2.155

ppi

CCi

G

WWPGORQ ×+−+= ….(26)

A cavitation correction fo ga d. Assuming choked flow into the throat annulus

around the power fluid jet, additional area required to pass the gas is:

M

r s is require

( )icig PGORWQA 24650/)1( −=

Converting o I-unit:

t S

( )( ) ( ) =−= icig WQA 6146.5)1)(10*434.5()0254.0( PGOR 24650/757.689452

))(550)(1(i

ci PGORWQ − ….(27)

he efficiency becomes: T

⎥⎥⎦

⎤

⎢⎢⎣

⎡

××

×⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

=×=pp

ii

dp

id

GQGQ

PPPPMN

)()(η ….(28)

55


The minimum area to avoid cavitation in square inches is:

( ) 24650/(GORA ⎥⎦

⎤⎢⎣

⎡−+= )1

6911

ici

iism PW

PGQ

In square meters:

( )( ) ( ) )0254.0(24650/757.68946146.5)1(3048.06911)

10*84013.11( 2

6 ⎥⎦

⎤⎢⎣

⎡−+= − ic

i

iism PGORW

PGQA

= ⎥⎦

⎤⎢⎣

⎡−+= ))(550)(1(28.0

ic

i

iism P

GORWPGQA ….(29)

In practice, it is necessary to use an equation of state like the idea as law orking

curves for pump efficiency. From the ideal gas law, air density is:

l g to predict w

,RTPM

air =ρ ….(30)

Standard conditions(SI):

olecular weight

R= 8.314472

P = 101325 Pa, atmospheric pressure

M= 28,97 g/mol, m

)()( 3

molKPam

⋅⋅

T= 288 K

he mass flow rate of air through pump at standard conditions in Field units is given as: T

⎥⎦⎤

⎢⎣⎡=⎥

⎦

⎤⎢⎣

⎡×⎥

⎦

⎤⎢⎣

⎡=×=

Dlbm

ftlbm

DftQQ airairakeiam 3

3

,int ρ

hen is given in iaQD

Mscf , we use: D

Mscf x 1000 = D

scf w

56


hence,

, where (flow rate of air through pump) is in iaQD

Mscf ⎥⎦⎤

⎢⎣⎡××=

DlbmQQ iaiam ,0763.01000

or standard conditions in SI, we have: F

33 225.11225288*314472.897.28*101325

mkg

mg

RTPM

air ====ρ

⎥⎦⎢⎣ s⎤⎡××=××=

kgQQQ iaairiaiam ,225.110001000 ρ ….(31)

Where iaQ is in s

.

Mm3

The mass flo

w can be expressed as:

powerfluidwater

airakeairiwater

QM

ρQQ

)()( ,int

powerfluidwater

airakeair

powerfluidwater

iwater

QQ

QQ

ρρ

ρρ

+ ρρ ,int=

+=

where ⎥⎦⎤

⎢⎣⎡= 3999mkg

waterρ and ⎥⎦⎤

⎢⎣⎡=airρ 3225.1mkg

term in the above equation simplifies to

The second

powerfluid

ia

powerfluid

ia QkgQQ 227.1)225.11000( ×××ρ

powerfluidwater

airakeair

Qs

kgQs

Q )999(),int =

×=

ρ

The first term simplifies to

(

)(water

iwater

ρ powerfluidQQρ = )(

Q

powerfluid

iQ

57


Hence, the equation for mass flow becom s:

e

powerfluidQairakeake QQ

M227.1,intint ×+

= ….(32)

The consumed input hydraulic power becomes:

….(33)

hydraulic power is expressed:

)( iawiidiawiw QQPPQQPQ

,)()( pdppdppC QPPQPQPE −=−=

while the transferred useful

(iiaidt QPE ))(() − +=−− ….(34)

ubstituting equation (32) into equation (34):

….(35)

Hence, the pump efficiency is:

+=

S

)227.1)(( iaiidt QQPPE ×+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛ ×+

=−

×+−=

−+−

=

==powecopowertransferreη

p

iai

iaiidiawiidt

QQQ

QQPPQQPPErinputnsumed

usefuld

227.1

)227.1)(())(( ….(36)

This efficiency restates the efficiency computed by equation (28), using:

pdppdpC QPPQPPE )()(

N

p

iai

QQQM 227.1×+

= ….(37)

58


From equation (36) we see that N and M gives us the efficiency:

MNEfficiency ×==η …

iao’s Jet Pump Model uses a functional form for

.(38)

J )(MfN = that is based on work by

on mass energy conservation principles.

implifying the typing of this function, we define two component elements:

Cunningham(3), who developed this function

S

[ ]222 )1/())(21(2 RRMRRB −−+= ….(39)

+= ….(40)

here R is the ratio of the nozzle to throat area).

sing equation (39) and (40), equation (21) becomes:

22 )1( MRC

(w

U

CKBK tdn )1()1( ++−+CKB td )1( +−

= ….(41)

Where is the effective friction factor at nozzle, and is the friction factor at throat and

diffuser.

is in this jet pump model set to 0.04. This value was selected by Jiao from optimization

based on high pressure data.

td−−+= ….(42)

here

N

nK tdK

nK

tdK is given by:

33.063,033.23 )())(10*67.3(1.0 RAWRRK p

Wp

dp P

PR =

59


Equation 42 is developed by Jiao, using regression analysis. The analysis was done by a

omputer program, performing a multiple linear least squares regression on the logarithms of

the variables and t side of the

quation simplifies to the constant 0.1, as AWR=0.

c

AWR (Air-Water-Ratio). For single-phase flow, the righpRR,

e

Converting the equation to SI-units:

33.063.033048.0

63.033.23 )15898.0()())(10*67.3(1.0 RAWRRptd−−+=

Rearranging:

−−+= ….(43)

K

33.063.033.23 )())(10*88.10(1.0 RAWRRK ptd

60


Nomenclature

iA Area of flow path at pump suction, sq. m

Area of flow path at nozzle exit, sq. m

th at throat, sq. m

Minimum flow area throat annulus to avoid cavitation, sq. m

Throat annulus area, sq. m

Air water ratio, Sm^3/Sm^3

nA

tA Area of flow pa

smA

sA

AWR

B Parameter used in computing N, dimensionless

Parameter used in computing N, dimensionless

Depth to the pump, m

Transferred useful hydraulic power, Watt

Consumed input hydraulic power, Watt

Gradient of pump intake fluid, Pa/m

Gradient of oil, Pa/m

Gradient of pump power fluid, Pa/m

Gradient of water, Pa/m

Friction factor at throat, dimensionless

Friction factor at diffuser, dimensionless

Friction factor at throat and diffuser, dimensionless

' Friction factor at nozzle (theoretical), dimensionless

Friction factor at nozzle (effective), dimensionless

C

pD

tE Ec

Gi

Go

G p

Gw

Kt

K

d

K

td

K

n

K

n

61


M

Dimensionless mass flow ratio

Mass flow rate through pump suction, kg/s

actor

te, Sm^3/s

Q d conditions, Mm^3/s

id through pump, Sm^3/s

low rate, Sm^3/s

R roat area

pressure,

mi

nm Mass flow rate through nozzle exit, kg/s

tm Mass flow rate through throat, kg/s N Dimensionless pressure recovery f

dP Pump discharge pressure, Pa P Pressure at throat entrance, Pa e

iP Pump intake pressure, Pa

s Static pressure, Pa P

p Power fluid pressure at nozzle entrance, Pa P

dQ Pump discharge flow ra

gQ In situ gas flow rate, Sm^3/s

iQ Flow rate at pump intake, Sm^3/s

iam Mass intake flow rate of air at standar

Flow rate of power flupQ

rQ Flow rate of returning fluid, Sm^3/s

scQ Maximum noncavitating f

Dimensionless ratio of nozzle area to th

dP / pP pR Ratio of discharge pressure to power fluid

iV Fluid velocity through pump intake, m/s

nV Fluid velocity through nozzle, m/s

tV Fluid velocity through throat, m/s

62


Appendix B

Sizing of Hydraulic Jet Pumps

he following text is a step by step procedure for sizing downhole liquid Jet Pumps. Most of

th equ tions u ed are echanical and momentum equations in

p end x A, “ erivat Model”.

1. Select the type of hydraulic jet pump installation:

nnulus will be the return conduit.

2. Compute the pump intake fluid gradient ( ,

Step by step procedure

T

e a s derived from basic fluid m

A p i D ion of the Jet Pump

If a casing type open system is selected, the a

iG )

oCCwi GWWGG )1()( −+×= , ⎥⎦⎤⎡Pa

⎢

Where is the water cut, and and are the water and oil gradients in Pa/m.

. omp e

the minimum throat annulus area to avoid cavitation. Using the equation:

⎣ m

CW wG oG

3 C ut , smA

⎥⎦

−+ ))(550)(1(i

ci

ii P

WP

A , ⎤

⎢⎣

⎡= 28.0sm

GORGQ [ ]2m

W Q e pump-inlet flow rate, iP is the pump-inlhere is th et pressure, and the fluid

gradient at the inlet in Pa/m. GOR is the gas-oil ration in Sm3/Sm3.

i iG

63


nt AA − 4. Using the manufacturers part list, select a value of

that exceeds , to avoid cavitation:

) >

Where s the area of flow path at nozzle

exit. Both values are in squa

Manufacturer tables are found as figure 5.

6. Select a reasonable value for a surface operating pressure,

The choice depends upon the available pressure of the surface pump, the jet pump

ance.

7. Estimate Nozzle pressure and flow rate, and

)()( −

smA

( nt AA − smA

tA i of flow path at throat and nA is the area

re meters.

5. Compute the dimensionless ratio of nozzle to throat area, R

tn AAR /≡

sP

setting depth, and other properties of the well, the fluid and the inflow perform

pP pQ

using the following equations:

, [ ]a Pestimatedfpppsp PDGPP += ×

p

ipnp G

PPAQ )(3.4 −= , [ ]sSm /3

Where is the friction loss in the power tubing given in Pa.

depends on oil viscosity, water cut, tubing length, production etc.

fpP

fpP

When estimating this value, it is usually possible to assume that fpP is approximately

1% of pP

64


8. Compute the frictional loss in power tubing, fpP

Velocity of fluid through tubing:

[ ]sec/,27.1)

2(

255

mQ

vQ

v p=→=π

2 dd

p

eters.

omputing the Reynolds number:

Where 5d is the tubing inside diameter in m

C

Re 5

μρvd ,=

where,

2100<RN , laminar flow

2100>RN , turbulent flow

For laminar flow we have:

2

2

Re64

udLfP

f

fp ρ=

=, [ ]Pa

For Turbulent flow we have:

2ufP ρ= 2 d

Lfp

where f is calculated from Haaland’s equation:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−=

nn

dk

nf

11.1

75.3Re9.6log8.11

alternatively the moody diagram can be used.

65


9. ompute the nozzle pressure,

Using the same equation as in point 7, swapping the estimated pressure loss

with the calculated pressure loss:

,)( −

C pP

[ ]PaPDGPP fpppsp ×+

is the fluid gradient in Pa/m is the pump setting deep in meters, and

ictional loss in Pa for the power fluid.

10. Compute the volumetric flow rate of the power fluid,

Using equa ppendix A:

=

where G p , pD

fpP is the fr

pQ

tion (23) from A

pnp G

ip PPAQ )(3.4 −= , [ ]sSm /3

11. Compute the flow rate of the returning fluid,

Using the relation:

+= , [Sm^3/Sm^3]

he returning fluid flow equals the power fluid flow + the intake fluid flow

12. Calculate the gradient of the return fluid,

nits in Pa/m:

rQ

Q pir QQ

T

rG

U

r

ppiir Q

QGQGG

)()( ×+×=

66


13. Calculate the return water cut,

r

iCcd

QWW )(= Q

14. Compute the gas-liquid ratio,

Gas production / Total return-liquid prod. = GLR in return line

15. ning fluid,

using the appropriate single phase or two-phase model, for

rCi QGORWQGLR /)1( −=

Compute the frictional loss in the retur frP ,

0≅GLR and GLR > 0

respectively. For the two- ecessary to select an appropriate two-phase

correlation, such as Hagedorn & Brown, Aziz, Govier and Fogarasi, Beggs and Brills,

r alternatively one might use gas-lift charts.

we do as follows:

The single ph d into two cases: laminar and turbulent flow. To

distinguish between the two cases we use the Reynolds equation:

phase case, it is n

o

For the single-phase

ase flow is divide

ρμ

hR

vdN =

where is the hydraulic diameter equal to hd 21 dd −

, laminar flow

, turbulent flow

2100<RN

2100>RN

67


For laminar flow with 0≅GLR , the total friction loss through the casing-tubing

owing equations:

Fluid velocity through casing-tubing annulus:

r /,)(*27.1 122

21

−−=

where is the casing inner diameter and is the outer diameter of the tubing.

The total friction loss through the casing-annulus for the laminar flow case:

annulus is calculated using the foll

[ ]smddQv

1d 2d

Re64

=

f

2uLfP ρ= 2 d

For turbulent flow at 0≅GLR , we use the following equation:

fr

2

2u

dLf Pfr ρ=

where f is calculated from Haaland’s equation:

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−=

nn

dk

nf

11.1

75.3Re9.6log8.11

lternatively the moody diagram can be used.

a

68


16. Calculating a reasonable pump discharge pressure,

This pressure equals the sum of the return fluid pressure gradient, the pressure friction

factor and the flowing wellhead pressure.

18. Compute , the dimensionless loss parameter for the throat and diffuser

using equation (43) from Appendix A:

10(1.0 RAWRRK ptd−−+=

l

ulated in point nr. 5, and is given from the relation:

dP

using the equation:

whfrprd PPDGP ++= )(

17. Set the dimensionless nozzle loss coefficient, nK equal to 0.04

Based on the empirical value from the Tulsa thesis

tdK

33.063.033.23 )())(10*88.

AWR equals GLR, which is ca culated in point 14.

R is calc pR

p

dp P

PR = , the ratio of the discharge pressure to the power fluid pressure

19. Compute M, the dimensionless mass flow rate,

Explained and calculated in Appendix A (equation 37):

p

iai

QM QQ 227.1×+

where is the volumetric production fluid flow rate in Sm^3/s, is the volumetric

gas flow rate at standard temperature and pressure in

=

iQ iaQ

sMm3

, and is the power fluid

flow rate in Sm^3/s.

pQ

69


20. imensionless parameters B and C, used to simplify the model

quation 39 and 40 from Appendix A:

Compute the two d

E

[ ]222 )1/())(21(2 RRMRRB −−+=

21. ow, calculate the dimensionless pressure recovery, N

and

22 )1( MRC +=

N

Equation 41 from Appendix A:

CKBKCKBN

tdn

td

)1()1()1(++−+

+−=

22. ecompute the nozzle pressure, from R pP

Rearranging equation 19 from Appendix A:

( )d

idp P

NPPP +

−=

23. ecompute the pump intake pressure, iP R

Rearranging equation 22:

)( dpdi PPNPP −−=

24. Recompute surface pump operating pressure,

sP

Rearranging equation from point 9:

fpppps PDGPP +−=

70


25. Calculate the non cavitation flow rate, scQ , in Sm3/s

( )sm

ntisc A

AA −Q=

26. ompute pump efficiency,

Q

C η , from

ion (38) in Appendix A: equat

MNEfficiency ×==η

27. Determine the power requirement of the surface pump

ower =

Assuming that the typical efficiency of triplex pumps is 90%, we have:

Power =

P sp PQ ×

spsp PQ

PQ××=

×1.1

9.0, [W]

1 HP = 740W

In horsepower:

6727401.1 spsp PQPQ ×

=××

HP =

28. different set of throats and nozzles.

ompare results of successive iterations to obtain the optimum combination:

the highest efficiency d lowest horsepower.

Repeat steps 3 to 27 for a

C

an

71

Pedersen 2006

Documents

Transcript of Pedersen 2006

situbondo 2006

komunika 20 2006

Modul Path 2006

BAB II LANDASAN TEORI 2.1 Kesadaran Multikultural 2.1.1 ...repository.uksw.edu/bitstream/123456789/14314/2/T1_132012017_BAB II... · 2.1.1 Pengertian Kesadaran Multikutural . Pedersen,

ISRO 2006-2015

emiten 2006

SALINAN LAMPIRAN IV PERATURAN MENTERI PENDIDIKAN … filePrakarya Prakarya Prakarya 2006, 2013 2006, 2013 2006, 2013 2006, 2013 2006, 2013 2006, 2013 2006, 2013 2006, 2013 2006, 2013

SPM OBGYN 2006

Prastowo 2006

Pcum 2006 cajigas

4. Outlook Perekonomian 2005 -2006 - bi.go.id · Outlook Perekonomian 2005 -2006 25 4. Outlook Perekonomian 2005 -2006 Prospek ekonomi Indonesia tahun 2005-2006 mengalami sedikit

PRIMARIA 1986-2006

PEDER OLSEN SOLHJEMSEIE - Korseberg · 2017-03-10 · PEDER OLSEN SOLHJEMSEIE Peder, født 6. desember 1812, var sønn av Ole Pedersen Tullut i Solhjem. (Vågå kirkebok 1811.1814,

LEMDIKANAS ---2006----

KRISTINE HOELGAARD PEDERSEN CV

SIM DEPDIKNAS 2006 20 Nopember 2006

Jateng DDA 2006

PERATURAN_AKADEMIK UGM 2006

ATURAN 2006

romsyatin 2006