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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.
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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 twophase 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 throatdiffuser 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 throatdiffuser friction factor,
which again depends on the gasoil ratio.
Calculations were performed on a North Sea well with a gasoil 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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 twophase
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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 onshore 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 subsurface 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
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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 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 casingtubing 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.
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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 highpressure 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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 TwoPhase 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 airliquid 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
throatdiffuser, 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
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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 oilfieldunits, but is presented here in SIunits. Conversion
from field to SIunits is a task specifically mentioned in the projectdescription, 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 6162) 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
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)
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 5658). 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 throatdiffuser,
respectively.
nK tdK
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 (AirWaterRatio). For singlephase 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 airwater ratio, equivalent to GOR in a gasoil system.
pR
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 716 are based on data from the thesisexperiment described
in Chapter 3.1 “Development of the Model”. They illustrate jet pump performance under
varying conditions.
Figure 7 is a plot of the throatnozzle loss parameter versus airwaterratio, for three fixed
values of (ratio of discharge pressure to power fluid pressure). The trend shows that an
increasing airwater ratio results in an increasing throatnozzle 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
throatnozzle 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 throatnozzle friction factor increase with increasing
airwaterratio.
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
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
Within the throat, pressure must remain above liquidvapour pressure to prevent throat
cavitation damage. Note that pressure drops below pumpintake pressure as produced fluids
accelerate into the throat mixing zone. If pressure drops below the liquidvapour 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 liquidvapour 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 stepbystep 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 SIunits.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 SIunits, 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
Reynoldsnumber 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 flowregime 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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 examplewell from the Tulsa thesis is hereafter called Well A.
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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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Review and Application of the Tulsa Liquid Jet Pump Model December 2006
16
16
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Reynoldsnumber 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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 34/10C16,
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 . Gasoil 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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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:
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006 Review and Application of the Tulsa Liquid Jet Pump Model December 2006
20
20
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
The annular friction loss for the returning fluid was calculated using PROSPER (table 4), by
setting the return conduit to “casingtubing 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).
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
throatdiffuser 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 jetpump 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.
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Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
2) Calculating power fluid ejection pressure:
)(21 22
prodsuctionprodprode VVPP −−= ρ
3) Pressure at the throatexit:
)]()([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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 gasoil 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 throatdiffuser 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 throatexit, the flow wi
be compressible in the entire throatsection.
The NTNU project model is not adjusted to in
im
to 0. This sets the nozzlethroat 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
9. Conclusion
From the calculations conducted in this project, it is clear that there is a difference between
p models when pumping a twophase 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 throatdiffus
loss factors.
30
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 TwoPhase 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 0.25Gasoil 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 0.35Gasoil 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 “casingtubing 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
igure 3: General Jet Pump Nozzle/Throat combinations (Allan, Moore, Adair, 1989) 6F
Figure 3.1: Pressure profile along Jet Pump crosssection (http://www.weatherford.com/weatherford/groups/public/documents/general/wft007479.pdf)
35
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
Figure 4: Comparison of different artificial lift methods (Jahn, F., Cook, M., Graham, M., 1998)2
36
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 experimentdata from thesis by Jiao1
38
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 experimentdata 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 experimentdata from thesis by Jiao1
39
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 experimentdata 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 experimentdata from thesis by Jiao1
40
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 experimentdata 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
Figure 13: N vs. M graph based on experimentdata from thesis by Jiao1
41
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
Efficiency vs. dimensionless mass flow ratio
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 experimentdata 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
Figure 15: N vs. M graph based on experimentdata from thesis by Jiao1
42
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
Efficiency vs. dimensionless mass flow ratio
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 experimentdata from thesis by Jiao1
Figure 17: The Gullfaks field is a part of the Tampenarea 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
13. Appendixes Appendix A DERIVATION OF THE JET PUMP MODEL IN SIUNITS 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 SIunits the gravitational constant is set to = 1 by choice.
The following derivations of the energybalancemomentum equations are based on the Tulsa
The following assumptions are made for the analysis:
1. Steady State flow
2. Uniform properties of a singlephase 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
The continuity equation states:
ρ
∇
.0
constVAmV
===⋅
Further, we define the following:
,
,/
p
i
p
i
n
i
tn
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 pressuredrop. 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 powersection, 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
[ ])(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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
NMPPmPPm
dpn
idi ⋅=−−
=)()(η ….(22)
Now, the singlephase 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 gasliquid
n (21), and regress on the pump energy loss
oefficients, and to predict a twophase 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 “addup” to bbl/D.
onverting this equation to take SIunits:
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
the SIvalues 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 SIunits:
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 gasoil ratio and water cut:
[ ]( ) [ ])/()1()/(8.21 2.1ppiCCii xGQGWWPGORQM ×+−+=
54
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
Conver ing the relations mentioned earlier and in addition:
ting to SIunits (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 Iunit:
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
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:
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
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
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 (AirWaterRatio). For singlephase flow, the righpRR,
e
Converting the equation to SIunits:
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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 pumpinlet flow rate, iP is the pumpinlhere is th et pressure, and the fluid
gradient at the inlet in Pa/m. GOR is the gasoil ration in Sm3/Sm3.
i iG
63
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
13. Calculate the return water cut,
r
iCcd
QWW )(= Q
14. Compute the gasliquid ratio,
Gas production / Total returnliquid prod. = GLR in return line
15. ning fluid,
using the appropriate single phase or twophase 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 twophase
correlation, such as Hagedorn & Brown, Aziz, Govier and Fogarasi, Beggs and Brills,
r alternatively one might use gaslift 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 singlephase
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
For laminar flow with 0≅GLR , the total friction loss through the casingtubing
owing equations:
Fluid velocity through casingtubing 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 casingannulus 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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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
Review and Application of the Tulsa Liquid Jet Pump Model December 2006
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