SPE 69406

Dynamic Flow Performance Modeling of a Gas-Lift Valve

J. G. Faustinelli, PDVSA EPM, D. R. Doty, University of Tulsa

Copyright 2001, Society of Petroleum Engineers Inc.

order to keep up the liquid production, an artificial lift method
This paper was prepared for presentation at the SPE Latin American and Caribbean Petroleum must be used.
Engineering Conference held in Buenos Aires, Argentina, 25–28 March 2001.
One of the most important artificial lift methods is gas-lift11.
This paper was selected for presentation by an SPE Program Committee following review of
information contained in an abstract submitted by the author(s). Contents of the paper, as
Gas is continuously injected from the annular space into the
presented, have not been reviewed by the Society of Petroleum Engineers and are subject to tubing through a gas lift valve. The gas - liquid ratio of the
correction by the author(s). The material, as presented, does not necessarily reflect any
position of the Society of Petroleum Engineers, its officers, or members. Papers presented at production fluid increases between the valve and the surface.
SPE meetings are subject to publication review by Editorial Committees of the Society of The reduced flowing pressure gradient allows a greater
Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper
for commercial purposes without the written consent of the Society of Petroleum Engineers is drawdown and thereby more liquid production.
prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300
words; illustrations may not be copied. The abstract must contain conspicuous For an efficient gas-lift design, a proper selection of
acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.
Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.
mandrels, gas-lift valves, and downhole equipment must be
matched with the inflow performance of the reservoir and the
volumetric availability of the injection gas. Production design
Abstract models that allow accurate calculations of the opening and
A new unified model that predicts the flow performance of closing pressures, and selection of port sizes play a vital role
nitrogen charged injection pressure operated gas-lift valves for the successful spacing of the valves and the unloading
has been developed and is able to predict the volumetric flow process.
rate more accurately than previous models. The new model The gas-lift valve is the most important component in the
exhibits orifice, throttling and transition flows which process because it acts as a flow regulator controlling the
encompass all of the flow regimes encountered in a gas-lift amount of gas passing into the tubing12. For this reason, the
valve. The model is based upon fundamental physical Tulsa University Artificial Lift Projects (TUALP) has directed
principles and has the advantage of describing the internal a sequence of studies1,2,3,4,5,6,7,8,9,10 in order to better predict a
flow mechanics occurring at the ball-seat and port locations. gas-lift valve’s flow performance.
The model predicts an isentropic process from the upstream
condition to the minimum flow area. The model also predicts a Valve Performance Modeling
pressure recovery associated with the sudden gas expansion Several gas-lift valve models have been developed in the past.
occurring beyond the minimum area of flow. This recovery A model such as the Nieberding model3 has three empirical
was verified by experimental data, and it has been included in coefficients to predict orifice flow, twelve to predict throttling
the new model. Previous models did not incorporate the flow and three to predict the transition between flows. Also,
pressure recovery effect, and in order to match the Acuña4 developed a model incorporating nine empirical
experimental data, it was necessary to empirically modify the coefficients for throttling flow. Each researcher used a
compressible flow theory used to develop these models. fundamentally different model to predict both orifice and
The new model is simpler (i.e. requiring fewer empirical throttling flow with the consequence that there is no unified
coefficients), more reliable, more accurate and more gas-lift valve model for all flow regimes. Another
theoretically consistent than previous models. The new model consequence, depending on the pressure and temperature
allows a feasible explanation of the fundamental physical conditions, is the possibility that these models would predict
behavior of the different flow regimes within a gas lift valve. orifice flow rates smaller than certain throttling flow9 rates – a
Experimental data, obtained from a dynamic test facility, show condition that it is not physically feasible. These models are
an excellent agreement between the data and the predictions based mostly upon experimental data obtained from a dynamic
generated using the new model. test facility and as a result do not necessarily capture the actual
physical phenomena occurring within the valve. In addition,
Introduction these models explicitly require that the flow performance data
An oil well producing by its own energy or natural flow does be gathered using only the constant injection pressure test
not require an artificial lift method. However, if the reservoir procedure.
pressure decreases, the liquid flow rate will be reduced and in
2 J. G. FAUSTINELLI, D. R. DOTY SPE 69406

Bertovic9 was the first researcher to develop a unified model Basic Flow Model. A new simple unified model is developed
that predicts orifice, transition, and throttling flow. His model to predict the gas volumetric flow rate through a gas-lift valve.
reduces the total number of empirical coefficients for all three This model is physically consistent and is capable of capturing
flow regimes to six. However, this model is not physically the essence of the compressible flow behavior. In order to
correct due to the improper use of the adiabatic equation from visualize the governing ideas of the model, four different
the minimum area into the expansion area. locations are described within the valve, see Fig. 4. These
Thus, the above observations strongly suggest the need to locations are primarily used for conceptual purposes and
develop a model that is both theoretically and physically should not be thought of as being at well defined physical
consistent which can predict the gas volumetric flow rate for locations. The conditions at each location are as follows:
any condition, independent of how the data are taken. Injection or inlet conditions at location
P1 = Piod
Preliminary Observations. Injection pressure operated gas- T1 = Tiod
lift valves exhibit orifice, transition and throttling flow. A1 = inlet area
Transitional flow behavior is observed as occurring between Ball-seat conditions at location
throttling and orifice flow, see Fig. 1. Transition flow can be P2 = Pball-seat
described as follows; for a constant injection pressure, and as T2 = Tball-seat
the production pressure is reduced from the injection condition A2 = dynamic area between the ball and the seat
towards zero, the gas flow rate increases, reaches a maximum, Port conditions at location
decreases, and then remains constant. The flow rate does not P3 = Pport
cease, even though the production pressure is reduced to T3 = Tport
atmospheric condition. Nieberding observed this type of valve A3 = dynamic port area
behavior in 5 out of 476 flow performance tests conducted on Production or exit conditions at location
the R-20 valve. Bertovic performed several transition flow P4 = Ppd
tests with the JR-STDN valve. Both investigators observed T4 = Tpd
that for the transition flow regime, the range of pressure gets A4 = exit area
wider as the size of the port gets smaller. Also, the use of the The minimum area is either at position (ball-seat) or
transition criteria in previous models often lead to large errors (port) and is designated by the subscript (m). The conditions at
in predicting the flow rate due to an inability to accurately the minimum area are:
predict transition flow. Am = minimum of {A2 or A3}
Pm = {P2 or P3}
Basic Concepts. Gas-Lift valve performance models are Tm = {T2 or T3}
developed using one-dimensional (1-D) compressible flow
theory. The 1-D models have the advantage of predicting a Model’s Assumptions. The assumptions used in the
physical process using only algebraic equations; however, real development of the model are as follows:
problems are three-dimensional. In order to develop the new 1. The mass flow rate is constant in time (the flow is
1-D model; consider a gas-lift valve as shown in Fig. 2. The assumed to be steady).
figure indicates a large area at the injection or upstream 2. The effect of gravity can be neglected at all positions.
condition, a minimum area located at either the port or the 3. In flowing from location to the minimum area, the fluid
ball-seat, and the production or downstream condition after a performs no external work.
sudden expansion below the port. 4. In flowing from location to the minimum area, the flow
The new model assumes the existence of pressure recovery is steady and axial, and the velocity profile at each section
from the minimum area into the expansion area. Pressure is relatively flat and normal to the walls.
recovery had been studied by Hepguler2, where he measured 5. The area at inlet location is much larger than the area of
the pressures in the port and in the expansion area, observing the minimum location either at location or .
that pressure recovery exists. It was also observed that the 6. There is no heat transfer between the fluid and the valve.
amount of pressure recovery for the orifice and throttling flow 7. There is no friction from location to the minimum area,
regimes is different (Fig. 3). Previous models did not therefore any change of state is isentropic.
incorporate pressure recovery and in order to match the 8. Any deviation from the behavior of an ideal gas is
measured data; it was necessary to artificially modify the captured by the compressibility factor (Z).
model’s critical pressure ratio to exceed the critical pressure 9. There is a pressure recovery from the minimum area to
ratio predicted by compressible flow theory. the exit area at location .
The new model takes into account the existence of pressure 10. Area A3 corresponds to the modified dynamic port area.
recovery, thereby allowing the natural assumption of 11. A2 is a variable area, and is a function of conditions
isentropic flow from the upstream condition to the minimum and and the bellows pressure (Pvc,Tb) at bellows
area. temperature (Tb).
SPE 69406 DYNAMIC FLOW PERFORMANCE MODELING OF A GAS-LIFT VALVE 3

12. The velocity at location is close to zero because A1 is see Fig. 6-a. The pressure at the minimum area is equal to the
much larger than Am. Location is assumed to be at pressure at the ball seat, P2, which is in-between the critical
stagnant conditions. pressure (Pm*) and the injection pressure (P1). Fig. 6-c plots
13. A2 is a function of the dimensionless stem position (N), the temperature versus entropy, showing the temperature
P R + P1(1 − R ) − Pvc, Tb moving downward from the upstream condition (T1) at
N= 3 …………………...…(1) location to the temperature located at the minimum area
Pvc,Tb
(T2) along a constant entropy line. It is assumed that there is
14. The value of R corresponds conceptually to the area ratio no pressure loss or recovery from location and and the
of the port to the bellows; however, the value of R is flow follows an isobaric path from location to . However,
calculated from dynamic data. there is a temperature recovery (T3) from location to ,
15. The value of R is assumed to be constant for each valve causing an entropy change, s3 > s2. The model also assumes
and port combination. that there is a pressure recovery from location to causing
16. If choked flow occurs at location , then no change of an additional increase in entropy, s4 > s3, along an unknown
internal energy exists from position to . Assuming an path.
isenthalpic process, then the temperature becomes Case 3. Critical conditions for orifice flow
constant from location to . Am = A3
17. If choked flow is encountered at location , then P3 Pm = Pm* = P3
becomes constant as P3* (when P4 is varied) along with the Tm = Tm* = T3
dimensionless stem position N (P3*). The value of P3* can Choked conditions exist at the minimum area (A3). Fig. 7-b
be calculated from the assumption that continuity exists plots the temperature versus entropy. As the flow moves
between locations and . downstream from the upstream condition at location the
temperature decreases at the minimum area at location ,
Physical Considerations. The mass flow through the valve is along a constant entropy line. Due to the critical condition at
assumed to be constant and steady from position into the location , the downstream condition at location will no
minimum area. The velocity increases with a corresponding longer affect location . There is an entropy increase from
increase in kinetic energy but with a decrease in the internal location to along an unknown path.
energy13 of the fluid represented by a decrease in temperature. Case 4. Critical conditions for throttling flow
The following explanation of the physical process occurring Am = A2
within the valve is given, which covers the five possible cases, Pm = Pm* = P2
and is summarized as follows: Tm = Tm* = T2 = T3
Case 1. Sub-critical conditions for orifice flow Choked conditions occur at the minimum area or location ,
Am = A3 P2 = Pm* and T2 = Tm*. Fig. 8-b plots the temperature versus
Pm = P3 entropy. As the flow moves from location to , the
Tm = T3 temperature moves isentropically downward to the
Pm > Pm* temperature, T2 = Tm*, located at the minimum area. Under
The area of the ball-seat is bigger than the area of the port, so assumption 16 given above, an isenthalpic process is assumed
the minimum area becomes the port area at location , see from location to , giving a constant temperature, T3 = T2.
Fig. 5-a. The pressure at the minimum area is equal to the The entropy increase from location to is only due to the
pressure at the port, P3, which is in-between the critical pressure difference. The existence of a non-choked condition
pressure (Pm*) and the injection pressure (P1). Fig. 5-c displays at location suggests a pressure recovery from location to
the temperature (T) versus entropy (s) showing the isobaric .
lines. Under the assumptions given above, the flow between Observation: although critical flow is reached at location ,
locations and is isentropic. Thus, the temperature and the ball seat area would continue to be reduced (until the valve
pressure at location decrease through location to the closes) with a further decrease of the downstream pressure.
temperature and pressure at location along a constant This is due to the fact that the dimensionless stem position (N)
entropy line. Then, there is some pressure recovery from the is a function of the port pressure (P3).
minimum area at location to the exit condition at location Case 5. Transition Flow
. The sudden expansion between the port section and the Am = A2
exit condition makes the entropy increases along an Pm = Pm* = P2
unknown path. Tm = Tm* = T2 = T3
Case 2. Sub-critical conditions for throttling flow P3 = P3*
Am = A2 This is the case where choked flow exists at both locations
Pm = P2 and at the same time. The minimum area is the ball-seat
Tm = T2 area. Transition flow is seen in Fig. 1. Fig. 9 plots temperature
Pm > Pm* versus entropy. It behaves the same as in Case 4; however,
The area of the port is bigger than the area of the ball seat, so choked flow is also reached at location (P3 = P3*).
the minimum area becomes the ball seat area at location ,
4 J. G. FAUSTINELLI, D. R. DOTY SPE 69406

Observation: if the downstream pressure at location is comparison among the new model, Nieberding’s modified
further reduced, the dimensionless stem position (N) becomes model, Acuña’s model and Bertovic’s model, observing that
constant (P3 = P3*), and the ball-seat area remains unchanged. the new model shows the best agreement with the data.
Discharge coefficients. The 1-D flow assumption requires that A more general comparison among models was made using
the flow properties are uniform across every plane all of the available data for the JR-STDN valve. The
perpendicular to the mean flow direction. The combination of comparison was made using the arithmetic mean error and the
the non-uniform flow and the vena contracta, formed at the absolute relative percent error.
minimum area, reduces the mass flow rate to a value smaller The arithmetic means of the absolute value of the individual
than that predicted for 1-D flow. The effect of non-uniformity experimental errors is defined as:
in the flow depends upon the specific flow geometry. For this 1 Nt
reason the model has two different discharge coefficients, one Error = ∑ Qgi experimental − Qgi model ……………(2)
N t i =1
for the ball-seat area (K2) and the other for the port area (Cd3).
The way that the discharge coefficients are incorporated into The absolute percent relative error is defined as:
the model is to assume that the discharge coefficients directly 1 Nt Qgi experimental − Qgi model
modify the area and not the mass flux. This concept unifies the E = ∑
N t i =1 Qgi experimental
*100 ………...(3)
model by allowing a single expression for the mass flux
(Equation A-9) to be used under all circumstances. These errors are summarized in Table 2. The new model uses
four coefficients: Cd3, K2, R, REC to cover all three flow
New model’s coefficients. Empirical coefficients must be regimes. Compare this with Nieberding’s model, which uses
found for a model to represent a particular gas-lift valve’s twelve coefficients for throttling flow, and three for orifice
behavior. flow, and three to predict the transition between flows.
The empirical coefficients are: Acuña’s model uses nine coefficients for throttling flow, three
Cd3: discharge coefficient used to calculate the dynamic port for orifice flow and three to predict transition between flows.
area (A3), refer to Equation A-6. Finally, Bertovic’s model which uses six empirical
K2: discharge coefficient used to calculate the dynamic port coefficients for all three flow regimes.
area of the ball seat (A2), see Equation A-5. Observations on the model. The new model has been
R: dynamic port to bellows ratio used in the calculation of the developed from basic physical principles, which allows it to
dimensionless stem position (N), refer to Equation 1. predict the influence of many different factors. It shows that
REC: coefficient used to calculate the pressure recovery from for smaller values of specific heat ratio the curve tend to bend
location (P3) to the production area or location (P4), see faster, reaching orifice flow sooner for similar upstream
Equation A-2. conditions.
The data used in this study were taken on the JR-STDN 1” For orifice flow, the flow at the ball-seat is always non-
injection pressure operated gas-lift valve9. Empirical choked; also, at the port location the flow is choked at the
coefficients were obtained for all available port sizes and for critical pressure (Pm*). In transition flow, the flow at the ball-
different values of the closing pressure. Table 1 gives the seat is choked at the pressure (Pm*) and the port is choked at a
values of the four empirical coefficients. The errors in Table 1 pressure calculated using the continuity equation (P3*). In
are calculated using the arithmetic mean of the absolute value throttling flow, the flow at the port is always non-choked and
of the individual flow rate errors for each test point. In the flow at the ball-seat may or may not be choked depending
addition, all the points are equally weighted; so the program upon the operating pressure condition.
can compute the coefficients for all flow regimes.
Conclusions
Results. Using the empirical coefficients listed in Table 1, it 1. A new model was developed which predicts gas-lift valve
is possible to illustrate the new model’s ability to predict the performance more accurately than previous models.
flow performance. A detailed solution procedure is given in 2. The model is theoretically consistent and allows a feasible
the Appendix, including the methodology used and the set of explanation of the fundamental behavior of gas-lift valve
equations used by the model. It can be seen in Fig. 10 that the flow regimes.
agreement between the data and the model is very good. For 3. This model is simple to use when compared to previous
this port size and closing pressure, the average absolute error models.
in predicting the flow rate was 5.1 Mscfd. In Fig. 11 the 4. The model uses fewer empirical coefficients than previous
agreement between data and model is shown to be very good. models in order to cover all flow regimes without loss of
For this port size and closing pressure, the average absolute predictive accuracy.
error in predicting the flow rate was 21.33 Mscfd. 5. The empirical coefficients for the new model do not
The new model can be compared with the performance of depend upon a specific data acquisition procedure. They
existing models: Nieberding’s orifice flow and modified can be obtained using any testing method, either constant
throttling model; Acuña’s throttling flow model; and injection pressure test procedure, or constant production
Bertovic’s model. Fig. 12 shows typical throttling data for the pressure test procedure.
same valve and port size. These experimental data allow a
SPE 69406 DYNAMIC FLOW PERFORMANCE MODELING OF A GAS-LIFT VALVE 5

6. The theoretical principles used in the development of this 3 = Port location

model are general and can be extended to any type of gas- 4 = Expansion or production location
lift valve. m = Conditions referred to the minimum area
Superscripts
* = Critical flow condition
Acknowledgments
The authors thank Zelimir Schmidt✝ and Roger Blais for the Greek
comments given to develop the model. We also wish to thank γ = specific heat ratio, (-)
the Tulsa Artificial Lift Projects (TUALP), its member ρ1 = Upstream or injection density, ( )
lbm
ft 3
companies and PDVSA EPM for sponsoring this work.
✝ Deceased
References
1. Biglarbigi, K., “Gas Passage Performance of Gas-Lift Valves”,
Nomenclature M. Sc. Thesis, The University of Tulsa, Tulsa, OK (1985).
a3* = Acoustic velocity at the port, (secft ) 2. Hepguler, G., “Dynamic Model of Gas-Lift Valve
A1 = Inlet or upstream area, (in ) 2 Performance”, M. Sc. Thesis, The University of Tulsa, OK
A2 = Area between the ball and seat, (in2) (1988).
3. Nieberding, M., “Normalization of Nitrogen Loaded Gas-Lift
A3 = Dynamic port area, (in2)
Valve Performance Data”, M. Sc. Thesis, The University of
Am = Minimum area, (in2) Tulsa, Tulsa, OK (1988).
Ap = Mechanical area of the port, (in2) 4. Acuña, H., “Normalization of One Inch Nitrogen Loaded Gas-
Bg1 = Upstream formation gas volume Lift Valves”, M. Sc. Thesis, The University of Tulsa, Tulsa, OK
( )
3
ft
factor, scf
5.
(1989).
Sagar, R. K., “Improved Dynamic Model of Gas-Lift Valve
Cd3 = New model experimental coefficient, (-) Performance”, M. Sc. Thesis, The University of Tulsa, Tulsa,
Error = Absolute error, (Mscf/d) OK (1991).
|E| = Percentage relative error, (%) 6. Rodriguez, M., “Normalization of Nitrogen Charged Gas-Lift
G = Gas gravity, (-) Valves Performance”, M. Sc. Thesis, The University of Tulsa,
gC = (
Conversion factor, secft − lbm
2
− lbf
) 7.
Tulsa, OK (1992).
Cordero, O., “Flow Performance of One Inch Fluid Operated
H = Universal gas constant, (lbmol
ft − lbf
− °R
) Gas-Lift Valves”, M. Sc. Thesis, The University of Tulsa, Tulsa,
OK (1993).
K2 = New model experimental coefficient, (-)
8. Escalante, S., “Flow Performance Modeling of Both Fluid and
m = Mass flow rate, (lbm
sec
) Pressure Operated Gas-Lift Valves”, M. Sc. Thesis, The
mm = Mass flow rate at the minimal area, (lbm
sec
) 9.
University of Tulsa, Tulsa, OK (1994).
Bertovic, D., “A Unified Model for Gas-Lift Valve Performance
MF = Mass flux at minimum condition, ( lbm
sec* in2
) Incorporating Temperature Effects”, M. Sc. Thesis, The
University of Tulsa, Tulsa, OK (1995).
MW = Molecular weight of the gas, (lbmol
lbm
) 10. Faustinelli, J., “Temperature and Flow Performance Modeling of
N = Dimensionless stem position, (-) Gas-Lift Valves”, M. Sc. Thesis, The University of Tulsa, Tulsa,
Nt = Number of data points, (-) OK (1997).
11. Brown, K., “Technology of Artificial Lift Methods”, Vol. 2ª,
P = Pressure, (psia)
Petroleum Publishing Co., Tulsa, OK, (1980).
Piod = Injection or upstream pressure, (psia) 12. Brown, K., “Gas-Lift Theory and Practice”, Prentice Hall, Inc.
Pvc,Tb = Valve closing pressure at bellows temperature, Englewood Clifts, N. J. (1967).
(psia) 13. Report of the American Society of Mechanical Engineers
Pvc,Tref = Valve closing pressure at reference temperature, (ASME), “Fluid Meters; their Theory and Application”, United
(psia) Engineering Center, New York, NY, (1983).
Qg = Volumetric flow rate of gas, (Mscf/d) 14. Zucrow, M., Hoffman J., “Gas Dynamics”, Volume I, John
r = Pressure ratio, (-) Wiley & Sons, New York, NY, (1976).
rcrit = Critical pressure ratio, (-)
R = Dynamic port to bellows area ratio, (-) Appendix – Model Solution Procedure
REC = New model experimental coefficient, (-) This section gives the equations and methodology used by the
s = Entropy, (Btu/lbm) model. A flow chart description of the model’s solution
T = Temperature, (°F) procedure is given in Fig. 13.
Velocity at the port, (sec )
V3 = ft The experimentally determined coefficients are: the
discharge coefficients (Cd3), the coefficient for A2 (K2), the
Z = Compressibility factor, (-) dynamic port to bellows area ratio (R), and the pressure
Subscripts recovery coefficient (REC).
1 = Upstream location The calculation follows the flow chart given in Fig. 13.
2 = Ball-seat location
6 J. G. FAUSTINELLI, D. R. DOTY SPE 69406

First. Calculate the pressure of the dome, port, and choked  2

 γ −1 

pressure and temperature:  P1
2
 Pm  γ  γ    Pm  γ 
MF1 =  * G *   *   * 1 −    .(A-8)
 T  Z *T  P1   γ − 1    P1  
Pvc ,Tb = Pvc ,Tref *  b  …………………………….……(A-1)  1 1  

 Tref 
MF = 1.099 * MF1 …………………………………….(A-9)
  P 
P3 = P4 * 1 − REC * 1 − 4   ………………………….(A-2) m
  Q g = 86.4 * …………………………………...(A-10)
  P1   ρ1 * B g 1
γ
 2 γ −1 The velocity at the port is given by:
Pm* = P1 *   …………………………………......(A-3) 53.33 * A 2 * MF
 γ + 1 V3 = ………………………………...(A-11)
P * A3
 2  G* 3
Tm* = T1 *   ………………………………………...(A-4) Z 3 * T3
 γ + 1 The acoustic velocity at the port is equal to:
H
Second. Calculate the ball-seat area, A2, and the dynamic port a *3 = γ * gc * T3 * ………………………………(A-12)
area, A3: MW
A 2 = K 2 * N(P3 ) * A port …………………………………..(A-5) Transition flow takes place when choked flow occurs at both
locations and at the same time. To calculate the choked
To calculate the dimensionless stem position (N) refer to
pressure, P3*, the continuity equation between location and
equation 1, where:
has to be used. Therefore, P3* is equal to:
A 3 = C d3 * A port …………………………………………(A-6)
 C1 − C1 * P1 * (1 − R ) 
 
 Pvc,Tb 
Third. Follow the logic stated below. P3* =  ………………………….(A-13)
If A2 is greater than A3, then  C1 * R C 2 * a 3 * G 
 − 
Amin = A3 P Z 3 * T3 
If P3 is greater than Pm*  vc ,Tb
Pm = P3 Where,
Calculate m and Qg (Equations A-7 and A-10) C1 = A port * K 2 * MF …………………………………(A-14)
Else (P3 is less than or equal to Pm*) C2 = 0.01875 * A 3 …………………………………..(A-15)
Pm = Pm*
Calculate m and Qg (Equations A-7 and A-10)
Endif
Else (A2 is less than or equal to A3)
Amin = A2
P2 = P3
If P2 is greater than Pm*
Pm = P2
Calculate m and Qg (Equations A-7 and A-10)
Else (P2 is less than or equal to Pm*)
Pm = Pm*
Tm = Tm* = T2 = T3
Calculate V3, a3* (Equations A-11, A-12)
Endif
If V3 is greater than or equal to a3*, then
Calculate P3* (Equation A-13)
P3 = P3* = constant
Calculate m and Qg (Eqs. A-7 and A-10)
Else (V3 is less than a3*)
Calculate m and Qg (Eqs. A-7 and A-10)
Endif
Endif
Combining the logic of the model with the flow chart (Fig. 13)
yields a better understanding of the calculation process.
The following are the equation used by the model.
mm = A min * MF …………………………………………(A-7)
SPE 69406 DYNAMIC FLOW PERFORMANCE MODELING OF A GAS-LIFT VALVE 7

TABLE 1 - NEW MODEL’S COEFFICIENTS FOR THE JR-STDN VALVE

Port PvcTref Cd3 K2 R REC Error

(in) (psig) (Mscf/d)
2/16 539.7 0.88 9.40 0.033 0.41 7.75
620.5 0.79 10.46 0.029 0.30 5.60
3/16 448.2 0.78 6.32 0.075 0.23 5.10
729.7 0.72 5.67 0.063 0.41 21.02
4/16 494.9 0.52 3.33 0.138 0.36 10.81
580.2 0.60 3.40 0.120 0.06 14.56
717.8 0.57 4.50 0.129 0.27 18.61
5/16 586.2 0.54 2.90 0.233 0.00 21.33
530.5 0.50 3.43 0.185 0.27 28.04

TABLE 2 - CALCULATED ERROR FOR JR-STDN 1’’ IPO GAS-LIFT VALVE.

Port PvcTref Error - Model Error - Bertovic Error - Acuña Error - Nieberd.
 E Error  E Error  E Error  E Error
(in) (psig) (%) (Mscf/d) (%) (Mscf/d) (%) (Mscf/d) (%) (Mscf/d)
2/16 539.7 10.26 7.75 16.71 7.52 28.88 23.58 29.31 23.86
620.5 7.25 5.60 9.91 5.64 26.16 26.83 26.59 26.88
3/16 448.2 7.09 5.10 23.56 6.88 61.33 40.59 34.76 28.88
729.7 17.36 21.02 38.96 29.04 47.51 79.65 54.62 91.05
4/16 494.9 8.06 10.81 14.62 17.32 57.84 59.29 23.27 38.77
580.2 9.27 14.56 18.17 21.90 32.82 52.53 33.06 65.19
717.8 12.67 18.61 13.64 15.95 30.54 44.97 28.75 56.20
5/16 586.2 9.43 21.33 15.35 26.86 23.70 82.06 30.12 60.01
530.5 10.05 28.04 17.37 26.83 43.17 93.83 46.83 98.72

Orifice

Transition
m

Throttling

P4 P1
Figure 1 - Different flow regimes that occur in the valve. Figure 2 – Distinct minimum areas in a gas-lift valve.
8 J. G. FAUSTINELLI, D. R. DOTY SPE 69406

Figure 4 – Points described in the valve.

Figure 3 – Pressure recovery trend measured by Hepguler.

Figure 5 – Case 1, non-choked flow where Am = A3

Figure 6 – Case 2, non-choked flow where Am = A2

SPE 69406 DYNAMIC FLOW PERFORMANCE MODELING OF A GAS-LIFT VALVE 9

Figure 7 – Case 3, choked flow at Am = A3

Figure 8 – Case 4, choked flow at Am = A2 and non-choked flow at A3

Figure 9 – Case 5, choked flow at Am = A2 and choked flow at A3

10 J. G. FAUSTINELLI, D. R. DOTY SPE 69406

500

400

Qg (Mscf/d)
300
Qg-TEST
200
Qg-MODEL
100

0
0 100 200 300 400 500 600 700
Ppd (psig)
Figure 10 – Model vs. Data, Port Size = 3/16’’ and PvcTref = 481 psig

1000
900
800
700
Qg (Mscf/d)

600
500 Qg-TEST
400
Qg-MODEL
300
200
100
0
0 200 400 600 800
Ppd (psig)
Figure 11 – Model vs. Data, Port Size = 5/16’’ and PvcTref = 586 psig

500

450

400

350

300 QgTest
Qg (Mscf/d)

QgNieberding
250 QgAcuna
QgBertovic
QgModel
200

150

100

50

0
200 250 300 350 400 450 500 550 600 650 700
Ppd (psig)

Figure 12 – New model compared with data and others models for throttling flow.
SPE 69406 DYNAMIC FLOW PERFORMANCE MODELING OF A GAS-LIFT VALVE 11

INPUT - DATA
P1, T1, P4, Aport, G, Tref, H, MW

Calculate
Pvc,Tb, P3, Pm*

yes no
A2 > A3

A3 = Amin A2 = Amin

P3 > Pm* P2 = P3

yes no

P2 > Pm*
NON-CHOKED CHOKED
P3 = Pm P3 = Pm*
yes no

Calculate Calculate NON-CHOKED CHOKED

m & Qg m & Qg P2 = Pm P2 = Pm*

Calculate Calculate
m & Qg V3 & a3*

yes no
V3 > a3*

TRANSITION FLOW

Solve for P3-choked

N(P3-choked) N(P3)

Calculate Calculate
m & Qg m & Qg

Figure 13 – Flow chart of the calculation procedure