Dynamic Model of GasLift Valve Performance

8/10/2019 Dynamic Model of GasLift Valve Performance

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Summary. A theoretical and experimental study was performed todevelop a dynamic model for gaspassage performance of a 1.5-in.,nitrogen-charged, bellows-operatedgas-lift valve. Performance curveswere obtained by using air for 0.25-and 0.50-in. ports with flow ratesreaching 2.5 MMscflD. Internal pressures and temperatures were measured during flow-performance teststo develop a dynamic model for bothorifice and throttling flow.

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Dynamic Model of Gas-Lift Valve PerformanceGokhan Hepguler SPE Union Pacific Resources Co., and ZellmlrSchmidt SPE R.N. Blais SPE and D.R. Doty SPE U. of Tulsa

Introduction

To design an efficient gas-lift installation,the production engineer needs reliable information on the performance of all systemcomponents, from the outer boundary of thereservoir to the separator. One critical component is the gas-lift valve. In a producingsystem, the gas-lift valve controls the pointof entry of compressed gas into the produc

tion string and acts as a pressure regulatorwhile the injection gas is controlled at thesurface choke. During the unloading process, the behavior of gas-lift valves becomesthe primary factor for reaching optimumsingle-point gas injection depth. Injectionpressure-operated valves are the most commonly used continuous-flow gas-lift valves.They consist of a nitrogen-charged domeand bellows assembly connected to a stemand ball that seat on a port (Fig. 1 .

The performance curves of injectionpressure-sensitive valves show two distinctflow regions (Fig. 2). In the orifice flowregion, at a constant injection pressure, theflow rate increases as downstream pressuredecreases during subcritical flow, but eventually critical flow occurs, where flow rateremains constant despite further decreasesin downstream pressure. On the other hand,in the throttling flow region, at a constantinjection pressure, the flow rate increaseswith decreasing downstream pressure untilit reaches a maximum and then decreaseswith decreasing downstream pressure. Fora given port size, the occurrence of orificeor throttling flow depends mainly on therelative magnitudes of the nitrogen pressurein the dome and the injection pressure.

One way to obtain reliable data in orificeand throttling flow regions is to perform

flow-performance tests on the gas-lift valvescurrently available with the valve treated asa black box and volumetric flow ratesreported as a function of valve-setting parameters and the differential pressure acrossthe valve. This data-acquisition method isextremely time-consuming because of thecombination of parameters affecting gaspassage performance of a valve. Modelingthe valve on physics principles allows a significant reduction in the number of testsneeded to characterize valve performance.

Copyright 1993 Society of Petroleum Engineers

This study investigates pressure and temperature distribution within the valve, internal valve mechanism, and forces acting oninternal elements of the valve. The paper explains the nature of the experimental dataand results obtained, defines the importantparameters that affect valve performance,and provides a model for both orifice andthrottling flow regions.

Experimental MethodsThe experimental study was performed witha 1.5-in. nitrogen-charged, injectionpressure-operated valve. The valve wascustom-made so that required temperatureand pressure measurements were possible;the general structure of the valve, however,was unaltered. Three sets of dynamic flowexperiments were conducted with the Tulsa U. Artificial-Lift Project's (TUALP) testfacility, which is capable of simulatingdownhole conditions (Fig. 3). In the initialset of experiments, the valve flow performance and the pressure and temperature distributions within the valve were determined.Table 1 gives a sample data set. Seven pressures and three temperatures were recorded during steady flow with pressuretransducers, dial gauges, and thermocouples(Fig. 1). For the second and third set of experiments, a valve stem was manufacturedso that it could be connected to an adjustablescrew assembly, which replaced the valve'snitrogen dome and bellows. These experiments revealed discharge coefficients and aneffective force around the ball as functionsof stem displacement. A micrometer measured the displacement of the stem from itsclosed position to within O OOI in. Ineffective-load tests, the valve stem incorpo

rated a miniature load cell having an accuracy of O 5Ibf (Fig. 4). In all experiments,air was the flowing fluid.

A separate experimental setup was usedfor static probe tests where the flow systemacted only as the air supply. To validate thedata gathered on the instrumented valve inthe dynamic model test facility, anothergroup of flow-performance tests was conducted in a separate test facility with an unmodified 1.5-in. valve that was the sametype and model as the custom-made valve.

Extensive data were gathered during theexperimental study. Refs. 1 and 2 give

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TABLE 1-GAS-LiFT VALVE FLOW-PERFORMANCE DATA WITHINTERNAL PRESSURES AND TEMPERATURES

RECORDED DURING STEADY FLOW

Valve N-15R instrumented, 0.25-ln. port)p 1=636 psig at 79FPvc =635 psig at 79FPtro =685 pslg at 79"FP nj = 700 pslg

Pd Td P PpData psig) (oF) psig) psig)

1 645.2 67.9 708.7 592,02 646.1 67.4 710.9 532.43 644.6 67.0 704.5 465.04 644.3 67.2 705.1 395,85 643.2 67.0 705.2 374.86 642.3 66.3 706.1 374.87 641.5 65.8 705.3 370.38 641.9 65.7 704.8 373.29 641.6 65.3 705.5 372.7

10 642.3 68.0 705.3 376.811 641.3 64.9 706.6 373.5

significant details on the experimental facility.

Preliminary Analysis

Under operating conditions, when there isno flow through the valve, the forces acting on the valve just before it opens aregiven b y 3

Pinj AB-Ap)+PprAp =PdAB . . . . . 1)

The valve will begin to open as

AB ApPinj = P d P p r . . . (2)

A B - A p A B - A p

The closing pressure of the valve, Pvc isdefined as the pressure at which the valvecloses when the injection and productionpressures are equal. Test-rack opening pres

sure, Ptro is the pressure at which the valveopens when the production pressure isatmospheric-Le. Ptro =Pinj in Eq. 1 whenPpr=O-psig gauge. However, the force balance given by Eq. 1 and pressure values obtained from it are valid only under staticconditions or when the flow rate is approaching zero. Under dynamic flow conditions, the gas-lift valve is a variable venturidevice with a flow area controlled by bothupstream and downstream pressures.

To develop a model for gas-lift valve performance, the dynamics of the compressible flow is applied at each stage of theanalysis. Gas flow through the valve is

assumed to be an adiabatic frictionless (isentropic) process, and the deviation of the actual performance of the valve fromtheoretical results is accounted for by a discharge coefficient.

Analysis and Results for theOrifice Flow Region

In the orifice flow region of the gas-lift valveperformance, the combined force resultingfrom the injection pressure and the pressurearound the ball on the stem is sufficient tokeep the valve fully open. The nitrogencharge pressure does not apply enough

JPT June 1993

Tp P3 Ppr T nj q(oF) psig) psig) OF) Mscf/D)

64.0 634.2 630.0 70,0 556.960.0 583.4 580.0 69.0 661.657.4 540.0 540.0 69.0 717.555.6 487.8 485.0 69.0 756.354.5 452.4 455.0 69.0 768.053.1 382.3 380.0 70.0 773.052.9 321.5 325.0 70,0 767.252.6 246.3 255.0 70.0 767.252.3 182.4 190.0 70.0 767.254.2 155.7 125.0 71.0 766.452.0 135.7 90.0 71.0 766.4

downward force to cause the valve stem andball to restrict the flow around the seat. Theflow area is equal to the port area of thevalve; therefore, analysis of the nitrogen

dome and bellows assembly connected to thestem does not enter into the model for thisflow region.

The experimental study is performed witha O.25-in. port with a O.3125-in. ball anda O.50-in. port with a O.6250-in. ball. Thedynamic model predicts valve performancein three steps: calculation of the flow rateby applying the energy equation to the conditions upstream of the valve and at the p0l1;prediction of the pressure gain downstreamof the port before the check valve by applying momentum and energy balances and thelaw of the conservation of mass; and computation of the pressure drop through the

check valve at high flow rates.

Calculation of the Flow Rate. The energyequation is applied to determine the volumetric flow rate. For our purposes, the system consists of the mass of air in a controlvolume bounded by the valve inlet ports andthe lower end of the valve seat port. Theequation to compute the volumetric flow ratebecomes

1 ~ j=I,240.3A t Y r;-;:;;;:- . . ., ,1-/34 TjZII g

. . . . . . . . . . . . . . . . . . . 3)

Eq. 3 provides the theoretical flow ratebased on /: P between the injection and portpressures. As expected, the experimental oractual flow rate is always less than the valuegiven by Eq. 3. Deviation from the theoretical rate results mainly from the neglectof the frictional losses and of the possibleeffects of the valve geometry. To accountfor this deviation, a discharge coefficient,the ratio of the actual and theoretical flowrates, must be included in Eq. 3. This discharge coefficient, Cd is calculated withthe experimental flow rate measured by theorifice meter run.

To design an efficientgas 11ft installation, theproduction engineerneeds reliableInformation on theperformance of all

system components,from the outerboundary of thereservoir to theseparator.

otT-- ' TRClGEN OME

P>----14-- STEM

S l ( ~ J t ~ ~ ~ L ~NLETPORT, Pinj Tlnj~ ~ I - - V A L V E P O R T ( S E A T

n ~ ~

- + - ~ - C H E C KVALVE

Fig- 1-Schematic of an injectionpressure-operated valve showing locationof pressure and temperature measurements.

ORIFICE FLOW

THROTTLING FLOW

CRITICAL FLOW-t- . . . . .

SUBCRITICAL FLOW

Fig 2-Gas-lift valve performance curves.

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7 1)1------

4I AIR COMPRESSOR2 PRESSURE VESSEL3 HEAT EXCHANGER4 SAFETY VALVE5 PRESSURE REGULATOR6 FLOW CONTROL VALVE

9

7 SURGE TANK8 ORIFICE FLOW METER9 DYNAMIC TEST BLOCK

1 EXHAUST AND MUFFLERII STATIC TEST BLOCK

Fig 3 Diagram of gas-lift valve test facility.

As Ref. 4 noted, the product of the discharge coefficient and the expansion factor,

Cd Y, for a given geometry is a function ofthe acoustic ratio, t:.plpl)/k, and the Reynolds number. However, the Reynolds numbers for this particular case are about 10 7

and have no effect on determining CdY.The plot of t:.pIPI)/k vs. CdY indicates alinear relationship for both port sizes. Aleast-squares linear regression yields

Cd Y= 0.53t3 -0 . 7l)[t:.pl Pinl)] -0.2413

+0.84 . . . . . . . . . . . . . . . . . . . . . . . 4)

The final form of the e::juation to calculate the volumetric flow rate through the

valve isdY t:.PPinj

q=1,240.3A t .'Y 1-134 TinjZinj Yg. . . . . . . . . . . . . . . . . . . 5)

Eq. 5 is valid for subsonic and transonicflow. Once critical flow is achieved, t:.pmust be the difference between the injectionand critical pressures, and there is only onevalue of Cd for a given port size. Any lower pressure downstream of the minimumflow area, which is the valve port area inan orifice flow region, has no effect on flowrate.

Temperature at the Valve Port. The gas

temperature at the valve port must be knownso that density can be calculated to determine the Mach number, N Ma at port con-

ditions before the pressure gain downstreamof the port can be computed. For each flow

rate, a temperature value was measured witha thermocouple probe. The measuredvalues, however, do not reflect the actualstatic temperature of the air flowing throughthe port. Hottel and Kalitinsky5 state that,when an air stream is partially stopped byan inserted temperature probe, the temperature increase caused by conversion of kinetic energy affects the probe reading. Theideal static temperature can be observed onlyif the measuring instrument is movingtogether with the gas. Th e relation betweenthe stagnation and true temperatures of aflowing gas is given by

Ts-Tt=v2/2gc]cp . . . . . . . . . . . . . (6)

The static temperature of the flowing gasat the port must be determined theoreticallywith the isentropic-process assumption andthe definition of density as

Pp ) k - l ) / kTpzp= . . Tlz l . . . . . . . . . 7)

Pm]

Pressure Gain Downstream of the ValvePort. Our experimental study showed thatthere is a pressure gain downstream of thevalve port before the check valve duringsubcritical orifice flow. F rom the valve portarea, the gas expands to a larger area before

the check valve, and this sudden expansionresults in a pressure gain during subcriticalflow. Once critical flow is reached, the flow

DJUST BLESCREW

SSEMBLY

..., - 4 + - - STEM

Fig 4 Diagram of valve used ineffective-load tests.

rate and pressure at the port remain constanteven though the downstream pressure con

tinues to decrease.The pressure increase resulting from anabrupt enlargement is analyzed by applyingconservation of mass, conservation ofmomentum assuming free shear flow), andconservation of energy assuming isentropic process). These conservation equationsare applied to the flow stations at the valveport and downstream of the port.

The conservation of energy equation maybe written with Mach numbers and temperatures:

2 + k - l ) N ~ a 3

2 '2 + ( k - I ) N Map. . . . . . . . . . . 8)

where N Ma = v .,Jkgcpl p . . . . . . . . . . . (9)

Combining the three conservation principles with the perfect gas equation of stateEOS) and Eq. 9 yields 6

N M a 3 . J 2 + k - l ) N ~ a 3

1 k N ~ a 3

N Map J 2 + k - I ) N ~ a p- - - - - - - - - - - - - - . . . . . . . . 10)

l + k N ~ a p+[ I-cp)/cp]

Rearranging Eq. 10 results in a quadraticequation in the form

ax 4 +bx 2 +c=O,

Our experimentalstudy showed thatthere is pressure

TABLE 2 THEORETICAL AND EXPERIMENTAL RESULTSIN ORIFICE FLOW REGION

gain downstream of thevalve port before thecheck valve duringsub critical orificeflow.

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ValveInstrumented N-15RUnmodified N-15RInstrumented N-15R

Port(in.)

0.250.250.50

E [1 OO qexpotlmental - q t n _ a l jiq expotlmantal

E*( )

-3.51.18.5

Absolute EoAl)

4.13.28.5

StandardDeviation

oAl)

4.04.18.8

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y

A n l e a n

t t t

O . . . . - - - - - l - ~ ~ , . . . . . ~ ~ . . . , . _ +

s e a t

IFig. 5 -F low chart of the computationalprocedure for orifice flow. Fig. 6 -F low geometry and active pressures around the ball during throttling flow.

with the coefficients

a=A2k2 + 1-k, b=2kA 2 - 2 and c=A,where

N Map--J2+ k-1)NltapA= .

1+kNltap + [ l-c/lcP]The positive root between zero and one

is the desired Mach number, N Ma3 , whichallows computation of the static pressureratio:

Pp 0.97N Ma3 2 + ( k - 1 ) N ~ a 3

P3 c >N Map 2 + ( k - 1 ) N ~ a p

11)

110.00

100.00

90.00

8 0 . 0 0 Pvc = 446 psi 9 @ 70 F7 0 . 0 0

60.00

50.00-


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I n i t i a l i z e d a t a

Assume Pp

C . l c u l . t . the c o . f f l c l . n t .. b 0 d of Eq. 24

S o l v e Eq.24 for \ item dil iplacement

Calcula te A min by Eq.14

Calculate C d Y from Eq. 25

Calcula te f l o w rate b y Eq. 5

C a l c u l a t e t e m p e r a t u r ea t valve seat . Eq.7

Find In-situ g a s veloc i ty u s i n g

the f low rate f rom Eq.5 a n dc o m p u t e N M a p f rom Eq.9

Solve Eq .10 for N M a a

C a l c u l a t e p r e s s u r e d r o p b e f o r et h e c h e c k valve, p 3 by Eq.11

N c o

Flow Area

Calcula te pressure d r o pa n d p " . by Eq.12 & Eq.13

Fig. 9 -F low chart of the computational procedure for throttling flow.

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Simultaneous solution of Eqs. 12 and 13provides the pressure drop and productionpressure.

Fig. 5 is flow chart of the procedure foranalysis of the orifice flow region. The theoretical results obtained from the dynamicmodel are in very good agreement with theresults of the experimental study. Table 2shows the percent absolute and averageerrors. The dynamic model successfully predicts the flow performance of the 1.5-in.

nitrogen-charged, bellows-operated gas-liftvalve in the orifice flow region. This modelcan be applied to all port sizes and other1.5-in. valves with similar geometry by performing a limited number of tests to verifyor adjust the discharge coefficient curvepresented.

Analysis and Results for theThrot t l ing Flow Region

In the throttling flow region of the gas-liftvalve performance, the downward force onthe stem and bellows assembly resultingfrom the nitrogen pressure in the dome becomes sufficient to depress the stem, thus

restricting gas passage through the valve.The throttling flow performance curves indicate that, as the production pressuredecreases, there is sufficient flow area initially to allow an increase in flow rate. Aftera maximum flow rate is reached, the flowrate declines approximately linearly as afunction of production pressure.

In the throttling flow region, critical flowoccurs after the maximum point on the flowperformance curve. Once critical flow isachieved, with a constant upstream pressure,the flow rate cannot be increased unless theminimum flow area is increased. Becausethe flow area decreases as production pressure decreases, critical flow must start at aproduction pressure on the downward slopeof the throttling flow performance curve.Critical pressure occurs at the minimumflow area just above the seat. The flow regime at the lower end of the port remainssubcritical after a normal shock.

From the standpoint of force-balance analysis, the area open for flow correspondingto an equilibrium position of the stem andball assembly stem displacement) is a function of 1) the combined upward force resulting from the injection pressure and theeffective pressure around the ball and 2) thecombined downward force applied by thenitrogen pressure acting on the bellows area

and the resistanceof

the bellows.Analysis of the throttling flow region focuses mainly on the parameters affectingstem displacement from the closed position.Once the stem displacement for given flowconditions is determined, the area open forflow and the discharge coefficient can be calculated. The remainder of the model for thethrottling flow region then uses the conceptsand equations discussed for the orifice flowregion.

The minimum flow area is represented bythe lateral surface area of a truncated rightcircular cone Fig. 6). The base of the coneis the edge of the valve seat ring. The top

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7 0 0 r _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

.ttlrex

800

500

400

300

Port = 0.25 in.P (experimentaL)=-610 psigvc

Pvc z: 605 psig

THEORETICALPvc = 610 psig

T P ) =79 Fvc

o-1lL EXPERIMENTALex

200

100

Pvc:: 615 pslg

o1 ~ o n O ~ ~ ~ ~ ~ ~ ~ ~ 4 0 ~ 0 ~ ~ 5 ~ O ~ 0 ~ ~ 6

PRODUCTION PRESSURE, PSIA

Fig. 1O Experimental flow performance curve and theoretical curves showing sensitivity of the model to valve closing pressure.

of the ring is determined by the position ofthe ball on the stem. This minimum flowarea is defined as

min =

7f(r p +Xmin)-v Ymin _Yp)2 +( r p -Xmin)2 .

. . . . . . . . . . . . . . . . . . (14)

Although there is an area between Points1 and 2 in Fig. 6 smaller than the minimum flow area given by Eq. 14, thatsmaller area must not be used. The analysis presumes ID compressible flow; therefore, the fluid streamlines must beperpendicular to the flow surface that determines the flow area.

The location of the point xmin, Ymin), obtained from the equation of the circle, is defined by

Ymin-h=--v r}-Xfrun 15)

J r2r2and xmin = b p . . . (16)( Yp - r b-Ys)2+r}During steady flow, stem displacement is

a function of the constant injection pressure,the nitrogen pressure in the dome, the bellows load rate, and the upward effectiveforce that results from a mean pressure valueacting around the ball. Although the valueof the constant injection pressure is known,the remaining parameters need to be determined by vigorous theoretical and/or experimental study. The following sections

JPT June 1993

explain the analytical and experimentalmethods applied to describe these parameters.

Prediction of the Nitrogen Dome Pressure. The nitrogen pressure in the dome un-der dynamic flow conditions is calculatedas a function of the stem displacement, valveclosing pressure (Le., the initial nitrogencharge pressure at flowing gas temperature),dome volume, and bellows area. The analytical method involves taking the pressuredifferential with respect to volume using theEOS for real gasses and transformingvolume to displacement 7 :

A BYsiV d

Bellows Load Rate. The literature containscontradicting definitions of the bellows loadrate on gas-lift valves. 7-9 For this analysis,the bellows load rate is defined as the incremental pressure difference between thegas opening pressure and the nitrogen pressure acting on the entire bellows area required for 1 in. of stem travel. Starting withthe force balance,

PgIAB-PdAB=ksYs+hys, (18)

and defining LB =ksiAB as the load rate andthe pressure constant that reflects hysteresis

as Phys =fhysIA B,

Pg-Pd=LBYs+Phys . (19)The load rate in Eq. 19 is determined from

static probe tests during which sufficient

time is allowed between measurements toensure the precise equilibrium position ofthe stem, though the valve reacted rapidly.The temperatures of both the nitrogen andthe air remained equal to the laboratory temperature.

Fig. 7 shows a plot of the pressure difference between the nitrogen and air pressureapplied in opposite directions over the entire bellows area vs. the displacement of thestem and ball from the closed position. Thesteep slope after a stem displacement of0.240 in. indicates that the valve is fullyopen. The significant result of the staticprobe tests is that the response of the bel

lows assembly depends on the direction ofthe change in pressure differential. This hysteresis is reflected by Phys in Eq. 19.

Phys is a function of the amount anddirection of change in the net force appliedon the entire bellows area. Ideally, for agiven net force resulting from the pressureapplied across the bellows area, there is onlyone stem displacement, regardless of theprevious lower or higher net force, and thevalve should be closed when that net forceis zero. The test data, however, have shownthat the stem displacement would be greater if the net force is applied in decreasingorder, exhibiting hysteresis.

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Effective Force on the Stem and Ball. Theupward opening force resulting from totalgas pressure acting o n the bellows assembly is not a directly measurable quantity during dynamic flow tests. In the force-balanceequations Eqs. 18 and 19), P g acts on theentire bellows area, A B under static conditions. Under dynamic flow conditions, thepressure equivalent to P g that acts on AB isa function of an effective pressure, P e , acting on some projected area, A mean , aroundthe ball and the injection pressure acting onthe remaining area. The value of P e isbounded by the injection pressure and thepressure at the valve port:

1 r pP e = J pdA . 20)

Ameanpinj

The difficulty in solving Eq. 20 is deciding on and justifying how the pressure profile changes between the injection point andthe valve port. Effective load tests are conducted to find a solution. In these tests, theload cell measures tension caused by a differential load given by

M=(Pe-Pinj)Amean, 21)

where Pinj>Pe>P p Analysis of the stem displacement shows

that Amean can be any area greater than theprojected area of the ball and less than orequal to the bellows area.

Analysis of the load cell data results in arelatively simple relationship for effectivepressure as a function of stem displacement,injection pressure, and valve port pressure:

A seat (Pmin +Pinj)(A b - A eat)Pe=P p - -

A mean 2Amean(AB-A b )

+Pinj 22)

Ameanwhere A seat = r X ~ i n 23)

Pmin at the point xmin,ymin) is assumed tobe equal to Pp during subcritical flow; however, at critical flow, Pmin must be equal tothe critical pressure. Also, test results conclude that the differential load correspondingto a pressure difference between the injection point and the valve port for a fixed stemdisplacement s independent of the flow rate,and the pressures of interest are static ratherthan stagnation pressures.

Stem Displacement. Because all the parameters that effect the stem displacement canbe expressed analytically as a function ofmeasurable variables, direct calculation ofstem displacement becomes possible. WithEqs. 15 through 23 combined, the stem displacement takes the form I

yff+ayl+bY;+CYs+d=O, 24)

where a=- C 4 - 2 L BABC3 - L B V di - A b C2)ILsAB,

b=(2C 3 C4 +LBABC} +LBABr)

+2L BC3Vdi +C 2Vdi +2ABC2C3)ILBAB

c=( - A B C I - C 4 C} - C 4 rJ -LBVdiC}

5 8

-LBVdirJ -2C 2C3 Vdi - A BC2C}

- A BC2rJ)IL BAB, and

d=(C I Vdi+C2C}Vdi +C2 VdirJ)ILBA B,

and where

-Pdi.-Phys

C3 y p r b

and C 4 =A bP d d ( 1 - Pdi Oz).zdi p

Eq. 24 is solved by the algebraic procedure given in Ref. 10 for the solution of aquartic equation and its resolvent cubicequation. Two of the four roots of Eq. 24are complex, and only one of the two remaining roots physically corresponds to thedesired stem displacement.

Discharge Coefficient. In the throttling flowregion, stem position is the main factor determining the flow geometry and flow area.CdY is expected to be a function of stemdisplacement.

Discharge-coefficient tests are performedto determine the discharge coefficient as afunction of stem displacement from theclosed position and of acoustic ratio,(ApIPinj)/k. The expansion factor is calculated with the minimum flow area for thefixed stem displacement. Fig. 8 is a plot ofCdYvs. acoustic ratio as a function of stemposition for 0.25-in. port. Linear regressionprovides Eq. 25 to be used in Eq. 5 for the

throttling flow region:

Cd

Y=6.02ys Apipi -1.23 Apipik k

-2.34ys + 1.05 25)Fig. 9 shows a detailed flow chart of the

computational procedure for the throttlingflow region. The dynamic model is very sensitive to small changes in stem displacement.But the stem displacement, in turn, is verysensitive to small variations in injection pressure, valve closing pressure, and bellowsbehavior. Therefore, the predicted flow performance in throttling flow is significantly

altered by small inaccuracies in measuredvalues of the injection pressure, valve closing pressure, bellows load rate, and thePhys observed during bellows-load-ratetests. Because Pinj and Pvc are measuredwith an experimental uncertainty of 5 psi,the theoretical performance curves aregenerated to examine how that uncertaintyis reflected in the model. As an example,Fig. 10 shows the effect of PvC Theoreticalcurves form a band when 5 psi is added toand subtracted from the measured pressurevalues. The lower curve coincides with theexperimental performance curve.

The theoretical model predicts the gaspassage performance of the valve inthrottling flow within the uncertainties of theexperimental study. All parame ters that account for the complicated flow geometry areanalyzed theoretically and experimentally.This model sets guidelines to obtain a moregeneral solution in throttling flow region forgas-lift valves with distinct flow geometry.

onclusions andRecommendations

1. The dynamic model accurately describes valve flow performance in the orificeflow region. It significantly reduces thevolume of data required to characterizeorifice flow; one simply needs to obtain discharge coefficients by performing a limitednumber of tests.

2. The model is general enough to be applied to all l .5- in. gas-lift valves that exhibitorifice flow if discharge coefficients are adjusted to account for geometric factors.

3. For throttling flow, the model predictsvalve flow performance by taking into account all parameters that affect the complicated flow geometry.

4. The unique theoretical model, tuned byempirical parameters, sets guidelines anddefines procedures to obtain a generalmechanistic model for different types of gaslift valves.

5. The consistency of the behavior of thebellows under dynamic flow conditions andover a time period must be investigated further. Further analysis of the response of thebellows, which is connected to the valvestem and ball assembly, is necessary tounderstand why small variations in individual parameters significantly affect valve performance.

6. Application of the model for I-in.nitrogen-charged valves needs to be investigated.

7. The equations that constitute the modelare convenient to program for both flowregions and can be incorporated into a design procedure. The complete model wastransformed into a FORTRAN program inRef. 1.

8. Most of the continuous-flow gas-lift installation designs use bellows-chargedinjection-pressure-sensitive valves with asmall port. These valves function as backpressure regulators to control casing pressure. Although the procedure given for thethrottling flow region may not always apply, the computational procedure given forthe orifice flow region can be incorporateddirectly into the design of most continuousflow gas-lift installations with injectionpressure-operated valves.

omenclature

A = area, L2 in.2Amean = area that P eff applies on, L2

in. 2Amin = minimum flow area, L2 in. 2A seat = area the port pressure applies

on during throttling flow,L2 in. 2

At = throat or port area, L2 in. 2

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Cp = specific heat at constant pressure,m Ll/ t 2 )/mT, Btu/lbm-oR

d = discharge coefficientCv = valve flow coefficient

E = average errorf hys = force constant that reflects

hysteresis, mLlt 2 , Ibfg c = proportionality constant

h = y coordinate of center ofcircle, L, in.

1 = mechanical equivalent of heat

k = ratio of specific heats,dimensionlessks = spring rate, m1t2, Ibf/in.L = load, mLlt 2 , Ibf

LB = bellows load rate, m/Llt 2 psi/in.

N Ma = Mach number, dimensionlessP = pressure, m/Lt 2 psia unless

stated otherwisePdi = initial dome pressure, m/Lt 2

psiaP e = effective pressure on the stem

and ball, m/Lt 2 psiaP g = gas pressure that applies on

entire area of bellows understatic conditions, m/Lt 2 psia

Phys = pressure constant that reflectshysteresis, m/Lt2, psia

Pinj = injection gas pressure, m/Lt2,psia

Prnin = pressure at minimum flowarea, m/Lt2, psia

Ppr = production pressure, m/Lt 2 psia

Ptro = test-rack opening pressure,m/Lt2, psia

Pvc = valve closing pressure, m/Lt2,psia

q = volumetric flow rate, L3 It,MscflD

r = radius, L, in.R

=universal gas constant,

mLl t2T, psi_ft3 Ilbm-mol- oRT = temperature, T, oR unless

stated otherwiseTinj = injection gas temperature, T, oR

Ts = stagnation temperature, T, ofTt = true static temperature, T, Ov = velocity, Lit, ft/secV = volume, L3, in. 3

Vdi = initial dome volume, L3, in. 3Xmin = x coordinate for minimum

flow area, L, in.Yrnin = Y coordinate for minimum

flow area, L, in.Y = stem displacement from closed

position, L, in.

Y = expansion factor,dimensionless

z = gas compressibility factor,dimensionless

Zinj = gas compressibility factor atinjection pressure andtemperature, dimensionless

) = ratio of square root ofminimum flow area tosquare root of totalupstream flow area

Y g = gas specific gravity,dimensionless

t: p = pressure differential, m/Lt2, psi

JPT June 1993

p = density, m/L3, Ibm/ft 3J> = area ratio, Ap lA3

Subscriptsb = ball on stemB = bellowsd = nitrogen domeP = gas-lift valve port1 = upstream2 = downstream3 = downstream of valve port

before check valve

cknowledgments

The support provided by the member companies of TUALP is gratefully acknowledged. Teledyne Merla deservesspecial thanks for fabricating significantcomponents of the test facility. We alsothank Camco for providing a test facility.

References

1. Hepguler, G.: Dynamic Model of Gas-LiftValve Perform ance, MS thesis, U. ofTulsa, Tulsa, OK (1988).

2. Hepguler, G. et al.: Instrumentation for theDynamical Modeling of Gas-Lift Valves ,paper ISA 88-0768 presented at the 1988 Inti.Instrumentation Symposium, Albuquerque,May 2-6.

3. Brown, K.E.: The Technology of ArtificiaLLiftMethods Petroleum Publishing Co., Tulsa(1980) 2a, 96-109.

4 Fluid Meters-Their Theory and Applicationsixth edition, report, ASME Research Committee on Fluid Meters, ASME, New YorkCity (1971).

5. Hottel, H.C. and Kalitinsky, A.: Tempe rature Measurements in High-Velocity AirStreams, J. Applied Mech. (March 1945)A25-A32.

6. Benedict, R.P. et al.: Generalized FlowAcross an Abrupt Enlargement, J Eng. forPower (July 1976) 327-34.

7. Decker, L.A.: Analytical Methods for De

termining Pressure Response of BellowsOperated Valves, paper SPE 6215 availableat SPE, Richardson, TX.

8. API Gas Lift Manual Vocational Training Series, API, Dallas, TX (1984) 6.

9. Winkler, H.W. and Camp, G.F.: DynamicPerformance Testing of Single-Element Unbalanced Gas-Lift Valves, SPEPE (Aug.1987) 183-90.

10. Beyer, W.H.: CRC Standard MathematicalTables twenty-sixth edition, CRC Press Inc.,Boca Raton, FL (1981) 9-12.

5 Metric Conversion Factors

ft3 x 2.831 685 E-02 m 3O OF-32)/1.8 Cin. x 2.54* E OO cm

Ibm x 4.535 924 E-01 kgpsi x 6.894 757 E OO kPa

*Conversion factor is exact.

Provenance

Original SPE manuscript, Dynamic Modelof Gas-Lift Valve Performance receivedfor review April 7, 1991. Revised manuscript received Sept. 21, 1992. Paper accepted for publication April 1, 1993. Paper (SPE21637) first presented at the 1991 SPE Production Operations Symposium held in Oklahoma City, April 7-9.

JPT

Authors

Blais Doty

Gokh Hepguler Is a petroleum englneer at Union Pacific Resources Co.in Fort Worth. Before Joining Union Pacific In 1989, be worked t Baker 011Tools In Houston. His teChnlcallnteresteInclude artificial-11ft methods, reservoirsimulation, multlphase flow, and computer systems and networks for,englnearing applications. Hepguler holds asand MS degrees in petroleum engineer.Ing from the U of Tulsa. ZeUmlrSchmidt Is professo r of petroleum engineering t the U of Tulsa and director of TUALP. He spent 10 years as aproduction engineer with INANaftaplinin Yugoslavia and has served as a consultant to various companles. Schmidtwas a 1987 Distinguished Lecturer andis a member of the Editorial ReviewCommittee. He holds an engineeringdegree from the U of Zagreb and MSand PhD degrees in petroleum engineerIng from the U. of Tulsa. Roger BlaiSinterim provost and ecedemlc vice president at the U of Tulsa, eamed,a BAdegree from the U 01 Minnesota and aPhD degree from the U of Oklahoma,both In physics. He has been at the U. ofTulsa since 1977 and has been associate director of TUALP since Its oundingin 1983. He chaired the Physics Dept.during 1986-88. Dale R Doty is associate professor of mathematical sciencesat the U of Tulsa and 8ssoclate directorof TUALP. He Joined the U of Tulsa In1975 and hes been Involved In research

and consulting In petroleum production.He holds BS. MS, and PhD degrees Inmathematics from Michigan State U.

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Documents

Dynamic Model of GasLift Valve Performance