Flowing and Gas Lift Performance Gilbert 1954

Flowing and Gas-lift Well Performance ABSTRACT The paper describes the tao-phase vertical-lift lunctlon, explains the hydraulics of natural flow, outlines t ~ o - p h a s eflow through orifices, sumnlarlzes methods for estimating individual well capabilities, and includes approximations for solutlon of naturalflow and gas-lltt problenls for tubing of the 1.66-, 1.90-, 2.375-, 2.875- and 3.50-ln. API sizes, and crude 011s in the gravlty range from 25 t o 40 API. INTRODUCTION Advances I n knowledge of the diHerent lifting methods do not lend theniselves to evaluation qulclily or i n slmple econoniic ternis. I n the aggregate, however, they constitute the necessary b a s ~ sfor in~provedl l f t ~ n gp o l i c ~ e sand profitabilities, wherever oil is r a ~ s e d . Production by natural flow rightly tops the llst of llftlng methods, inasmuch a s it produces more 011 than all other methods con~blned. It proceeds with n ~ l n i n ~ ucost n ~ In relatlve absence of operating difficulties; and 1s relinquished finally In an atnlosphere charged with regret, and supercharged with explet i v e s intended to fortify the conclusion that the stoppage 1s an irreversible act of Providence. Nevertheless, production men have been haunted for years by the thought that a more defin~teknowledge of flowlng performance would suggest means of resuming flow after premature stoppages, permit more effective well control, more appropriate How-string selections, and serve In general to Increase the proportion of oil quantities economically recoverable by natural flow. bevelopment of organized information on v e r t ~ c a l flow has been s o far a matter of slow growth. A presentatlon in 1930 of the baslc theory by the late Professor boctor J. Versluys' prov~dedan initial impetus for current developments, but h a s been applied only to a liniited extent because of practical difficulties in evaluating factors which appear in the Versluys differential.

* N. V . De Bataafsche Petroleum t

'

Maatsch~pp~, The Hague, The Netherlands Presented at the sprlng meet~ngof the P a c ~ f i cCoast D ~ s t r ~ c t , D ~ v l s ~ oofn Product~on,Los Angeles, May 1954. References are at the end of the paper.

An Interesting attempt to solve the problem of t testing flow through short two-phase vertical l ~ f by (67-ft) tubes was reported in 1931 by 1. V. Moore and H, 1). Uilde.' Failure of this project to provide the deslred generalization seems attr~butableto use of tube lengths s o short that representative condltions were not attained. kemler and Poole,' in a paper on flow~ngwells presented before the American Petroleunl Institute I n 1936, developed a limited correlation between gas-llquid ratlo and pressure drop per unit of tublng length, and explained a niethod of estinratlng flowing Ilfe. The work of C. J. May and A. Laird4" resulted in vertical-l~ft peneralizations well adapted to predlct results within a restricted range of condltlons. The interesting paper by Poettn~an and Carpenter6 appeared subsequent to the tinie of derivation of the material here presented. "Gas-Llft Principles and Practices" by S. E'. Shaw,' the pioneer consultant on vertlcal flow, provides an lnterestlng discussion of gas-lift history and methods with correlations which, though limited in scope, were none the l e s s useful. Shaw's observation, that power lunctions may be applied in approximat~ngthe relationships between nlininiuni gaslift Intake pressures and glven liquid production rates, has been used here. The excellent paper by L. C. Babsone added considerably to knowledge of vertical Bow, particularly in the range lor gas-llquld ratios greater than 2.0 Mcf per bbl. T o a large extent the present paper i s a result of reviewing Babson's data and work after addlng a considerable fund of depth-pressure information involving gas-liquid ratios l e s s then 2.0 blcf per bbl. Thus, Shah, Babson, and the late L. N. R,lerrill mentloned by Babson, provided the prior hork nlainly used In the appended correlations of vertlcal flow. It will be noted that no distinction is made here between gas-lift and natural flow. In the gas-lift range covered by Babson, n ~ i s tHow and annular flow predominate and no perceptible diHerences are to be expected. hhere foam How exists, one would be led to expect son~ewhatsteeper gradients for natural

FLOWING AND G A S L I F T WELL PERFORMANCE flow than for gas-lift wlth the same total gas-l~quid ratios, because for natural flow more of the g a s would be in solution, and the ratios are small enough that the g a s in solution could be an appreciable part 01 the total available. I n any c a s e , i t may only be sald that no fimi differences between gas-lift and two-phase natural flow have been observed; and the results here presented involving foam flow are derived from flowing-well data and would therefore tend to represent foam-flow g a s requirements conservatively. The present purposes are to explain flowing and gas-lift well performance i n the light of what is now known of vertical flow, and to provide procedures and empirical correlations which have been usefully applied over a period of years i n solution of practical problems. It i s believed that the p a p h i c a l procedures suggested are ~ l d e l yadaptable. I'he gradient data, however, should be used with care and are oflered in the hope that they may encourage development of more specific data in forms readily adaptable to field use.

A PRIhlARY DISTINCTION T o develop an understanding of the behavior of flowing and gas-lift wells, l t is essential first to recognize that there i s one s e t of conditions affecting f l o ~of gas-liquid mixtures into a bore hole, an entirely different s e t of conditions affecting flow of mixtures from the bottom to the top of the well, and a third s e t affecting flow through beans at the s u r face. We may designate the first a s "inflow performance, 1 9 the second a s "vert~cal-lift perforn~ance," and the third a s "bean performance." Inflow Perfornlance Knowledge of individual well inflow performance i s a basic necessity in equipping and operating 011 wells for n~aximumprofit under any s e t of imposed conditions. Rigorous determination of the inflow performance of a well at any s t a t e in i t s producing llfe would involve: 1. Measurement of the static pressure in the well at the midpoint of the producing interval. 2. Measurement of the mass rate of inflow of each fluid phase corrected to midpoint conditions for each of a s e r i e s of steady operating pressures measured at the midpoint. Fortunately, such detailed deternlinatlons are seldom necessary; but one or more very complete determinations in a particular field can prove helpful in de-

127

ciding what short-cuts and approximations are locally practicable. The liquld inflow performance of a well at any stage in i t s decline may be defined by ~ t static s pressure (in pounds per square inch) and maxirnunl inflow rate when the increment of inflow per unit of pressure or productivity index (PI, measured in barr e l s per day per pound per square inch) i s a constant. The gas-liquid ratio (GLR measured in thousand cublc feet per barrel)may be, for practical p u r poses, constant. Under such simple conditions the inflow perforn~ancecan be depicted a s shown in Fig. 1. Individual-well inflow-performance trends may be depicted a s shown in Fig. 2. T h i s latter type of graph i s useful in predicting both the timing and character of lifting changes needed to mantain production and implement the local reservoir policy. The relation between the midpoint pressure and the liquld inflow rate (inflow performance relationship, or IPH, a s mentioned later) i s not always a straight line a s shown in Fig. 1, but may be concave to the origin a s shown in Fig. 3. Even l f this curvature i s marked, it is posslble by study of a group of such curves from a given field tp develop a locally applicable nrethod of predicting the IPH from the well's static pressure and a single drawdown deterniination. Gas-liquid ratio control i s a principal factor i n development of reservoir policy. Hatios in practice are affected by cumulated withdrawals; and, at any one stage of withdrawal, they are altected by production rates, a s has been pointed out by H. J. S u l l i ~ a n . Generally, ~ of all factors bearing upon future well performance, the gas-liquid ratio is the least predictable.

Lu

Y

i? 5

2000

z Z 0 4 z Lu

w3 LU cd

I-

a

2

1000

200

400

boo

800

PRODUCTION IN BID

a

Fig. 1-Diagram of lnflaw Performance

W. E. GILBERT

128

@)

STATIC PRESSURE

@

@

MAXIMUM GROSS INFLOW RATE

@(CUT Ctzi%

GASILIOUID RATIO CURVE)

. 0

CUMULATED OIL WITHDRAWALS

Fig. 2-Graph

of watercut curve shown in F'lg. 1 generally indic a t e s water influx predon~inantlyfrom a relatively low-pressure source, and the cut curve in Fig. 3 a predominantly high-pressure source. Cut-cumulative curves such a s the one shown in Fig. 2, are of well-known value in estimating future oil recoveries fronr wet wells. In selecting the best operating gross rate for an individual wet well, it IS sometin~eshelpful to plot the approximate water IPH from the gross IFH and two or three cuts a s indicated in E'ig. 4. However, if s p e c i d t e s t s are necessary for this purpose, it i s desirable to remember that temporary use of abnornlally high inflow rates can induce a permanent in@ GROSSLIOUID IPR

of Individual-well Inflow-performance Trends

DETERMINATIONS

3 CORRESPONDING WATER RATES

There have been instances of gas-oil ratio reduct ~ o nfollowing a change in the tubing setting depth In flowing wells. However, the practice i s not recommended, especially i f it involves killing the well with water, inasmuch a s the change In tubing depth can have only a very minor effect of the drawdown opposite gas s a n d s for a given liquid rate; and if the water reduces gas production, i t i s likely, at the same time, to reduce the well's future oil potential.

O

DEFINING WATER IPR

MAXIMUM GROSS

LIQUID INFLOW RATE

hater can be a useful agent In an oil reservoir; but on reaching the well bore, i t s usual tendency i s to mininilze flowing life, ultimate recoveries, and total profits. The operator i s fortunate when water production can be economically excluded. The type Fig. 4-Procedure

for Outlining the Water IPR

MOO GASlLlOUlD RATIO = 1 5 MCFIBBL

2000

1OOO

PRODUCTION IN BID

F i g . 3-Diagram

of Inflow Performance with Curved IPR

crease in water productivity under some circunistances, e.g., if the water shut-off i s insecure, or if there is a wide disparity between water and oil viscosities, gentle formation dips, and edge water close to the bore. Lilfierential depletion i s progressive during sustained flowing periods wherever the ratio of lateral permeabilities to vertical permeabilities i s large; and interflow through the bore between producing layers takes place durlng any subsequent periods of shutdown unless a suitable mud i s spotted in the producing interval. Thus any water inflow from a relatively high-pressure source tends to seek out and enter the more depleted oil layers during a shutdown period, and with permanent injury in some fields to effective oil pernleabilities. After long pe-

FLOWING AND GAS-LIFT W E L L P E R F O R M A N C E

n o d s of flow, a two- or three-week shutdown can cause from 20 to 40 percent permanent reduction of the inflow capacity of a well tapping a depletiontype reservoir even if the water cut i s no greater than 5 percent. Thus, differential depletion i s a factor requiring close consideration in many fields if the operator's equity in operating wells i s to be protected. Generally, the rate of liquid inflow increases a s the operating pressure in a well i s reduced, and the absolute maximum'liquid inflow would result if zero absolute pressure could be maintained at the bottom of the well. T h i s condition, of course, cannot be attained, and l e a s t of all in a flowing well, because a pressure drop in the tubing from the bottom to the top of the well is necessary to sustain verti:a1 flow. A large pressure at the bottom of a well facilitates vertical outflow but discourages lateral inflow. In this sense, the pressure requirements for inflow and those for vertical lift are opposed to one another, and, in particular c a s e s , when an effective compromise can no longer be made by adjustment of controllable factors, flow must cease. For any steady flowing condition, the sum of 1 , the effective pressure dfop from the drainage radius to the bore, 2, the pressure drop i n the vertical column, and 3, the pressure drop across tbe bean (or orifice) at the surface is substantially equal to the difference between the well's static pressure and the flow-line pressure. Vertical-lift Performance All we need know about two-phase vertical flow i s how much pressure i s required to lift the well liquid a t a given rate from a given depth with a give n gas-liquid ratio through tubing of a given size. T h i s problem i s more complicated than the problem of single-phase flow in surface pipelines because we are dealing w ~ t hthe flow of a gas-liquid mixture, and because the input pressure must be sufficient not only to overcome flow resistance in the pipe and the bean a t tbe surface, but must in addition be sufficient to support the total weight of the compressible mixture in the pipe. In single-phase horizontal flow, the total pressure drop for a given flow rate can be represented a s so. many pounds per square inch per thousand feet of length. No such convenient can be used for vertical two-phase flow because the pressure drop per unit of length i s not constant, but increases with depth. For this reason, in proceeding to systematize field information on vertical flow for pressures l e s s than the bubble-point pressure, we find there i s a different

129

depth-pressure gradient for each s i z e of pipe, each rate of liquid flow, and each gas-liquid ratio.* Depth-Pressure G r a d ~ e n t s

.

The depth-pressure gradient i s the basic unit of two-phase vertical flow, and solution of individual well problems i s largely a matter of having available a representative family of gradient curves covering suitable ranges of tubing s i z e s , rates of liquid flow, and gas-liquid ratios. h e are not concerned here with any detailed analy s e s of the physical phenomena which cause the pressure gradients observed or derived from practical field information. However, it is of interest to know that a single gradient curve represents a sequence of different types of flow. I hus, starting a t the upper end of a gradient like that for 200 bbl per day in 2.875-111. tubing w ~ t h1.0 Mcf per bbl gasliquid ratio, a s shown at A in Fig. 5, mist flow will predominate a t the lowest pressures, modified progressively by an upwardly moving oil film which clings to the inside surface of the pipe and increases in thickness with depth. T h i s film, combined with mist flow in the center of the pipe, has been described a s annular flow. A s relative depths Increase, the film becomes s o thick and wavy that it occasionally bridges across the section, resulting In slug f l o ~ .At still greater depths, slug flow merges into foam flow, and this finally merges into singlephase flow a t the pressure beyond which all of the gas is in solution. Fig. 5 illustrates the use of a gradient curve in determining the tubing pressure from the intake pressure, or the intake pressure from the tubing pressure, for a given production rate and ratio for a well of any depth. For instance, as shown a t t) in Fig. 5, 5,000ft wells with 2.875-in. tubing producing 200 bbl per day with 1.0 Mcf per bbl will have an intake pressure of about 440 psi if the tubing pressure at the surface is one atmosphere (zero gage and, a s shown a t C, they will have an intake pressure of about 1,750 psi if the tubing pressure i s 800 psi. Similarly, as shown a t D in Fig. 5, an 8,000-ft well for the same conditions having an intake pressure of *Grad~entspresumably are a l s o affected by many other factors lncludlng 11qu1d surface tenslon, v ~ s c o s ~ tand y gravlty, flowlng temperatures, g a s gravlty, and gas-l~quids o l u b ~ l ~ t ~Howes. ever, there 1s a reasonably c l o s e correspondence between results whlch have been obta~nedIn the 11ght-011 (25 t o 40 API) fields of Long Beach, Santa F e . Domlnguez, Ventura, Canal, and T e n S e c t ~ o n ,and several forelgn fields, vv~thoutad~ustlng for such factors. Also, d has not been found necessary to correct grad~entsfor water cuts. However, the grad~entsare Inadequate to predlct the effects of emulsions.

W. E. GILBERT

130

2 4

6 8 10 12

PRESSURE

14

PRESSURE IN HUNDREDS OF P S 1

Fig. 5-Use of Gradient Curves

1,390 psi will have a tubing pressure of 200 psi. Thus by using the gradient for the desired production rate and ratlo and interpolating when necessary, we nlay estimate either the tublng pressure or the intake pressure in a well of any depth when one of these pressures i s known. Empir~calgradient curves for 2.875-111. tubing are glven in Fig. 23 through 27,*t and slmilar families of g a d i e n t curves for 1.66-, 1.90-, 2.375- and 3.50in. tubing are given in Fig. 28, 30, 31, 32, and 33.*$ l'he 2.875-1n. gradients are based upon correlation of data with gas-liquid ratios ranglng from 0.4 to 2.0 Mcf per bbl from 8,000-ft wells in Ten Section F ~ e l d , combined with Babson's generalizations in the gaslift range. Errors attributable to the 2.875-in. curves for 100,200,400, and 600 bbl per day should not exceed 15 percent. The gradients for 50 bbl per day in the 2.875-in. s l z e and for 50, 100, 200,400, and 600 bbl per day In the other s i z e s are based upon information which is far l e s s con~plete. However, they are offered In the belief that they reliably indicate the relative characteristics of each s i z e and are not likely to lead to nl~sapplications if conservatively employed. Further, it may only be s a ~ dthat they represent a fair correlation and interpretation of the 'lnformation readily available. The Two-phase Vertical-lift Functlon

Using the process outlined in Fig. 2, and a s e t s , a s those on Fig. 23 through of g r a d ~ e n t . c u ~ esuch *In each plate, group A includes gradients for gas-llquld ratios whlch a r e ' l e s s than the optimum, and group B includes gradients for ratlos greater than the optimum. The gradient for the optlmum gas-llquid ratlo (called optimum because it prcvldes the lowest pressure for the given rate of flow) 1s at the bottom of group A and I S marked wlth an m o w pointing t o the ratio. The grad~entshave been dlvided Into two groups slmply to avold crossing of Ilnes.

t See p.

145-149, ~ n c l .

$ See p. 150, 154-157. incl.

'

27, the t ~ o - ~ h a sfunct~on e for any particular depth and tubing-outlet pressure may be constructed. For example, confining attent~onto 8,000-ft wells, 2.875in. tubing, and zero gage tubing pressure, the function may be represented either in terms of intake pressure and gas-liquid ratio, a s shown at A in Fig. 6, in terms of Intake pressure and production rate, a s shown at B in Fig. 6, or in terms of intake pressure, production rate, and gas-liqu~dratio, a s shown a t C in Fig. 6. All three q a p h s represent equivalent information: At A i s the type of graph used by Babson; the fornl of illustration B is interesting because the ordinates and abcissae are the same a s those for the inflow performance (1PH)of a well, and the two types of data may thus be superimposed. They are cons~dered following in connection with estimates of the duration of flowing life. Several general characteristics of two-phase vertical flow may be observed at C In Fig. 6. In particular, it may be noted that: 1. For any constant gas-liquid ratlo there i s a rate of flow which requires minimum intake pressure. Also, this rate of Bow for minimum pressure and the minimum pressure itself both increase a s the gasliquid ratio i s decreased a s indicated by curve 1. (These observations are of Interest in connection with flowing wells because of the tendency of flowing wells to have a more or l e s s constant gas-liquid ratio a t any one tlme.) 2. For any constant rate of flow, there i s a gasliquid ratio which prov~desminimum intake pressure. 1his minimum intake pressure i s directly related to the rate of flow, while the gas-liquid ratio for minimum intake pressure 1s inversely related to the rate of flow a s ~ndicatedby curve 2. ( I h e s e o b servations are of interest in connection w ~ t hgaslift w h ~ c hpermits control of gas-liquid ratios.) The form of the two-phase function i s largely the result of the Interaction of flow resistance *and s l ~ p page of g a s through the oil, the resistance factor being least important when slippage i s greatest and vlce versa. It i s more or l e s s obvious that the column pressure, being the result of the weight of the mixture, i s greatest at low gas-liquid ratios. However, it i s l e s s obvious that for any gas-liquid ratio and depth there i s a rate which requires minimum lifting pressure with lower rates requiring more lifting pressure because of slippage, and h ~ g h e rrates requiring more lifting pressure because of resistance. $

13 1

FLOWING AND G A S L I F T WELL P E R F O R M A N C E

0 1

2

3

GAS I LIOUID RATIO IN M C F 1 B

-

4

I

5

LlOUlD PRODUCTION IN B I D ----a

Fig. 6-The Two-phase Vertical-lift Function for 2.875-in. Tubing Set (Tubing Pressure = Zero P S I Gage)

at

8,000 F t

132

W. E. GILBERT

For the reader who finds difficulty with this s e t of facts, the following explanation may prove helpful: a. Imagine for this purpose a 2,000-ft length of 2.875-in. API tubing mounted vertically on the face of a precipice in the Hocky Mountains with s end where the a wind-swept platform at ~ t upper reader may observe results, and a special-type pump station a t i t s lower end capable of continuously injecting oil of, say, 0.85 gravlty and g a s with a constant ratlo of, say, 1.2 hlcf per bbl. l a s 0.01 bbl per day of oil b. Taklng the i n ~ t l a rate with the 1.2 Mcf per bbl ratlo, the liquid level would appear a t the upper platforn~after about 115 days (2,000 0.01 x 172.7 = 115.8). Out of the oil column the observer would s e e g a s bubbling a t the rate of about cu in. per s e c (1,200 x 0.01 x 1,728 + 1,440 x 60 = 0.24); and the pump a t the lower end would be operating at an input pressure approximating the l ~ q u i dgradient pressure, amounting to 736 psi (2,000 x 0.433 x 0.85 = 736.1). kiost of the gas punlped Into the tubing durlng the 115-day period would have slipped out of the column by the time the level of the mixture reached the observation platform. c. Now, I £ pumplng of the same mixture were maintained a t 400 bbl per day, because the l ~ q u l d alone would fill the pipe in l e s s than 45 min, (2,000 x 1,440 400 x 172.7 = 41.7+), there would be very llttle tlme for slippage of g a s through the oil and the 480 hlcf per day of g a s (400 x 1.2 = 480) accompanying the oil would soon produce an oil mlst around the top of the pipe, the gas-oil volun~eratio at that point belng about 213 volun~esper volume (480,000/400 x 5.614). The input pressure would be about 150 psi. d. Last, with an i n j e c t ~ o nrate of 4,000 bbl per day, slippage would be obviated by extreme turbulence and the input pressure, increased by resistance, would be about 230 p s ~ . Similarly, if we hold the liquid rate constant and increase the gas-llquid ratlo, the intake pressure decreases, from a starting point with zero g a s when the pressure 1s the sum of the l l q u ~ dwelght and the l i q u ~ dflow r e s ~ s t a n c e ,to a mlnlmum, and then increases steadily for greater ratios due to increasing resistance accon~panyinghigher fluid v e l o c i t ~ e s . '1 he observations sun~marizedin Par. 1 and 2 apply In general to two-phase v e r t ~ c a lflow from any

Y1

rm 200 m LIQUID PRODUCTION IN B / D

F i g . 7-comparison from

5,000

I 400

of T u b i n g S i z e s for V e r t i c a l - l i f t

F t w i t h Z e r o T u b i n g Pressure

depth wlth an educter of any s l z e or type and any given outlet pressure. l h e relative effects of the difierent tubing s l z e s are indicated In Fig. 7 (which was constructed from the gradlent curves of Fig. 23-28,30-33). In general, the smaller tubing s u e s offer the advantage of lower Intake pressures at comparatively low rates of flow, and therefore tend to prolong the flowing l ~ f eof low gas-l~quidratio wells. The smaller s i z e s , however, limit rates of Bow, especially for the higher gasliquid ratios. Annular flow i s not treated here. However, it may be mentioned that the poorer an annulus i s as a flow section for slngle-phase flow, the more effective it will be for two-phase flow In minimlzlng gas s l ~ p p a ~ e and in improving g a s utilization in the slippage range a s compared with a circular flow string of equal s e c t l o n d area.

Two-phase Bean Performance "Bean" i s the 011-country term for the orifice used on the tublng outlet to control the production rate of a flowlng well. Product~onmen are accustomed to

FLOWING AND G A S L I F T WELL P E R F O R M A N C E

selecting bean s i z e s for particular wells on a trialand-error b a s i s and no correlation for two-phase flow through beans has been generally applied. The following approximation i s derived from regularly reported daily individual well production data:

P

435 r 0 5 4 6 B =

s

1.89

wherern: P = tubing outlet pressure, in psig r = gas-liquid ratio, in klcf per bbl = gross liquid, i n bbl per day 5 = bean s i z e , in sixty-fourths of an inch ' Approximate solutions for any one of the lour variables when the other three are known, may be made by use of Fig. 29. Directions for use of this chart with examples are given on p. 151. The constant (435) used in the formula i s based upon Ten Section data with beans of the types shown in Fig. 29, the results of one sanipling from the Ten Section Field being shown in Fig. 8. An error of '/,,,-in. in bean s i z e can effect an error of 5 to 20 percent in pressure estimates. In many c a s e s , gas-liquid ratios are reported bnly to the nearest 50 or 100 cu ft per bbl and are frequently difficult to determine because of fluctuations which occur In many wells. Spot readings of pressure at heading wells are not representative of daily averages. For reasons of thls kind, no fornrula could be expected to maintain a 100 percent correspondence with observed data at individual wells even though it correctly correlated the variables involved. As an exan~pleinterpreting the forn~ula,the p e r formance of ''/,-in. beans 1 in. in length i s illustrated 13

6

15

9

11

18

7

ACCURACY

0

20

7 / 40

60

80

100

PERCENT OF READINGS

Fig. 8-Indicated

Range of Accuracy of

Bean-performance Formula (Ten Sect~onData)

LlOUlD PRODUCTION IN B I D

Fig. 9-Performance

of

''4, -in.

Pressures More than

70

Percent Greater than the

Bean for Tubing

Lead-l ine Pressure

in Fig. 9. In the type of formula used, it i s assumed that actual mixture velocities through the bean exceed the speed of sound, for which condition the downstream, or flow-line, pressure h a s no eflect upon the tubing outlet pressure (i.e., pressure on the upstream side of the bean). 1hus, the formula applies for tubing pressures at l e a s t 70 percent greater than the line pressure. Fo; tublng pressures l e s s than 70 percent greater than the line pressure, the bean s i z e indicated by the formula will be too small for the given conditions. fio study has been made of two-phase bean performance for tubing pres-. sures in the range from zero to 70 percent greater than line pressure. Field men usually try to avoid operation in this range because fluctuations of line pressure affect the well's operation. Flowing-well Performance Individual well problems in natural flow may be analyzed quite simply by graphical means, a form which has general application being shown in Fig. 10. In this figure, curve A represents the inflow p e r formance of a well with tubinglntake pressures plotted against liquid production rates. The gradient curves B for 2.0 hlcf per bbl gas-liquid ratio were then plotted starting with the known intake pressures for 0 , 50, 100, 200, and 400 bbl per day. The intersections of these gradients at the surface establish the tubing pressures which the well will sustain, and the tubing-outlet performance may then be plotted a s shown by curve C . The depth vs. pressure diagram of Fig. 10 illustrates the part

W. E. GILBERT

134

played by gradients in flowlng-bell perforniance. liowever, curve C may be obtpned dlrectly iron, curve A by subtracting the total gradlent pressure for each of several rates of flow. Supenniposlng the perfornlance curve I) for a ''/,,-in. bean, we conclude that the well should flow a t about 96 bbl per day with a tublng pressure of about 850 psi. Cons ~ d e r i n gcurves C and L), a s shown i n Fig. 11, it will be seen that there are p o s s ~ b l eequllibr~un~ eaamAL W

U DATA

S I A K FUESlR€ # PSI = ZYI) INROW RATE IN BID = YX) W l W U D RATK) IN M C F I B B L = 2 UOUDWGT INPSIPER 100 Fl = 3 8

nu

PRODUCTION IN BID

illustrating Equilibrium Conditions a t the Well Head (For same w e l l as shown I n Fig. 10)

Fig. 11-Diagram

ratlo. Both the tubing- outlet curve C and the bean curve 13 change with each change in the gas-liquid ratio after the manner shown a t B in F'ig. 12, but for each ratio there IS a stable equil~briumpoint. The relationship between bean s i z e and production rates i s of the type shown at A in Fig. 12. Two similar curves derived from wells where several bean s i z e s were used over a short period of time (6 months or l e s s ) are shown In Fig. 13. Neither of these wells could be expected to maintain steady flow a t rates of l e s s than 50 bbl per day, although they both would produce satisfactorily a t h ~ g h e r rates. The reason for t h i s limitation l i e s in the fact that the differential pressure, available to increase product~onwhen the rate of flow temporarily drops below the stable equilibrium rate, progressively dlmlnishes a s the bean s i z e i s reduced below the bean s i z e which provides maximum tubing pressure. Thus, considering the bean-performance intersections with the tubing-pressure curve for 1,600 cu ft per bbl at B In Fig. 12, a ''/,,-in. bean provides Fig. 10-Diagram

of Flowing-well Performance

points, 1 and 2. Intersection 1 provides a stable flowing condition because, if a tendency develops to increase the flow rate, the bean imposes more pressure resistance than the well can sustain; and if a tendency develops to reduce the flow rate, the well develops a higher tubing pressure than the bean requires. In each of these c a s e s a temporary displacement of the flowing rate generates a press sure differential which returns the well to ~ t equilibriumproducing condit~on. Intersection 2 i s an unstable equilibrium point because any temporary displacement brings into action a pressure d i f f e r ential which increases the displacement, and causes the well either to flow faster or to die, depending upon the direction of the initial displacenlent. T h i s limited explanation assumes a constant gas-liquid

0

100

200

300

400

PRODUCTION IN B I D

Fig. 12-Effect

of Gas-liquid Ratio on E q u i l i b r i m Production Rates

FLOWING AND G A S L I F T WELL PERFORMANCE 16 ~

$

yzlJ

,

~

12

N ~ Z Ggz a z