Lockhart&Martinelli Improved

J. Eng. Sci., King Saud Univ., Vol. 14 (2), pp. 423-435 (1408/1988)

Improvingthe Lockhartand MartinelliTwo-PhaseFlow Correlationby SAS

Adel Hemeida and Faisal Sumait

Petroleum Engineering Department, College of Engineering,King Saud University, Riyadh, Saudi Arabia

The correlation of Lockhart and Martinelli in its present form cannot be used tostudy a large set of data because it requires the use of charts and hence cannot besimulated numerically. A correlation between Lockhart and Martinelli parameters</>and X lor a two phase pressure drop in pipelines was developed using theStatistical Analysis System (SAS). This enabled the development of a computerprogram for the analysis of data using the Lockhart and Martinelli correlation. Fielddata from Saudi flowlines were then analysed using the program and the resultsshow that the impoved Lockhart and Martinelli correlation predicts accurately thedownstream pressure in flowlines with an average percent difference of 5.1 andstandard deviation of 9.6%.

Nomenclature

Apd

Cross sectional area of pipe, sq. ftPipe diameter, inThe absolute roughness, inFriction factorPipe length, ftReynold's numberPressure, psiThe reduced pressureUpstream pressure, psiDownstream pressure, psiFlow rate, cu ft/secGas flow rate, scf/hr.The multiple correlation coefficient, dimensionless

e

fL

NReP

Ppr

PI

Pz

q

QgR

~.Y'-"~I w~ , dl .i.}5'.(""\ t. II) .)l:J\ ~...w\'?- L~i MI ,~..dl i}J\ ~

f!

I

!iI!

i

t!!

423

424 Adel Hemeida and Faisal Sumait

tT

The reciprocal pseudo reduced temperatureAverage temperature, ORSuperficial liquid velocity, cu ftjsec.Superficial gas velocity, cu ftjsec.Lockhart and Martinelli parameterThe reduced densityCompressibility factorSpecific gravityDifferenceDensity, Lbmjcu ftViscosity, cpLockhart and Martinelli parameter

VsL

VsgX

yZ

Yt1

p

J1

cp

Subscripts

gL0TP

gas phaseliquid phaseoil

two-phaseviscous flow mechanismturbulent flow mechanism

v

Introduction

Many of the two-phase flow correlations [1-4J are in use, but the Lockhart andMartinelli correlation [4J, because of its simplicity, has been widely used in industry topredict two-phase flow in horizontal pipes especially for low gas and liquid flow ratesand small pipe sizes. The pressure drop resulting from two-phase flow, is predicted bycalculating the single-phase pressure drop for each phase as ifeach of them was flowingalone through the pipes; and correcting these values for two-phase flow by Lockhartand Martinelliparameter cpo The parameter cp dependsupon whetherthese phases areflowing viscously or turbulently. This coefficient is obtained from standard chartsdeveloped by Lockhart and Martinelli using experimental data obtained at low liquidflow rates and small pipe diameters. Other correlations [1, 2J, can be used in numericalsimulation and in the study oflarge sets offield data. In this work, it was undertaken totackle this deficiencyin the Lockhart and Martinelli correlation and hence use it in theanalysis of large sets of field data. The Statistical Analysis System (SAS) was used todevelopthe equationsto calculatethe Lockhart and Martinelliparameter cp for liquidand gas in their respective flow regimes. A computer program was then developed toanalyse large sets of data using the improved Lockhart and Martinelli correlation.

Development

The computer program flow chart is shown in Fig. (1).One of the subroutines is tocalculate the oil formation volume factor and the gas solubility at operating pressures

Journal of Eng. Sci., Vol. 14. No.2 (1988). College of Eng., King Saud Univ.

Improving the Lockhart and Martinelli Two-Phase Flow Correlation bySAS

ASSUME LIQUID-PHASE

FLOW ONLY, CALCULATE

qL' vsL' PL' (NRe)L' fL' t.PL

ASSUMe GAS-PHASEFLOW ONLY, CALCULATE

qg' Vsg' Pg' (NRe)g' ~Pg

~P CALCULATE

(APL).5,~PTP(9)'~TP(L)9

Fig.(1). Flow chart for improved Lockhart and Martinelli correlation.

~f--" .:..illI ~ , d\ ~. (-"It 'A) JI.:!I~..w\ 'f:-<- L~I ~\ ,~.dl (}.oJI ~

425

I.III

:t,ij

1

I

I~

426 Add Hemeida and Faisal Sumait

and temperatures by the Lagrangian interpolation method [5J, while the other is tocalculate the compressibility factor of gas using the Hall-Yarborough equation [6, 7J,which were developed by using the Starling-Carnahan equation of state:

0.06125 Ppr t e -1.2(1 -c)2z=y

The value of reduced density (y) can be obtained from the solution of the followingequation using the Newton-Raphson iterative technique:

(l)

2 3 4

f=-0.06125Pprte-1.2(1-')2+y+~ +y 3+Y -(l4.76t-9.76t2+4.58t3)y21- y)

+(90.7 t-242.2 t2+42.4 t3)y<2.18+2.821)=0 (2)

Fig. (2)shows the subroutine for calculating the gas compressibility factor. By assumingthe gas phase flowing alone, downstream pressure is calculated by Weymouth'sequation [7]:

P =[p2 4.64 x 1O-1°Q;YgTZLJ

O.5

2 1 d16/3(3)

Therefore, the pressure drop is:

/1P = upstream pressure (P 1)- downstream pressure (P 2) (4)

The pressure loss is calculated by the equation [8]:

/1PL = 43.5fqlLPL (5)

wherefis the friction factor determined from the equation for Moody friction factor [7Jin terms of Reynolds number and relative roughness (Table I).

Reynold's number for each phase is obtained as follows:

dv P-(NRe)L = 123.9~

flL(6)

where qLVsL= A

p

and (N Re)g = 123.9dvsgPgflg

(7)

qwhere v =---L

sg Ap

The type oflow mechanism isdetermined from Table (2).The X parameter is calculated

Journal of Eng. Sci.. Vol. 14, No.2 (1988). College of Eng., King Saud Univ.

Improving the Lockhart and Martinelli Two-Phase Flow Correlation by SAS 427

CALCULATE

PR, TR, l/TR

CALCULATEA, S, C, D

YES

YES

( RETURN )Fig. (2). Flow chart for calculating gas compressibility factor.

°.Y'-'.:.illl < G:- , dl '-:-D.(-"'\i. .;\) <)1.:11oMI ,~ LI)I..\.60\ ,~..dl pWI ~


Table (I). Equations for calculating friction factor [7].

Flow mechanism NR< f

laminar (N R<< 2000)

(2000 < N R<< 4000)

64/ N Re

0.5/(N Re)O.3critical

transition [ (200~ 1.16

J4000 < N Re < ;-)

[NRe>C~dr16J

1

,fl = 1.14- 210g~

I

[

e 9.34

J,fl =1.14-2 log d+ NRe,fl

turbulent

Table (2). Reynolds number and flow mechanism

Liquid Gas Flow mechanism

>2000<1000>2000<1000

Liquid turbulent, gas turbulentLiquid viscous, gas viscousLiquid turbulent, gas viscousLiquid viscous, gas viscous

>2000>2000<1000<1000

from the equation [4]:

x = (APL)O'5

APg

Figure (3) is used to obtain the Lockhart and Martinelli parameter </>.The curves for</>Lare used when the pressure gradient for the liquid is used and vice versa. Thetwo-phase pressure drop is then calculated as:

(8)

.(AP

) -</>2 (AP

)AL TP AL LorG

The following equation was derived to calculate the parameter </>using SAS software:

(9)

</>= exp [2.303a + b Ln (X) + 2.~0 (Ln X)2 ]where a,b,and c are constants and they have been selected according to the type of fluidand flow mechanisms (Table 3).

(10)

Table (4) shows a sample of the computer outputs. The validity of the program wastested. The results obtained, showed a percentage error of 1.8 in pressure drop. Thesquare of the multiple correlation coefficients for </>'sare close to unity and hence theequation derived can replace the charts with high accuracy.

Journal of Eng. Sci., Vol. 14, No.2 (1988). College of Eng.. King Saud Univ.

!,iri~

!~""~11e.

it,

t

ii


1000.0 1000.0

~"'"

~

5 100.0

~I;;~---..:-'

1

:~

100.0

"&

~ 10.0 10.0..EE/1.

1.00.01 0.1 1.0 10.0

Parameter X =J(:~Y(~:)G=ff!i;

1.0

100.0

Fig. (3). Correlation for multi phase flow (after Lockhart and Martinelli).

Field Data

Tests were done on 101Saudi flowlines of 4,6 and 8 inches diameter [9]. The pipelength varies from approximately 2000 to 35000 ft and flow rate varies fromapproximately 400 to 18000STBjD. Most of the PVT data were taken from theexperimental PVT analysis, such as formation volume factor, gas solubility, gas-liquidratio and average specific gravity of the gas. API gravity and viscosity of oil were takenfrom actual field data. Gas viscosity was taken from reference [10] as a function of gas

:'y<--'.!.illl4 6:- , xJ.1~ . (.J> ,t ./\) Jl:!1 : wl '? LI)I ..u.;;.1,;c, xJ.1 ijJl ~

I " I

i I !I

I I , "";

I ; ,

I I' ii'i 1 i Iii<t>lvl ;

,--<t>lIvI II I

q,gtt'

i ' ! " 'I <t>lvv I ! i <t>glv

"'o I I

I 4>gvl

"'90'

" .."I

,'" gvv 0/'I

''''.. " i I i I I ! 0' I i,,I ='-

" ! j, I'LUfi1 ;

,I ;

I

'" gllI I ,--<t>lit ,

I<" <t>Ivl<t>glv <t>II v

<t>gvl <1>'vv

'f' gvv Ilil I, i II

I 11111 ;

Table (3). Values of a, b, and c of equation (10) for different flow mechanisms

Parameter q, a b c R2

gvv 0.4625 0.5058 0.1551 0.996576gvt 0.5673 0.4874 0.1312 0.999395gtv 0.5694 0.4982 0.1255 0.999324gtt 0.6354 0.4810 0.1135 0.999928Lvv 0.4048 0.4269 0.1841 0.984280Lvt 0.5532 -0.4754 0.1481 0.999905Ltv 0.5665 -0.4586 0.1413 0.999718Ltt 0.6162 -0.5063 0.124 0.992167

'-c;::..,;:::~~.,t"r1;:::

~

~w0

."....

~Dependent variable: Y

Table (4). Computer output for the determination of the coefficients for the rPgn equation. SAS general linear models procedureC/J

~.

Source DF""

<:~N

Model

Error

Corrected total

28

10........

\QCoCo

'--

Source OF

XIX2(]

£.~~""

~t"r1;:::~

Parameter Estimate

Intercept (a)X (b)X2 (c)

0.462515780.505783790.15507377

'";:'

~

C/JI::>;::i::>..

,....--;:::;:;.

;:t...

Sum of squares Mean square F value PR>F R-square C.V. i::>..

4.84029187 2.42014593 1164.35 0.0001 0.996576 6.8427 ::J::""

0.01662831 0.00207854 ROOT MSE Y MEAN ::!4.85692018 0.04559100 0.66627273

""

-- .----------- I::>

Type I SS F Value PR>F DF Type III SS F Value PR>F I::>;:::i::>..

4.26274560 2050.84 0.0001 1 4.56368887 2195.62 0.0001 ...,0.57754627 277.86 0.0001 1 0.57754627 277.86 0.0001

I::>

(;;..-----------------

Tror HO: PR> -T- STO error or C/JParameter =0 estimate ;::

::!.--..-----_._- .--

.22.49 0.0001 0.0205625746.86 0.0001 0.0107940916.67 0.0001 0.00930302

).J"-" .:ill \ .< ~ , ,J.:J,\~ . (-", t '1\) Jl:JI )..wl ,~ ~~1 ~I ,~J.:J,\ i~1 ".l.;-


gra vity. Its average value was estimated as 0.0127 cp after correction for the existance ofN2, CO2, and H2S gases.

Results and Discussion

The computer output using calculated gas compressibility factor, gas solubilityand oil formation factor at operating pressures and temperatures shows a high degreeof accuracy in results with an average percentage difference of 1, 2 and 0.5%respectively.

Table (5) shows samples of test results obtained using improved Lockhart andMartinelli correlation. The statistical results yielded an average percentage error of5.1 %, standard deviation 9.6% and average absolute error 8.4% for downstreampressures.

Table (5). Lockhart and Martinelli correlation - test analysis

Test Diam. Flow rate P2(Measd.) P2(Calc.) %Err press. drop % Err P2

6 4.0260 941 276 278.4262 35.37472 -0.897183710 4.0260 8647 231 195.1303 -20.15150 15.528000011 4.0260 4513 208 178.8734 -49.36710 14.003170012 4.0260 4370 204 181.5468 - 36.80856 11.006480013 4.0260 4022 201 176.8641 -48.27182 12.007920014 4.0260 3892 203 178.4189 - 53.43724 12.108930015 4.0260 2216 195 185.4497 - 53.05745 4.897610016 7.8264 15440 480 433.1257 - 51.51025 9.765485017 7.8264 15522 480 438.1752 -43.56753 8.713506018 7.8264 12212 462 431.6255 -46.02198 6.574570019 7.8264 7184 428 435.0644 14.71748 -1.650559020 7.8264 2408 421 449.3134 83.27458 -6.725262021 7.8264 16882 380 321.3018 -45.85797 15.446890023 7.8264 7793 288 269.0238 -42.16941 6.588968025 7.8264 3667 269 284.9198 48.24184 -5.918141026 4.0260 12884 326 261.9651 - 23.20105 19.642600027 4.0260 7804 288 243.6818 - 32.34903 15.388260028 4.0260 4176 273 254.3103 - 36.64644 6.846038029 4.0260 1618 271 267.7656 - 32.34446 1.193519030 4.0260 984 264 262.6232 - 34.42096 0.521527038 6.0000 3533 183 174.5437 - 35.23472 4.620945039 6.0000 2540 186 186.6878 3.619967 - 0.369779540 6.0000 6750 195 172.7394 -32.261820 11.415720041 6.0000 4570 193 178.1591 -43.649860 7.689607042 6.0000 3590 186 173.5823 -59.131900 6.676179044 6.0000 11508 435 378.5863 -29.691430 12.968670045 6.0000 8086 436 386.5228 -48.987350 11.347990051 6.0000 3837 264 252.3550 - 23.290090 4.411004052 6.0000 12562 294 262.8994 -10.798830 10.578440066 6.0000 3812 307 296.6782 -86.014780 3.362142067 6.0000 3112 304 301.6867 -17.795080 0.760971975 6.0000 11008 302 265.5043 - 38.825210 12.084670076 6.0000 3812 267 260.5042 -38.210560 2.432880086 6.0000 14319 422 366.5192 - 77.056630 13.1471000


The effects of some important variables such as pipe size, liquid flow rate andliquid Reynolds number, on percentage errors in downstream pressure calculations,were studied. The results are presented in Figs. (4 and 5). The minimum percentageerror in downstream pressure prediction at different liquid flow rates for 4 in, 6 in and8 in pipes, are presented in Fig. (4).As liquid flow rate increases the percentage errorincreases. The rate of increase in percentage error is higher for smaller diameter pipesbecause of the increase in superficial liquid velocities. It can be seen that at low liquidflow rates the absolute value of percentage error decreases to zero and then starts torise. It should be noted that the variation in percentage error gradually fades. Sincea small range in pipe diameters was considered in the Lockhart and Martinellicorrelation (0.0586to 1.017in) it was safe to assume that the accuracy in pressure dropcalculations was a function of flow rate irrespective of the prevailing flow pattern. Forlarge pipe diameters, the flow pattern changes for a constant flow rate which leads tomore erroneous results when using the above Lockhart and Martinelli correlation. Inother words the ratio of the fraction of the space occupied by the liquid decreases withthe increase in pipe diameter. When going to larger diameters, radial static pressuredifference may exist and may produce differencesbetween static pressure drops of theliquid and gas phases which also lead to erroneous results.

Since Reynolds number is used to calculate cp,according to the flow mechanism,Reynolds number for the liquid phase was calculated and plotted against the

30. 4 in

0 6 in

X 8 in

20cr:acr:cr:UJ 10

0

I-ZUJUcr:UJa..

0

_10 LI QUID F LOW RATE, 1000 BB L I DAY

_20

Fig. (4). Effect of liquid flow rate on percentage errors in downstream pressure prediction at various pipediameters.

Journal of Eng. Sci., Vol. ]4, No.2 (1988). College of En!}., King Saud UniL'.


30

20a:0a:a:UJ 10

00

.....

zUJua:UJa..

00

0

. .REYNOLDS NUMBER FOR LIQUID PHASE

_10

_20

Fig. (5). Effect of Reynolds number on percentage errors in downstream pressure prediction.

percentage errors in downstream pressure prediction. Fig. (5)shows that the increase inReynolds number causes an increase in percentage errors. This reflects the effect ofliquid superficial velocities on the pressure drop calculations. As the liquid superficialvelocity increases the percentage error increases.

Conclusions

The Lockhart and Martinelli correlation was improved by allowing predictions of<pto be made without the use of charts. The program developed accordingly was usedto study a large set of field data and the following conclusions were drawn:

(1) The equations for parameter <p,developed by the SAS method, are in goodagreement with Lockhart and Martinelli experimental graphs. Therefore, the programthus developed proved its accuracy and capability of analysing large sets of data.

(2) As liquid flow rate increases, the percentage error in downstream pressureprediction increases. The range of increase is higher for smaller diameter pipes.

References

[1] Beggs, H.D. and Brill, J.P. (1973) A study of two-phase flow in inclined pipes. J PT(May), 607.

[2] Dukler, A.E., Wicks, M. and Cleveland, R.G. (1964) Frictional pressure drop in two-phase flow, (A)

~.Y'--".!.ill I "'-'lo:- ," \.J,I.\}5'. (..., i . A) jl:.J1~..wl '? ..I)I -ilihl . "-: GI "\}.JI ~- '-


Comparison of existing correlations for pressure loss and holdup, (B) An approach through similarityanalysis", A.ICh.E. Journal (Jan.), pp. 38-51.

[3] Eaton, B.A.; Andrews, D.E.; Knowles, C.R.; Silberberg, I.H. and Brown, K.E. (1967) The prediction offlow patterns. liquid holdup and pressure losses occuring during continuous two-phase flow inhorizontal pipelines, J.PT (June): 815-828; Trans. AIME, 24 (June 1967).

[4] Lockhart, R.W. and Martinelli, R.C. (1949) Proposed correlation of data for isothermal two-phase,two-component flow in pipes, Chemical Engr. Progress, 45, N-l:39--48

[5] Gerald, Curtis F. (1978) Applied Numerical Analysis, Reading, 2"d Ed. Massachusetts: Addison-Wesley Publishing Company.

[6] Yarborough, L. and Hall, K.R. (1977) How to solve equation of state for Z-factor, SPE Reprint No. 13,I, p. 233. Texas.

[7] Ikoku, Chi u. (1980) Natural Gas Engineering, A System Approach. Tulsa, Oklahoma: PennwellPublishing Company.

[8]

[9]

Katz, D.L. (1959) Handbook of Natural Gas Engineering, New York: McGraw-Hili Book Company.

Caponegro, Frank. (1981) Two-phase flow data on wells and flowlines, SPE Paper No. 1655. G, Univ.Microfilm Int., Michigen, U.S.A.

[10] Brown, K.E. (1977) The Technology of Artificial Lift Methods. 1, Tulsa: Petroleum PublishingCompany.

(Manuscript Received: 5.7.1987; Accepted: 4.11.1987)

Journal of Eng: Sci., Vol. 14, No.2 (1988). College of Eng.. King Saud Univ.

Lockhart&Martinelli Improved

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