Vertical 2 Phase Flow

7/27/2019 Vertical 2 Phase Flow

1/6

Fig.l

P-C bi-phase friction 10ss

'--"

~$U)

~11O-1'

e

.. ,g~

--

.

10 3f'1f = 66(Mlq.. I dlr3

Pressure traverse found

for 2-phase vertical flow

10-2

G.c. BORGIAG. GOITARDI

University of BolognaBologna, Italy

program using assigned P-V-T andother properties are first developed.Two theories are incorporated into fueprogramo

Then the treatment of muItiphaseflow of liquid and gas as flow of asingle phase with combined propertiesby an energy balance will be analyzed.Friction losses are accounted for dif-

ferentIy in this approach.The resuIt is a second program for

the HP 67/97calculator.Laws controlling two-phase flow in

10-3TIME-CONSUMING calculations for

pressure traverse in vertical pipe withflowing fluids have been streamlinedinto two programs for fue HP 67/97hand-held calculator.

Using existing theories for such flow,fue flow equations and a calculator

10

Mt(qon)

d1

102

Fig.2 Fig.3

Pressure traverses by th P-C method Pressure traverses by the M-K method

........

o o

qon = 60 cu m/ddi = 6.2 cm

qon = 60 cu m/d

di = 6.2 cm

22

Pressure P. bars

E

8.~ 4

- 4E:~

15.Q)

o

E:~

15.Q)

o

66

818

1100 150 25050

50

OGJ

OIL & GAS JOURNAL - SEPT. 15, 1980 191


2/6

Fig.4

Friction 10ss factor (Tek)

00:1

vertical pipe are complicated becauseof the variation in flowing fluids'specific volume with pressure andtemperature changes. Slippage lossesand the variety of flow patterns thatliquid and gas may have during up-ward flow also complicate the picture.

Results from theoretical proposalsare incomplete, but in fue flow regionof practical importance, equations of

satisfactory accuracy have evolved.Designed for the HP 67/97 calcu-lator, this program evaluates pressuretraverses in pipe for assigned P-V-Tproperties of fue fluids, the oi! rafe,the GOR, and ID of the pipe.

Flow equations. When the average

Flow equations

Fig.5

Pressure traverses (Tek)

temperature along fue pipe is known,the pressure traverse is evaluated byintegration of a differential equationoi, the type dp/dh =f(p), where: p= pressure; h = depth; and f(p) is afunction of the pressure that variesaccording to the different theories andto the P-V-T characteristics of fluids.

By using the theory implying fueeasiest mathematical formulation ofthe problem, the engineer can deter-mine the pressure traverse along avertical pipe by laborious hand calcu-lations, or by using' gradient curvetables.

TheHP 67/97 calculator was pro-

grammed to salve the flow equations(Table 1). Equation (1) is the pres-sure gradient obtained by Poettmannand Carpenter (P-C) in differentialform, valid for an infinitesimal lengthof flow string.1

P-C equation (1) was obtained on

Table1

192

Fig.6

Pressure traverses

-./

the basis oi an energybalance,wher~ ~fue energy los8 was calculated by th

Fanning equation. The two-phase tota1'"

Nomenclature

Bg, gas umefactor, cu mIs cu mBo, oil e factor, cu mIs cu m

STOBt, bi-phase volume factor, cu mIs

-.0Imeter ID, mphase loss factor, dimen-

sionlessMt, bi-phase ,mass factor, kg/cu m

STOpressure, bar absoil volumetric flow rata, cu mIsec

gas volumetric flow rata, cum/secRs, gas solubility, s cu mIs cu m STORgo, gas oil ratio, s cu mIs cu m STOg, acceleration. of gravity, bar abs

(sq m/kg)oil density, kg/cu mgas density, kglcu maverage temperatura along thepipa, K.

Z, gas compressibility factor,Isionless

to standard

~

/

n,

Tabte 2Runge-Kutta integration

Given a differential equation of thetype dy/dx = f(x,y),the solutionis:

= Yn++ (ko+2k+2k2+k3)/6Xn+ Ax

xf(xn,y.)k = Axf(x.+Ax/2,y.+ko/2)

k2= Axf{x.+Ax/2,y.+k/2)k3 = AX f{Xn+AX, y.+k2)

/

OIL & GAS JOURNAL - SEPT.15,1980


3/6

Program listing

Table3

J

energy loss factor was determined byP-C by analyzing production data fromGiland gas wells.

Equation (4) is a good approxi-mation (correlation factor about 0.99)for the two-phase loss factor deter-

mined by C-P (Fig. 1).The variation of the Gilvolume fac-

tor Bo and of the gas solubility factorRs can be established at fue mean

flowing temperature by diagramsbased on laboratory experiments.

The curves Bo = Bo(p) and Rs= Rs(p) can cIten be approximatedby straight liDes, equations (8) and(9), with significant deviations at lowpressure only. In this program fuegas compressibility factor Z = Z(p)has algo been approximated by astraight line, equation (11).

The two-phase mass factor Mt is

expressed as a function of Gil and

194

gas densities and of the GOR atstandard conditions, equation (3). Thetwo-phase volume factor Bt is ex-pressed as a function of tbe Gil vol-ume factor Bo, tbe gas volume factorBg, the gas solubility Rs, and Rgo,

the GOR, equation (10).Equation (1) is a function of pres-sure only, if temperature is assumedto be constant and the Gil rate, fueGOR, the ID of the pipe, and tbe P-V-Tcharacteristics of gas and oil' areknown.

Tbe differential expression of pres-sure gradient in equation (2) by Mur-aviev and Krylov (M-K) was obtainedby correlations of laboratory experi-mental data.2

Tbe first and second terms in equa-tion (2) represent fue loss due to tbecoexistence of two flowing pbases.Tbe tbird and fourth terms account

'-'

J

for the friction losses tbat would arise

sbould gas and Gilbe flowing alone.Knowledge of pbysical and thermo-

dynamic fluid cbaracteristics permitsexpression of fue rigbt-band sirle ofequation (2) as a function of pressure

only. Equati~ns (5) to (12) are validfor constant values of tbe Gil rate,fue GOR, tbe average temperaturealong tbe pipe, and fue pipe ID.

The pressure traverse along fue pipeis from numerical integration of dif-ferential equations (1) and (2), usingtbe four-step Runge-Kutta integrationmetbod (Table 2).

Using the programo Table 3 is tbeprogram listing. The steps to be ex.ecuted from tbe instructions (Table ' 4) are:

Load the program cardoPress key A to initialize registers.Enter data by pressing key B.

GIL & GASJOURNAL- SEPT.15, 1980

*

4 InilializeRCL7

-51 RCL9 3 924 _nn-_n_n_n x,y?

211 TOa 2214TX - 4 RTN

T ' 35 45 Enler "L"Ld

data 8 36 B

23R 6 36

*IBIe I '1 131-Seii-------STOI35 46 Ihe melhod

mn--n-----n -- --- n n -- ---

-55,-:-

ST02 35 2 The Runge-ST05 35 5

RS2 2 CHS -22-24 Kutta RCLA 361

,d 23161inlegralion

+ -55ST03 35 P 16-51

melhod R 7 36 7-]2

+ -55 RCLE BIRCL3 36 3 RCL6RCL4 36. -5,

-55---=3s

RCLE I 3615RCL2 36-35...ti- 5

CLPRTX

7 Prinlo Bo

C 7 +T -1 resulls STO

RTN

" 16- MIR LB

,

i....'S..""".Y"'MnOG CO.."". ,.o. ,.".. """",oro+

,,"T 1 3RTN 24 mn nn_n_n, 21SBb 231612

RCLO 3614'ro -2

x' 53 yxand

1 1Poettmann RCL 364 ,"o RCLl 36 1

and el 36. - - 5e STO 35

Carpenler' -35

"" yx + -55RCL0 '36

Krylovformula RCLl 36 1- ,

+ - 5RCL1 36 1 K

-35 RCLO 36 14RCL 36 1 1

... , -24formula

-- __n__--_-- K*LBL1 1 1 ,

23R 61 ,

X 53 7 7Pi 16-24 EEX -23K 5 05

H -, -24 -35' -55 +

1 X 52 1- ". +

C B 12 C 33 ,

, -3 RC 6,

'" Muiaviev -

'- , -35

xR 3 B

,ro +- -

- --- -

LABELS FLAGS SETSTATUS"'tial" "oterd." e PO ""

D 'u"d o TRIG DI SsSTART FLAGS

'Be Me

" u,.d 'u"d '

o' D"I I

000 DEGO FlXOP C , MK ' , . , 'DO GRADO SCIO

, " , g B RADO NG20'e


4/6

Table 4

User instructions

Select the calculation method by en-. tering cpfor P-C or 1 for M-Kmethods.

Press key C.Start the program by pressing key

D.

At the end of each integration step,the program prints the couple of

values p and h. Thereafter there isa pause of about one secando Duringthis pause it is possible to change thelength of the integration, step .:lh byexecuting these instructions:

Press key R/S.Enter the new .:lh.STO6.

Press key R/S.This program option may be used

to reduce execution time of the cal-culation. The execution time of each

integration step is about 30 secondsfor the C-P method and about 50 sec-onds for the M-K method.

Input data used to calculate the

pressure traverses of Figs. 2 and 3using the C-P method and M-K meth-od, respectively, are shown in Table5.

To test the program, the input data ..and the results obtained for two ex-

amples of ca]culation may be used(Table 6).The program uses metric units and

pressure expressed in bars absolute.

Single phase approach

Using a third theory a second pro-gram for the HP 67/97handheld cal-culator can be developed.

Similar to the earlier work, fuetheory of M. R. Tek treats multiphaseflow of liquid and gas as the flow of asingle phase with combined proper-ties.4 Using an energy balance, the

196

J

Table 6

,,- K

1~~.~0

~.~0'"105.~~ n"1611.~~ ..."327.~~ n..

1.15 ""1.5~~"~~~1-~3..,

".~~ .,**I.b~ *"1.~1 H'

-2.0~000000-03",3.410000000-03H'

U20.00 ***0.>5 ..*

500.0~ *""6.944444444-04 **'9.UI000~000-05 '"

0.lb "H'

.10'00 ""

0.00 "'.'10\.00 ""lb00.00

327.00 ""LIS'"

l.500~00000-03,'"10.00..,0.80,..1.00".

-c .00000000-03,..3.;10000000-03",

"20.00...0.Y',H'

500.00",e.944444444-04 '"9."'0000000-05 ".,

0..06.",

109.35'"100.00 ... 0

"7.96 ""100.00 ..,

110.99 "',200.00 ,..

113.79."200.00...

114.09""300.00n.

110.32,..300.00".

153.69'*.14".00 ... 173.19 , ..1400.00...

E7.03...1\00.00,.,

178.5" '"1000." n,

ID4.0I..,lb00 00 no

theory leaves out of consideration thecomplex mechanisms caused by thecoexistence of numerous flow pat-tems, the variation of the flowingfluids' specific volume with pressureand temperature, and the slippagelosses.

Friction losses are accounted for byusing the concept of a "two-phase ffactor," by Poettmann and Carpenter(P-C). It is in the form of an em-pirical correlation as a function of the"viscosity-less" Reynolds' number pvd(density)(velocity) (diameter) of theflow pipe.

Tek finds a two-phase total energyloss factor by using field data fromseveral flowing and gas-lift wells. The ' /concept of the "two-phase Reynolds'number function" and of the massratio of gas to liquid are used.

Similar to the work of P-C, the two-

OIL & GAS JOURNAL - SEPT. 15, 1980


5/6

"-'

-'"

'-"

phase friction factor is a function of

the liquid and Gil viscosity.The pressure traverse is evaluted

Flow equations

dp/dh = Mt/Bt (g + 32f qon2

Program listing

by integrating a differential equationof the type dp/dh = f(p,T). Depth andpressure along the pipe (h and p) areas used earlier.

The function f(p,T) is a function ofpressure and temperature which var-ies according to the P"V-T character-istics of the flowing fIuids. Tempera-ture is assumed to be constant in this

programoEquations. The HP fJl/97 was pro-

grammed to salve the flow equations(Table 7). The expression of the pres-sute gradient, equation (1), is validfor an infinitesimal length of flowstring. It comes from the well-knownflow equation, neglecting change ofkinetic energy of the fluido

Equation (2) is an approximationfor the two-phase loss factor proposedin a diagram by Tek (Fig. 4).

Equations (4) and (5) are the

Reynolds' numbers that gas and Gilwould assume flowing alone in thepipe. The gas/liquid ratio, equation(9), varies along the string accordingto the pressure.

Linear pressure approximations-equations (12), (13), and (14)-are forthe Gil volume factor Bo, the gassolubility Rs, and the gas compressi-bility factor Z. -

Two-phase mass factor Mt is afunction of the Gil and gas densitiesand GOR at standard conditions,equation (11). -

The two-phase volume factor Bt isexpressed as a function of the Gilvolume factor Bo, the gas volumefactor Bg, the gas solubility Rs, andRgo, the gas/Gil ratio, equation (10).

The assumption of constant temper-ature along the pipe makes the righthand sirle of equation (1) a function

Table 8

OIL & GAS JOURNAL - SEPT. 15, 1980 199

.""t ...,,""'" ""'. . .'M"M'. ". . .. t. .. K'V. ,"""""""" """.'N"" .tE.. ........",..,..""",..,"".""' .,",.....c..."..yO"'" ...,....t..

o"' "LBLA 11 + 052 7 7 11 RCLl 3646 ,," RCLC 36

STOl 3546 Initialize 059 -35 11 RCL7 36 7 l7l 35

4 CLX 51 - ss Method 116 -35 172 RCL6 36 06No005 RTN 24 nhhh..hmn-- 061 6 06 117 RCL6 36 6

173

0 06 "L BLB 2112 062 24 118 + -ss 174 RCL5 36 05

007 PRTX -14 063 STo' 35 ss , ll9 -35 175 62008 SToi 3545 Enter data 064 sPC 16-11 'w RCLl 36 46 176 1 1

, 1521 16 26 46 065 RCL' 36 0' 121 RCL3 36 3 1RT 24

m_mm_hnnn66 PRTX -14 Print 122 -35 178 ex 33

11 OLBLC 21 13 067 RCL8 3 6 0 8 123 RCL2 3 6 0 2 1791 X 52

12 RCLC 36 13 068 PRTX -14 resulls 124 + SS '"" yX 31

13 STO1 35 1 69 PSE 1651 125 P;" 16-51 181 :qy-41

14 4 4 "" RCL7 36 7 126 + -55BI

182 RCL5 3 5

15 Pi 16-24 71 X> 2 16-44 127 STO 35 1835 5 The

16 24 072 GTOe 2 2 1 6 1 5 128 RTN 24 184+ 5

17 RCLE 3 6 1 5 73 RCL8 3 6 0 8 129 'LBL2 21 02 185 RCL536 5

18 " 35 74 X>02 16-44 ,," G58b 2 3 1 6 1 2 186 -2RCLD 3614 075 GTOD 2 2 1 4 131 RCLl 36 01 187

1 X 5

' -24 76 RTN 24 132 EEX -23 188 yx Tek0 8 e 211615 133 3 3 189 -35

022 RCL8 3612 78 RCL8 36 8 4 " 35' LOG 1632

23 24 79 ""5 16-51 191 62

024 STD8 3512 "'o RCL0 36 00 1'6 9 "9 192 1 01025 2 0 2 081 S 16-51 137 RCLD 3614 193 RCL5 36 05 formula026 RCLO 3614 82 X>Y? 16 34 138 4 4 194 -35

027 24 83 GTOD 22 14 195 - 45028 5 0 5 084 RTN 24 '" 196 4 4

29 yX 31 85 '-LBLd 211614 141 -24 197 " 35RCLE 3 615 086 RCL' 36 09 142 eX 33 198 CHS 22

31 Pi 16-24 087 + -SS 143 ' 24 199 ex

032 24 88 STOl 3546 144 yx 31 "" 2 233 " G58 23 2 145 LSrX 1663 21 234 " 35 "" RCL7 36 7 146 X' 53 22 -35

035 STOE 3515 091 -35 147 -35 23 RCL0 36 00

036 "L8LD 2114 092 RTN 24 _mnhnn._h. 148 EEX 23 204 XZ 5337 RCL9 36 8L 149 3 3 205 3538 STD 35 46 94 PS 1651 , , 24 26 RCLE 361539 GS82 23 2 95 RCL5 36 05 TD6 6

/-LosZ 7 x -35

RCL7 36 7 The 96 RCLl 36 47 152 RCLA 36 11 208 9 941 ST+8 35-SS 8 97 -35 153 RCLO 36 14 29 8 08

42 -35 98 RCL4 36 4 154 - 45 ,," EEX -23043 T02 35 2 99 + -55

RS1SS nS 16-51 2l1 6

44 2 2

Runge-Kutta

'"" STOD 351 156 RCL' 36 09 212 CHS -22045 ' -24 11 CRS 22 . 6 1 213 + -SS046 GSBd 231614 12 RCLA 1 158 " 35 214 RCLA 36 11

47 ST03 35 3 13 + -SS 2 1651

48 2 2 14 3 3,..

RC B 36 12Ng-24 15 4 04 161 -35 217 -35

GSBd 231614 inlegration 16EEX -23 614 218 RCL8 36 08

051 ST04 35 4 1 S 16-51 219 + -SS

52 G58d 231614 18 CHS 2 "" no 16-512 6 2 19 35 6 -35 221 "

54 + SS ,," RCLl 36 1 166 ReLS 36 8 222 CL 36SS RCL3 36 3 III -35 6 S 1651 223 24

56 ReL4 36 4 112 RCLl 36 46 16 + SS 24 RTN 24REGISTERS lABElO FlAGS SETSTATUS

Bt l' 'o d l' k o 13 k1 l ' k2;

K l' '" l' oh l' h l' p 'Initi"iz BEntee d,t,e StoctO

", dE o

FlAGS TRIG DISP

hm,,!" T" 1" '1 1" '2 l"b 1"

1"e, 1" e2"

pon 1" Pgn

'Bt "sed

'"sed "sed

' ONO"

DEGo I F IX '"o o O

b2 o , , T E K3 , , , O O G RA OO S Ol O

,'go lB "sed le "sed

o "sed E ",ed l' "sed ; , , , , 3' O O

R AO O N G2O3 . O'h


6/6

InstructionsTable 9

of pressure only. The form is deter-mined by specifying the P-V-T char-acteristics of flowing fluids-equations(11) to (15)-the ID of the pipe, theGilflow rate, and the GORat standardconditions.

Equation (6) shows the relation be-

tween dead Gil viscosity (P,Od) andsaturated Gil viscosity (P,os).5 Equa-tions (7) and (8) are good approxima-tions for the function of Gil solubilityA(Rs) and B(Rs).5

The pressure traverse along the pipeis obtained by numerical integratio nof the differential equation (1). Thefour-step Runge-Kutta integrationmethod allows good approximationusing large steps.

The programo Table 8 i s theprogram listing. The instructions tellhow to use the program (Table 9).These steps must be executed:

200

Load the program cardoPress key A to initialize registers. .Enter data by pressing key B.Start the program by pressing key C.At the endof each integration step,

the program prints the couple ofvalues p and h. There is a pause of

about one second, during which timeitis possible to change fue length ofthe integration step Ah by executingthese instructions:

Press key R/S.Enter the new integration step.STO7.Press key R/S. This program option

may be used to reduce the executiontime of the calculation, which is about60 seconds for each integration step.

A test case may be helpful in test-ing the program (Table 10).

Caution must be taken in the choice

of the gas/Gil ratio, Rgo. If during the

Test caseAh 100.00ho 0.00Po 105.00hmax 1600.00Tav 327.00a, 1.15a2 1.50 (10}-3

b,

b2

10.00

0.801.00

--0.002820.00

0.95500.00

2.50 (10}-53.00 (10)"30.0626.944(10)-4

108.57100.00112.19200.00115.86300.00119.57

400.00151.591200.00155.92

1300.00160.34

1400.00164.84

1500.00169.42

1600.00

C,C2ponpgnRgo

/Lg/Lo.di

qon

Table 10

mmbarsabsmKcumIscumSTOcum/(scumSTOxbarsabs)scumIscumSTO

scum/(scumSTOxbarsabs)dimensionless(barsabs)-'kg/scumkg/scumscumIscumSTOPalsecPa/secmseum/sec

--

~

calculation the physically impossible '-"condition of Rs(pRgo is met, theprogram will go into error.

Some pressure traverses obtained

from fue program are shown (Fig. 5).A comparison is shown for the pres-SUTetraverses calculated by the P-C,M-K (Muraviev and Krylov), and Tekmethods (Fig: 6). The Tek nethodgenerally gives average gradients ofpressure greater than the other two.6

Nomenclature is defined in the ac- '-companying box. The internationalsystem of measure (SI) has been usedin this programo Pressure is in barsabsolute.

References

1. Poettrnann, F. H., and Carpenter,P. G., "The rnulti-phase flow of gas, oil.and water through vertical fIow strings,"API Drilling and Production Practice, Dal-Ias, 1952.

2. Muraview, 1. M., and Krylov, A. P.,"Ekspluatatsiya neftyanykh rnestorozhdeniy,"Gostoptekhizdat, Moscow, 1949.

3. Szilas, A. P., "Production and trans-pOrt of oil and gas," EIsevier, Budapest,1975.

4. Tek, M. R., "Multiphase flow of water,oil and natural gas through vertical flowstrings," JPT, October 1961, pp. 1029-36.

5. Chew J. and Connally, C. A., Jr., "Aviscosity correlation for gas-saturated crude ~-r-oils," Trans. AIME, 1959, 216, p. 23-25.

6.0rkiszewski, J., "Predicting two-phasepressure drops in vertical pipe," JPT, June

1967, pp. 829-38.OIL & GASJOURNAL- SEPT.15,1980

STEP INSTRuenONSINPUT

KEYSOUTPUT

DATA/UN'TS DATA/UNITS

1 I ni tia Hze [L] CJ 0.00- 2 Enter Llh n Llh [LI CJ Llh

3 Enter ho n ho C!::JCJ ho4 Enter po po IT::J CJ po5 Enter hmax n hmax C!::JCJ hmax6 Enter Tav Tav C!::JCJ Tav7 Enter al a, C!::Jc=J al8 Enter a2 a2 QDCJ a29 Enter b1 b1 IT:J CJ b1

10 Enter b2 b2 [I] CJ b211 Enter c c [LI CJ c112 Enter c2 c2 [I] CJ c213 Enter Pon Pon [I] CJ Pon14 Enter Pgn Pgn [I] CJ Pon15 Enter Reo RgO [LI CJ RgO16 E nt er U g Ug [I] CJ Va17 Enter Uod Uod C!::Jc=J Uod18 Enter di di [I] CJ di19 Enter qon qon IT:Jc::J qon20 Start u::Jc::J P1' h1

CJCJP2, h2

CJCJ .......CJ c::J Pn, hn

21 Return to step 1 for a new case CJCJCJCJCJCJCJCJCJCJCJCJ

(O)Steps 2, 3 and 5: if Llh

Documents

Vertical 2 Phase Flow