-- ---Fig.l
.
P-C bi-phase friction 10ss '--"
3f'1
10 f
= 66(Mlq.. I dlr3
Pressure traverse found
~
$
U)
for 2-phase vertical flow
~11O-1' e
.. ,g
~
program using assigned P-V-T and other properties are first developed. Two theories are incorporated into fue programo Then the treatment of muItiphase flow of liquid and gas as flow of a single phase with combined properties by an energy balance will be analyzed. Friction losses are accounted for differentIy in this approach. The resuIt is a second program for the HP 67/97calculator. Laws controlling two-phase flow in
G.c. BORGIA G. GOITARDI University of Bologna Bologna, Italy
10-2
10-3
102
10 Mt(qon) d1
--
TIME-CONSUMING calculations for pressure traverse in vertical pipe with flowing fluids have been streamlined into two programs for fue HP 67/97 hand-held calculator. Using existing theories for such flow, fue flow equations and a calculator
Fig.3
Fig.2
Pressure traverses by th P-C method
Pressure traverses by the M-K method o
o
qon = 60 cu m/d di = 6.2 cm
qon = 60 cu m/d di = 6.2 cm
2
2
E
8 °. ~
-4
4
E:
E:
~
15. Q)
~
15.
o
Q)
o
6
6
8
8 1
50
1
50
100
150
250
Pressure P. bars
........
OGJ
OIL & GAS JOURNAL
-
SEPT. 15, 1980
191
Fig.5
Fig.4
Friction 10ss factor (Tek)
Pressure traverses
(Tek)
Fig.6
Pressure
traverses
-./
00:1
temperature along fue pipe is known, the pressure traverse is evaluated by integration of a differential equation f(p), where: p oi, the type dp/dh pressure; h depth; and f(p) is a function of the pressure that varies according to the different theories and to the P-V-T characteristics of fluids. By using the theory implying fue easiest mathematical formulation of the problem, the engineer can determine the pressure traverse along a vertical pipe by laborious hand calculations, or by using' gradient curve tables. TheHP 67/97 calculator was programmed to salve the flow equations (Table 1). Equation (1) is the pressure gradient obtained by Poettmann and Carpenter (P-C) in differential form, valid for an infinitesimal length of flow string.1 P-C equation (1) was obtained on
=
vertical pipe are complicated because of the variation in flowing fluids' specific volume with pressure and temperature changes. Slippage losses and the variety of flow patterns that liquid and gas may have during upward flow also complicate the picture. Results from theoretical proposals are incomplete, but in fue flow region of practical importance, equations of satisfactory accuracy have evolved. Designed for the HP 67/97 calculator, this program evaluates pressure traverses in pipe for assigned P-V-T properties of fue fluids, the oi! rafe, the GOR, and ID of the pipe. Flow equations. When the average
=
=
Table1
Flow equations
the basis oi an energy balance,wher~
~
fue energy los8 was calculated by th Fanning equation. The two-phase tota1'"
Nomenclature Bg, gas umefactor, cu mIs cu m Bo, oil e factor, cu mIs cu m STO Bt, bi-phase volume factor, cu mIs -.0 Imeter ID, m
phase loss factor, dimensionless Mt, bi-phase ,mass factor, kg/cu m STO pressure, bar abs oil volumetric flow rata, cu mI sec gas volumetric flow rata, cu
~
m/sec
Rs, gas solubility, s cu mIs cu m STO Rgo, gas oil ratio, s cu mIs cu m STO g, acceleration. of gravity, bar abs (sq m/kg) oil density, kg/cu m gas density, kgl cu m average temperatura along the pipa, °K. Z, gas compressibility factor, Isionless to standard n,
/
Tabte 2
Runge-Kutta integration Given a differential equation of the f(x,y),the solution is: type dy/dx
=
= Yn+¡+ (ko+2k¡+2k2+k3)/6 Xn+ Ax Áxf(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)
192
OIL & GAS JOURNAL
- SEPT.15, 1980
/
/
Table3
Program
listing
J i....'S..""".Y"'MnOG RCL7 RCL9 x,y? TOa RTN "L"Ld 8
4 Inilialize -51 24 _nn-_n_n_n 211 - 4 35 45 Enler data
TX T '
3
9
,
2214 'ro 36 B
24 21 SBb 231612 RCLO 3614 -2 x' 53 1
* IBIe STOI
I
23 36
R 6
,d ST03
35 5 -22 361 -55 16-51 36 7 -]2
.
*LBL1
RCLE RCL2 CL PRTX C7 T
o + STO RTN
7 Prinl - 1 resulls
I
16-
R LB
,
1
-24
1 X
-55 52 -
,
3
-
5
35 -35 -55 '36
RCLl
36 1
K RCLO 1
36 14 1
Krylov
formula
K
, , 7 EEX 5 H
7 -23 05
-
-35 +
".
+ C , RC ,
12 -3
'"
36
STO ' + RCL0
-- __n__--_--
53 16-24
,
CB
and
36 36 1
5
61
1
3 6
Muiaviev
-
'x
"
,
1 -35
1
'
Bo
formula
23
X Pi K
-55 ---=3s 3615 36 -35 ...ti - 5
-
36 -24
,
BI
-5,
36
RCL
...
RCLE RCL6
-55 36 3 36
el
Carpenler
+ RCL1
RS
R
+ RCL3 RCL4
RCL RCLl
,"o
and
yx
,-:-
ST05 CHS RCLA + P R 7
yx
-
-55
35 2 The Runge2 Kutta -24 23161 inlegralion 35 melhod
nn_n_n
e
""
ST02 2
mn
1 Poettmann 4
.
mn--n-----
'1 131-Seiééi------35 46 nIhe -- ---melhod n n -- ---
"""",oro
,,"
3
T1 RTN
,."..
,.o.
CO.."".
+
*
R
,ro
-
, +
3
-35 B -
MI
- --"'tial" P C
LABELS "oterd." e PO "" D START ' u"d ' 'u"d Be Me " u,.d , MK ' ,
,
"
'e
.
o '
, ,
FLAGS
'-'
-
SETSTATUS
FLAGS 000
TRIG DEGO
DISs FlXO
'DO GRADO SCIO B I RADO I NG20 o'g D"
1
J energy loss factor was determined by P-C by analyzing production data from Giland gas wells. Equation (4) is a good approximation (correlation factor about 0.99) for the two-phase loss factor determined by C-P (Fig. 1). The variation of the Gilvolume factor Bo and of the gas solubility factor Rs can be established at fue mean flowing temperature by diagrams based on laboratory experiments. The curves Bo Bo(p) and Rs Rs(p) can cIten be approximated by straight liDes, equations (8) and (9), with significant deviations at low pressure only. In this program fue gas compressibility factor Z Z(p) has algo been approximated by a straight line, equation (11). The two-phase mass factor Mt is expressed as a function of Gil and
=
=
=
194
gas densities and of the GOR at standard conditions, equation (3). The two-phase volume factor Bt is expressed as a function of tbe Gil volume factor Bo, tbe gas volume factor Bg, the gas solubility Rs, and Rgo, the GOR, equation (10). Equation (1) is a function of pressure only, if temperature is assumed to be constant and the Gil rate, fue GOR, the ID of the pipe, and tbe P-V-T characteristics of gas and oil' are known. Tbe differential expression of pressure gradient in equation (2) by Muraviev and Krylov (M-K) was obtained by correlations of laboratory experimental data.2 Tbe first and second terms in equation (2) represent fue loss due to tbe coexistence of two flowing pbases. Tbe tbird and fourth terms account
for the friction losses tbat would arise sbould gas and Gilbe flowing alone. Knowledge of pbysical and thermodynamic fluid cbaracteristics permits expression of fue rigbt-band sirle of equation (2) as a function of pressure only. Equati~ns (5) to (12) are valid for constant values of tbe Gil rate, fue GOR, tbe average temperature along tbe pipe, and fue pipe ID. The pressure traverse along fue pipe is from numerical integration of differential equations (1) and (2), using tbe four-step Runge-Kutta integration metbod (Table 2). Using the programo Table 3 is tbe program listing. The steps to be ex. ecuted from tbe instructions (Table ' 4) are: Load the program cardo Press key A to initialize registers. Enter data by pressing key B. GIL & GASJOURNAL- SEPT.15, 1980
"
I 4
Table 4
User instructions
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.",
"7.96 "" 100.00 ..,
109.35 '" 100.00 ...
110.99 "', 200.00 ,..
113.79 ." 200.00 ... 110.32 ,.. 300.00 ".
114.09"" 300.00n.
153.69'*. 14".00 ...
E7.03... 1\00.00,.,
I
0
173.19, .. 1400.00... 178.5" '" 1000." n, ID4.0I..,
lb00 00 no
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 is a pause of about one secando During this pause it is possible to change the length of the integration, step .:lh by executing 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 calculation. The execution time of each integration step is about 30 seconds for the C-P method and about 50 seconds for the M-K method. 196
Input data used to calculate the pressure traverses of Figs. 2 and 3 using the C-P method and M-K method, respectively, are shown in Table 5. To test the program, the input data .. and the results obtained for two examples 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 program for the HP 67/97 handheld calculator can be developed. Similar to the earlier work, fue theory of M. R. Tek treats multiphase flow of liquid and gas as the flow of a single phase with combined properties.4 Using an energy balance, the
theory leaves out of consideration the complex mechanisms caused by the coexistence of numerous flow pattems, the variation of the flowing fluids' specific volume with pressure and temperature, and the slippage losses. Friction losses are accounted for by using the concept of a "two-phase f factor," by Poettmann and Carpenter (P-C). It is in the form of an empirical correlation as a function of the "viscosity-less" Reynolds' number pvd (density)(velocity) (diameter) of the flow pipe. Tek finds a two-phase total energy loss factor by using field data from several flowing and gas-lift wells. The ' concept of the "two-phase Reynolds' number function" and of the mass ratio of gas to liquid are used. Similar to the work of P-C, the twoOIL & GAS JOURNAL - SEPT. 15, 1980
/
of the type dp/dh
= f(p,T).
Depth and
pressure along the pipe (h and p) are as used earlier. The function f(p,T) is a function of pressure and temperature which varies according to the P"V-T characteristics of the flowing fIuids. Temperature is assumed to be constant in this programo Equations. The HP fJl/97 was programmed to salve the flow equations (Table 7). The expression of the pressute gradient, equation (1), is valid for an infinitesimal length of flow string. It comes from the well-known flow equation, neglecting change of kinetic energy of the fluido Equation (2) is an approximation for the two-phase loss factor proposed in a diagram by Tek (Fig. 4). Equations (4) and (5) are the
Flow equations dp/dh = Mt/Bt (g + 32f qon2
-,,"'~
Reynolds' numbers that gas and Gil would assume flowing alone in the pipe. The gas/liquid ratio, equation (9), varies along the string according to the pressure. Linear pressure approximationsequations (12), (13), and (14)-are for the Gil volume factor Bo, the gas solubility Rs, and the gas compressibility factor Z. Two-phase mass factor Mt is a function of the Gil and gas densities and GOR at standard conditions,
by integrating a differential equation
phase friction factor is a function of the liquid and Gil viscosity. The pressure traverse is evaluáted
equation (11).
-
The two-phase volume factor Bt is expressed as a function of the Gil volume factor Bo, the gas volume factor Bg, the gas solubility Rs, and Rgo, the gas/Gil ratio, equation (10). The assumption of constant temperature along the pipe makes the right hand sirle of equation (1) a function Table 8
Program
listing
.""t ...,,""'" o"' "LBLA 7 2 4 005 006 007 008
, ""-'
,
11 12 13 14 15 16 17 18
3546 51 24 2112 -14 3545 16 26 46 24
OéLBLC RCLC STO1 4 Pi
21 13 36 13 1 35 4 16-24 24 36 15 35 3614 -24
RCLE " RCLD
RCL8
022 23 024 025 026 027
5
31 032 33 34 035 036 37 38 39
" STOE "L8LD RCL9 STD GS82 RCL7
8 -35
35-SS
35 2 2 -24
043 44 045
T02 2
046 47 48
GSBd ST03 2
231614 35
051 52
GSBd ST04 G58d
231614 35 231614
'
'go
T"
""
71 072 73 74 075 76
78 79
"'o 081 82 83 084
85 086 087 88 "" 091 092 94 95
The
96 97 98
99
'"" 11
Runge-Kutta
12 13 14 15 16 1 18
4
inlegration
6
2 SS 36 3 36 4
l' 'od
hm,,!"
067 068 69
3 2 -24
2 + RCL3 ReL4
Bt
,
35 3515 2114 36 35 46 23 2 36 7
ST+8
54 SS 56
Enter data
l' ko
1" '1 lB "sed
,,"
RCL8 ""5 RCL0 S X>Y? GTOD '-LBLd RCL' + STOl G58 RCL7 RTN 8L PS RCL5 RCLl
l"b1
36 08 -14 1651 36 7 16-44 22 16 15 36 08 16-44 22 14 24
RCLA + 3 4 EEX
1 -SS 3 04 -23
3646
11 116 117 118 ll9 'w 121
RCL7
36
RCL6 +
36
resulls
,," 131 132
RCLl RCL3
-35 36 02 SS 16-51 -55 35 24 21 02 23 16 12 36 01 -23
1'6 137 138
9 RCLD 4
_mnhnn._h.
,
RS
141 142 143 144 145 146 147 148 149
-24 33 24 31 1663 53 -35 23 3 24 6
EEX 3 , TD6
152 153 154
RCLA RCLO
36 36
1SS 156
nS RCL'
45 16-51 36 09
158
"
,..
-
.
6
RC B
19
RCLl
"
'
""
/-Los
l' oh 1" e2
b2 1" e, o "sed E ",ed
6 166 6 16
l' h "
pon
l'
l' p 1" Pgn "sed 'h
-35 36 8 1651 SS
ReLS S +
'Initi"iz
BEntee
o ;
, ,
'
Bt
d,t,e
1 ex 1 X yX :qy RCL5 5 + RCL5
LOG 191 192 193 194 195 196 197 198 199
1 RCL5
-
RCL0 XZ RCLE x
", d ' "sed 3
,
,
,
"sed
2 -35 36 00 53 35 3615 -35
9 8
+ RCLA
9 08
o '
, 3
-23 -22 -SS 36 11 1651
RCL8 +
-35 36 08 -SS
no
16-51
" CL
221 222 223 24
formula
4 35 22
EEX 6 CHS
217 218 219
Tek
45
" CHS ex 2 2
""
E
The
5
-35 1632 62 01 36 05 -35
4
2l1 212
lABElO O
33 52 31 -41 3 5 5 5 36 5
-2
213 214 2
Ng
36 05 62 1
1 X yx
189
,,"
Stoct "sed , TEK
,
No RCL5
208 29
2
35 36 1 -35 36 46
35 36 06
21 22 23 204 205 26 Z 7
1 35
36 12 -35 614 16-51
161 S
CHS
11 14
RCL6
181 182 183 184 185 186 187 188
"9 3614 4
eX ' yx LSrX X'
36
178 179
BI
...,....t..
05
RCLC
'""
3 35
3
"
l7l 172 173 174 175 176 1
7 -35 6 -ss -35 36 46 36 3
RCL2 + P;" + STO RTN 'LBL2 G58b RCLl EEX
133 4
.,",.....c..."..yO"'" ,,"
'"
1651 36 05
-55 351 22
RCLl
122 123 124 125 126 127 128 129
24
211614 36 09 -SS 3546 23 2 36 7 -35 24
11
Print
211615 36 8 16-51 36 00 16-51 16 34 22 14
RCL4
l' '"
1" '2
35 ss , 16-11 36 0' -14
36 47 -35 36 4
+ STOD CRS
Method
06 24
RTN
K
l' k2 "sed
RCL8 PRTX PSE RCL7 X> 2 GTOe RCL8 X>02 GTOD RTN 0é8 e
RCLl
13 k1 le
STo' sPC RCL' PRTX
III 112 REGISTERS
;
ss
-35 6
.,"".""'
.tE.. ........",..,..""",.
+
061 062 063 064 065 66
m_mm_hnnn
""".'N""
K'V.,""""""""
059
31 3615 16-24 24
"
41 42
-
Initialize nhhh..hmn--
05
yX RCLE Pi
29
...t...
3612 24 3512 02 3614 24
STD8 2 RCLO
028
..«'M"M'.".
11 7
STOl CLX RTN "LBLB PRTX SToi 1521 RT
'
'-'"
""'.
36
RTN FlAGS
24 24
FlAGS
ONO" o o O , O O ' O O 3 .
O
SETSTATUS TRIG
DISP
GRAOO RAO DEG oO
SOl O NG2O FIX
I
'"
""'-"
OIL & GAS JOURNAL - SEPT. 15, 1980
199
Table 9 Table
Instructions
Ah ho Po
STEP
INSTRuenONS
INPUT DATA/UN'TS
KEYS
[L] CJ
1
InitiaHze
- 2
Enter
Llh
3
Enter
ho
4
Enter
po
5 6 7 8
Enter Enter Enter Enter
hmax Tav al a2
9 10
Enter Enter
b1 b2
b1 b2
11 12 13 14 15
Enter Enter Enter
c¡ c2 Pon
Enter Enter
Pgn Reo
c¡ c2 Pon Pgn RgO
16
Enter
Ug
Ug
n n
Llh ho
po
n
hmax
Tav a, a2
17
Enter
Uod
Uod
18 19
Enter Enter Start
di qon
di qon
20
[LI CJ C!::JCJ IT::J CJ
C!::JCJ C!::JCJ
C!::Jc=J QDCJ IT:J CJ
Return
to step
1 for a new case
if Llh
(O)Steps 2, 3 and 5:
of pressure only. The form is determined by specifying the P-V-T characteristics of flowing fluids-equations (11) to (15)-the ID of the pipe, the Gil flow rate, and the GORat standard conditions. Equation (6) shows the relation between dead Gil viscosity (P,Od) and saturated Gil viscosity (P,os).5 Equations (7) and (8) are good approximations for the function of Gil solubility A(Rs) and B(Rs).5 The pressure traverse along the pipe is obtained by numerical inte gratio n of the differential equation (1). The four-step Runge-Kutta integration method allows good approximation using large steps. The programo Table 8 is the program listing. The instructions tell how to use the program (Table 9). These steps must be executed:
200
OUTPUT DATA/UNITS
0.00
Llh ho po
hmax Tav al a2 b1 b2 c1
[I]
CJ
[I] [I] [I]
CJ CJ CJ
c2 Pon Pon
[I]
CJ
Va
[I]
CJ
IT:Jc::J
di qon P1' h1
CJCJ
P2, h2
CJ c::J CJCJ CJCJ CJCJ CJCJ CJCJ CJCJ
Pn, hn
[LI CJ
[LI CJ
C!::Jc=J
u::Jc::J CJCJ
21
10
Test case
RgO
Uod
.......
CJ r=J
CJ c::J CJ c::J CJCJ c::J c::J c::J c::J c::J CJ CJCJ c::J CJ CJCJ
Load the program cardo Press 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 of values p and h. There is a pause of about one second, during which time itis possible to change fue length of the integration step Ah by executing these instructions: Press key R/S. Enter the new integration step. STO7. Press key R/S. This program option may be used to reduce the execution time of the calculation, which is about 60 seconds for each integration step. A test case may be helpful in testing the program (Table 10). Caution must be taken in the choice of the gas/Gil ratio, Rgo. If during the
100.00 0.00 105.00
m m barsabs hmax 1600.00 m -Tav 327.00 °K cumIs cumSTO a, 1.15 a2 1.50 (10}-3 cum/(scumSTO x barsabs) 10.00 b, s cumIscumSTO 0.80 b2 scum/(scumSTO x barsabs) 1.00 C, dimensionless --0.002 C2 (barsabs)-' 820.00 pon kg/scum 0.95 pgn kg/scum 500.00 scumIscumSTO Rgo 2.50 (10}-5 Palsec /Lg 3.00 (10)"3 Pa/sec /Lo. di 0.062 m 6.944(10)-4 qon seum/sec 108.57 100.00 112.19 200.00 115.86 300.00 119.57 ~ 400.00 151.59 1200.00 155.92 1300.00 160.34 1400.00 164.84 1500.00 169.42 1600.00
calculation the physically impossible condition of Rs(p»Rgo is met, the program will go into error. Some pressure traverses obtained from fue program are shown (Fig. 5). A comparison is shown for the presSUTetraverses calculated by the P-C, M-K (Muraviev and Krylov), and Tek methods (Fig: 6). The Tek Ínethod generally gives average gradients of pressure greater than the other two.6 Nomenclature
is defined in the ac-
companying box. The international system of measure (SI) has been used in this programo Pressure is in bars absolute.
,r
'-"
----
,
'-
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, DalIas, 1952. 2. Muraview, 1. M., and Krylov, A. P., "Ekspluatatsiya neftyanykh rnestorozhdeniy," Gostoptekhizdat, Moscow, 1949. 3. Szilas, A. P., "Production and transpOrt of oil and gas," EIsevier, Budapest, 1975. 4. Tek, M. R., "Multiphase flow of water, oil and natural gas through vertical flow strings," JPT, October 1961, pp. 1029-36. 5. Chew J. and Connally, C. A., Jr., "A viscosity correlation for gas-saturated crude ~-r-"""" oils," Trans. AIME, 1959, 216, p. 23-25. 6.0rkiszewski, J., "Predicting two-phase pressure drops in vertical pipe," JPT, June 1967, pp. 829-38.
OIL & GASJOURNAL- SEPT. 15,1980
