Page 62 - Petroleum Production Engineering, A Computer-Assisted Approach

Guo, Boyun / Petroleum Production Engineering, A Computer-Assisted Approach 0750682701_chap04 Final Proof page 53 22.12.2006 6:07pm

WELLBORE PERFORMANCE 4/53
and dp f F m 2 L
r
144 ¼ þ , (4:46)
r
dz 7:413 10 D r L y 2 L
10
5
X 1 ¼ log [(N L ) þ 3]: (4:36)
where m L is mass flow rate of liquid only. The liquid
Once the value of parameter (CN L ) is determined, it is used holdup in Griffith correlation is given by the following
0:1
N vL p (CN L ) expression:
for calculating the value of the group , where 2 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
N 0:575 0:1 2
p N D
a
vG
1
p is the absolute pressure at the location where pressure y L ¼ 1 4 1 þ u m 1 þ u m 4 u sG 5 , (4:47)
gradient is to be calculated, and p a is atmospheric pressure. 2 u s u s u s
The value of this group is then used as an entry in the
second chart to determine parameter (y L =c). We have where m s ¼ 0:8ft=s. The Reynolds number used to obtain
found that the second chart can be represented by the the friction factor is based on the in situ average liquid
following correlation with good accuracy: velocity, that is,
2
2:2 10 m L
y L N Re ¼ : (4:48)
¼ 0:10307 þ 0:61777[ log (X 2 ) þ 6]
c Dm L
2
0:63295[ log (X 2 ) þ 6] þ 0:29598[ log (X 2 ) To speed up calculations, the Hagedorn–Brown cor-
relation has been coded in the spreadsheet program Hage-
4
3
þ 6] 0:0401[ log (X 2 ) þ 6] , (4:37) dornBrownCorrelation.xls.
where Example Problem 4.4 For the data given below, calculate
and plot pressure traverse in the tubing string:
0:1
N vL p (CN L )
X 2 ¼ : (4:38)
N 0:575 0:1
p N D
vG
a
Tubing shoe depth: 9,700 ft
According to Hagedorn and Brown (1965), the value of Tubing inner diameter: 1.995 in.
parameter c can be determined from the third chart using Oil gravity: 40 8API
N vG N 0:38 Oil viscosity: 5 cp
a value of group L . Production GLR: 75 scf/bbl
N 2:14 N vG N 0:38 Gas-specific gravity: 0.7 air ¼ 1
D
We have found that for L > 0:01 the third chart
N 2:14 Flowing tubing head pressure: 100 psia
D
can be replaced by the following correlation with accept- Flowing tubing head temperature: 80 8F
180 8F
Flowing temperature at tubing shoe:
able accuracy:
Liquid production rate: 758 stb/day
c ¼ 0:91163 4:82176X 3 þ 1,232:25X 2 Water cut: 10 %
3
Interfacial tension: 30 dynes/cm
3
4
22,253:6X þ 116174:3X , (4:39) Specific gravity of water: 1.05 H 2 O ¼ 1
3
3
where
N vG N 0:38
X 3 ¼ L : (4:40)
N 2:14 Solution This example problem is solved with the
D
N vG N 0:38 spreadsheet program HagedornBrownCorrelation.xls. The
However, c ¼ 1:0 should be used for L # 0:01. result is shown in Table 4.3 and Fig. 4.4.
N 2:14
D
Finally, the liquid holdup can be calculated by
4.4 Single-Phase Gas Flow
y L
y L ¼ c : (4:41)
c The first law of thermodynamics (conservation of energy)
governs gas flow in tubing. The effect of kinetic energy
The Fanning friction factor in Eq. (4.27) can be deter- change is negligible because the variation in tubing diam-
mined using either Chen’s correlation Eq. (4.5) or (4.16). eter is insignificant in most gas wells. With no shaft work
The Reynolds number for multiphase flow can be calcu-
device installed along the tubing string, the first law of
lated by
thermodynamics yields the following mechanical balance
2 equation:
2:2 10 m t
N Re ¼ , (4:42)
2
y L
Dm m (1 y L ) dP þ g dZ þ f M n dL ¼ 0 (4:49)
L
G
r
where m t is mass flow rate. The modified mH-B method g c 2g c D i
uses the Griffith correlation for the bubble-flow regime. Because dZ ¼ cos udL, r ¼ 29g g P , and n ¼ 4q sc zP sc T , Eq.
ZRT
The bubble-flow regime has been observed to exist when (4.49) can be rewritten as pD 2 T sc P
i
( )
l G < L B , (4:43) 2 2 2
zRT dP g 8f M Q P sc zT
sc
þ cos u þ dL ¼ 0, (4:50)
2
where 29g g P p g c D T 2 P
5
g c
i sc
u sG
l G ¼ (4:44) which is an ordinary differential equation governing
u m
gas flow in tubing. Although the temperature T can be
and approximately expressed as a linear function of length L
through geothermal gradient, the compressibility factor z
u 2
L B ¼ 1:071 0:2218 m , (4:45) is a function of pressure P and temperature T. This makes
D
it difficult to solve the equation analytically. Fortunately,
which is valid for L B $ 0:13. When the L B value given by the pressure P at length L is not a strong function of
Eq. (4.45) is less than 0.13, L B ¼ 0:13 should be used. temperature and compressibility factor. Approximate so-
Neglecting the kinetic energy pressure drop term, the lutions to Eq. (4.50) have been sought and used in the
Griffith correlation in U.S. field units can be expressed as natural gas industry.