Page 59 - Petroleum Production Engineering, A Computer-Assisted Approach

Guo, Boyun / Petroleum Production Engineering, A Computer-Assisted Approach 0750682701_chap04 Final Proof page 50 22.12.2006 6:07pm

4/50 PETROLEUM PRODUCTION ENGINEERING FUNDAMENTALS
r air ¼ density of air, lb m =ft 3 to use. The Guo–Ghalambor model can be expressed as
g g ¼ gas-specific gravity, 1 for air follows:
V m ¼ volume of mixture associated with 1 stb of oil, ft 3
2
B o ¼ formation volume factor of oil, rb/stb 144b(p p hf ) þ 1 2bM ln (144p þ M) þ N
2
B w ¼ formation volume factor of water, rb/bbl 2 (144p hf þ M) þ N
R s ¼ solution gas–oil ratio, scf/stb
b
p ¼ in situ pressure, psia M þ N bM 2
c
p
ffiffiffiffiffi
T ¼ in situ temperature, 8R N
z ¼ gas compressibility factor at p and T. 144p þ M 144p hf þ M
tan 1 p ffiffiffiffiffi tan 1 p ffiffiffiffiffi
If data from direct measurements are not available, N N
solution gas–oil ratio and formation volume factor of oil 2
can be estimated using the following correlations: ¼ a( cos u þ d e)L, (4:18)
0:0125API 1:2048 where the group parameters are defined as

p 10
R s ¼ g g (4:14)
18 10 0:00091t 0:0765g g q g þ 350g o q o þ 350g w q w þ 62:4g s q s
a ¼ , (4:19)
4:07T av q g
" # 1:2
0:5
g g
B o ¼ 0:9759 þ 0:00012 R s þ1:25t (4:15)
g o 5:615q o þ 5:615q w þ q s
b ¼ , (4:20)
4:07T av Q g
where t is in situ temperature in 8F. The two-phase friction
factor f 2F can be estimated from a chart recommended by
Poettmann and Carpenter (1952). For easy coding in com-
T av q g
puter programs, Guo and Ghalambor (2002) developed c ¼ 0:00678 , (4:21)
the following correlation to represent the chart: A
f 2F ¼ 10 1:444 2:5 log (Drv) , (4:16) 0:00166
d ¼ (5:615q o þ 5:615q w þ q s ), (4:22)
where (Drv) is the numerator of Reynolds number repre- A
senting inertial force and can be formulated as
5 f M
1:4737 10 Mq o e ¼ , (4:23)
(Drv) ¼ : (4:17) 2gD H
D
Because the Poettmann–Carpenter model takes a finite- cde
difference form, this model is accurate for only short- M ¼ cos u þ d e , (4:24)
2
depth incremental h. For deep wells, this model should
be used in a piecewise manner to get accurate results (i.e.,
the tubing string should be ‘‘broken’’ into small segments c e cos u
2
and the model is applied to each segment). N ¼ 2 , (4:25)
2
Because iterations are required to solve Eq. (4.8) for ( cos u þ d e)
pressure, a computer spreadsheet program Poettmann- where
CarpenterBHP.xls has been developed. The program is
available from the attached CD. A ¼ cross-sectional area of conduit, ft 2
D H ¼ hydraulic diameter, ft
Example Problem 4.2 For the following given data, f M ¼ Darcy–Wiesbach friction factor (Moody factor)
calculate bottom-hole pressure: g ¼ gravitational acceleration, 32:17 ft=s 2
L ¼ conduit length, ft
Tubing head pressure: 500 psia p ¼ pressure, psia
Tubing head temperature: 100 8F p hf ¼ wellhead flowing pressure, psia
Tubing inner diameter: 1.66 in. q g ¼ gas production rate, scf/d
Tubing shoe depth (near q o ¼ oil production rate, bbl/d
bottom hole): 5,000 ft q s ¼ sand production rate, ft =day
3
Bottom hole temperature: 150 8F q w ¼ water production rate, bbl/d
Liquid production rate: 2,000 stb/day T av ¼ average temperature, 8R
Water cut: 25% g g ¼ specific gravity of gas, air ¼ 1
Producing GLR: 1,000 scf/stb g o ¼ specific gravity of produced oil, freshwater ¼ 1
Oil gravity: 30 8API g s ¼ specific gravity of produced solid, fresh water ¼ 1
Water specific gravity: 1.05 1 for freshwater g w ¼ specific gravity of produced water, fresh water ¼ 1
Gas specific gravity: 0.65 1 for air
The Darcy–Wiesbach friction factor (f M ) can be
Solution This problem can be solved using the computer obtained from diagram (Fig. 4.2) or based on Fanning
program Poettmann-CarpenterBHP.xls. The result is friction factor (f F ) obtained from Eq. (4.16). The required
shown in Table 4.1. relation is f M ¼ 4f F .
Because iterations are required to solve Eq. (4.18) for
The gas-oil-water-sand four-phase flow model proposed
by Guo and Ghalambor (2005) is similar to the gas-oil- pressure, a computer spreadsheet program Guo-Ghalam-
borBHP.xls has been developed.
water three-phase flow model presented by Poettmann
and Carpenter (1952) in the sense that no slip of liquid
phase was assumed. But the Guo–Ghalambor model Example Problem 4.3 For the following data, estimate
takes a closed (integrated) form, which makes it easy bottom-hole pressure with the Guo–Ghalambor method: