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DESCRIPTION
SURFACE PRODUCTION
Citation preview
Surface Production Operations
1
Surface Production OperationsHassan Hassanzadeh
EN B204Mhhassanz@ucalgary.ca
Flow in Wellbores and Gathering Systems
Learning objectives
• Perform fluid flow calculations in pipes and
2
• Perform fluid flow calculations in pipes and wellbores
• Line sizing for oil and gas pipelines
• Two-phase flow calculations
Single-Phase Liquid Flow
1. Single-phase liquid flow exists in an oil well only when the wellhead pressure is above the bubble-point pressure!
2. Water disposal or production wells
+∆+∆=−=∆
Dg
Lufu
gz
g
gppp
c
F
cc
22
21
2
2
1 ρρρ
∆P = pressure drop, lb /ft2
2
3
∆P = pressure drop, lbf/ft
2
p1= pressure at point 1, lbf/ft2
p2 = pressure at point 2, lbf/ft2
g = gravitational acceleration, 32.17 ft/s2
gc = unit conversion factor, 32.17 lbm-ft/lbf-s2
ρ = fluid density lbm/ft3
∆z = elevation increase, ftu = fluid velocity, ft/sfM=Moody friction factorfF = Fanning friction factor (fM=4fF) L = tubing length, ftD = tubing inner diameter, ft
∆z
1
θ
Lcosθ=∆z
Single-Phase Liquid Flow (cont.)
LDAg
qf
g
gppp
c
M
c
+=−=∆
2
2
212
cosρ
θρ
LDg
uf
g
gppp
c
M
c
+=−=∆
2cos
2
21
ρθρ
which can be written in flow rate as
whereq = liquid flow rate, ft3/secA = inner cross-sectional area, ft2
4
A = inner cross-sectional area, ftwhen changed to U.S. field units, becomes
5
25
21 1015.1cos433.0d
LqfLppp oM
o
γθγ −×+=−=∆
where
p1 = inlet pressure, psip2 = outlet pressure, psiγo = oil specific gravity, water = 1.0q = oil flow rate, bbl/dayd = pipe inner diameter, in.fM =Moody friction factor
0.433=62.4/144
1.15x10-5=(5.615/24/3600)2x 62.4/(2x32.17)x125/144
q = fluid flow rate, bbl/dayρ = fluid density lbm/ft3
d = tubing inner diameter, in.µ
Single-Phase Liquid Flow (cont.)
d
qN
µ
ρ48.1Re =
The Fanning friction factor ( fF ) can be evaluated based on Reynolds number and relative roughness. Reynolds number is defined as the ratio of inertial force to viscous force.
5
For turbulent flow where NRe > 2,100, the Fanning friction factor can be estimated using empirical correlations such as following one:
d = tubing inner diameter, in.µ = fluid viscosity, cp
dµ
Re
16
NfF =
d/δε =
For laminar flow where NRe < 2,000, the Fanning friction factorFanning friction factor is inversely proportional to the Reynolds number, i.e.:
Chen’s (1979) correlation
where
+−−=
8981.0
Re
1098.1
Re
149.7
8257.2log
0452.5
7065.3log4
1
NNfF
εε
is the absolute relative pipe roughness
1.48=(1000/62.4)(1/39.37)/(24*3600*6.2898)/(π/4)/(1/39.37)2/0.001
Darcy–Wiesbach friction factor
6
Darcy–Weisbach friction factor is 4 times larger than the Fanning friction factor, so attention must be paid to recognize which one of these is meant in any "friction factor“ chart or equation being used.
Allowable working pressure for pipes
It is desirable to operate a pipe at a high pressure in order to achieve higherthroughputs. This is, however, limited by the maximum stress the pipe canhandle. The maximum allowable working pressure is given by (ANSI, 1976)
( )( )Yctd
SEctP
−−
−=
2
2
0
max
7
( )Yctd −− 20
t = pipe thickness, in.c = sum of mechanical allowances, corrosion, erosionS = allowable stress (minimum yield strength for the pipe material)E = longitudinal weld joint factor (equal to 1 for seamless and 0.8 for fusion-welded, spiral-welded, and 0.6 for butt-welded)Y= Temperature de-rating factor, 0.4 up to 900 °F, 0.5 for 950 °F, and 0.7 for 1000 °F.d0 = pipe outer diameter, in.
Allowable flow velocity in pipes
ρ
Cve =
High velocities in pipes can cause pipe erosion problems, especially for gasesthat may have a flow velocity exceeding 70 ft/sec. The velocity at whicherosion begins to occur is dependent upon the presence of solid particles andtheir shape. The following equation can be used as a simple approach:
Where
8
ZRTp
C
ZRTpM
Cv
g
e/97.28/ γ
==
Whereve = erosional velocity, ft/secρ= fluid density, lb/ft3
C = a constant ranging between 75 and 150, in most cases C is taken to be 100
( )5.0
2435.1012
=
ZT
pdq
g
sceγ
where (qe)sc is in MSCFD, d is in inches, p is in psia, and T is in °R
Recalling gas density relation:
Equivalent diameter and Reynolds number in field units for flow of gases in pipes
µ
ρvDN =Re
==
perimeter wetted
flow of area44 he Rd
For cross-sections other than circular, an equivalent diameter, de, defined as four times of the hydraulic radius, Rh, is used instead of d
For a flow conduit with a square cross-section (axa): de=a
For flow through a casing-tubing annulus: d =d -d
9
perimeter wetted For flow through a casing-tubing annulus: de=dci-dto
(note that 1cp = 6.7197x10-4 lbmass/ft-sec)
In field units, flow rat is commonly expressed in MSCFD, p in psia, T in oR, viscosity in cp.
[ ] [ ]( )( )( )( )
( )( )( )( )( ) d
q
T
p
d
q
T
p
cp
d
d
q
cp
d
d
Bq
vdN
scg
sc
scscg
sc
sc
sc
g
gsc
g
gsc
g
µ
γ
µ
γ
πµ
π
ρ
ρ
ρ
µ
πρ
µ
ρ39.710
732.10360024121000
144497.281488
107197.6
12
124
360024
1000
107197.6
12
124
360024
1000
4
2
4
2
Re =×
=××
×
=××
×
==−−
Horizontal gas flow in pipes
=
Dg
uf
dl
dp
c
F
22 ρ
Assuming horizontal, steady-state, adiabatic, and isothermal flow of gas withnegligible kinetic energy change
Substituting for ρ = pM/ZRT and
=
2
4
DpT
ZTpqu
sc
scsc
π
10
∫∫ −=⇒
−= dl
DgRT
MTpfqdp
Z
p
DTp
pTZq
ZRT
pM
Dg
f
dl
dp
csc
scFsc
sc
scsc
c
F
522
22
4222
2222 32162
ππ
Here, we assumed T is constant (isothermal flow). Otherwise, an averagetemperature is commonly used instead of T. Two types of averages used arearithmetic average Tav=(T1+T2)/2 and the log-mean temperature given by:
( )21
21
/ln TT
TTTav
−=
Horizontal gas flow in pipes (cont.)
( ) ( )[ ] ( )( )( ) ( ) ( )TLZf
dpp
p
Tq sc
scγ
52
1
2
2
2
2522
187.8576
14412/117.32732.10243600/1000
−
=×
( ) ( )( ) ( )TLZf
Dpp
p
gRTq
D
TLZfq
gRT
ppp
avgFsc
cscsc
avgFsc
csc
sc
γ
γ
π
52
2
2
1
2
22
5
2
22
22
1
2
2
187.8576
97.2832
2
−
=⇒
−=
−−
Assuming an average gas compressibility factor and integrating gives:
Any consistent system of units can be used. When gc = 32.17 lb-ft/ lbf-sec2, qsc in MSCFD, p in psia, T in oR, d in inches, L in ft, and R = 10.732 psia ft3/lbmole, the equation becomes:
11
( ) ( )[ ]TLZfp avgFsc
scγ2187.8576
( )
( )TLZf
dpp
p
Tq
TLZf
dpp
p
Tq
avgsc
scsc
avgFsc
scsc
γ
γ
52
2
2
1
52
2
2
1
634.5
2.8177
−
=
−
=
Thus
This equation is called Weymouth EquationWeymouth Equation, is the general equation for steady-state isothermal flow of gas through a horizontal pipe.
Note that fF is Fanning friction factor
Note that f is Moody friction factor
( )TLZf
dpp
p
Tq
avgsc
scsc
γ
52
2
2
1C−
=
d
Weymouth EquationWeymouth Equation
( )TLZ
dpp
p
Tq
avgsc
scsc
γ
3/162
2
2
15027.13−
=
3/13/13/1
008.0032.0
4
1032.0
ddf
df F ==⇒=
WeymouthWeymouth equation proposes the following relationship for the moody frictionfactor as a function of pipe diameter d in inches:
Panhandle A EquationPanhandle A Equation
This equation assumes that f is a function of Reynolds number as follows:
where, qsc in MSCFD, p in psia, T in oR, d in inches, and L in ft.
12
1461.01461.01461.0 Re
0192.0
Re
0768.0
4
1
Re
0768.0==⇒= Fff Substituting for fF in
( )TLZf
Dpp
p
Tq
avgFsc
scsc
γ
52
2
2
12.8177−
=
( )( )
5.0
5.207305.05.0
2
2
2
1
5.0
20
0192.0
8177.2
gg
scg
avsc
scsc
D
D
q
TLZ
pp
p
Tq
γµ
γ
−
=
07881.0
61821.246060.053940.0
2
2
2
1
07881.1
16491.32
ggavsc
scsc
D
TLZ
pp
p
Tq
µγ
−
=
The Panhandle A Equation is most applicable to large diameter pipelines, at high
flow rate
where, qsc in MSCFD, p in psia, T in oR, d in inches, and L in ft.
03922.0.003922.0 Re
000898.0
Re
00359.0=⇒= Fff
Panhandle B EquationPanhandle B Equation
This equation also assumes that f is a function of Reynolds number as follows:
Substituting for fF in ( )
TLZf
dpp
p
Tq
avgFsc
scsc
γ
52
2
2
12.8177−
=
020.0
530.249.051.0
2
2
2
1
02.1
1364.109
ggavsc
scsc
d
TLZ
pp
p
Tq
µγ
−
=
The Panhandle B Equation is most applicable to large diameter pipelines, at high
values of Reynolds number
ClinedinstClinedinst EquationEquation
13
ClinedinstClinedinst EquationEquation
∫∫ ∫ == rrpcrpcpcr dpZppdppZppdpZp )/()/()/( 2
∫∫ −= dlDgRT
MTpfqdp
Z
p
csc
scFsc
522
2232
πL
dgRT
MTpfqdpZpdpZpp
csc
scFsc
p p
rrrrpc
r r
522
22
0 0
2 32)/()/(
1 2
π=
−∫ ∫
5.0
0 0
5.05 1 2
)/()/(3.985
−
= ∫ ∫
r rp p
rrrr
Favgsc
scpc
sc dpZpdpZpLfT
d
p
Tpq
γ
where, qsc in MSCFD, p in psia, T in oR, d in inches, and L in ft.
This equation takes into account the non-ideal behaviour of gas
General pipeline gas flow equation
are
14
Average pressure in a gas pipeline
( )TLxZf
dpp
p
Tq
avgF
x
sc
scsc
γ
522
12.8177−
=
( )( )xTLZf
dpp
p
Tq
avgF
x
sc
scsc
−
−
=
12.8177
52
2
2
γ
∫∫ −= dlDgRT
MTpfqdp
Z
p
csc
scFsc
522
2232
π x L-x
L
Integrating once over “l” form 0 to Lx
Integrating once over “l” form Lx to L
15
( )xTLZfp avgFsc − 1γEquating these two equations gives:
( )( )
( )x
pp
x
pp xx
22
1
2
2
2
1
−=
−
− This suggests that ( )[ ] 5.02
2
2
1
2
1 ppxppx −−=
( ) ( )[ ]
++=−−== ∫∫
21
2
21
0
5.02
2
2
1
2
1
03
2
pp
ppdxppxpdxxpp
LL
xav
Rearranging and multiplying both numerator and denominator by (p1-p2):
−
−=
−
−
++=
2
2
2
1
3
2
3
1
21
21
21
2
21
3
2
3
2
pp
pp
pp
pp
pp
pppav
Pipeline efficiency and transmission factor
The pipeline equations developed so far assume 100100%% efficientefficient conditions. Inpractice, even for single-phase gas flow, some water or condensate may bepresent, which accumulate in low spots in the line over long periods of time.Some solids, such as pipepipe--scalescale andand drillingdrilling mudmud, may also be present. Toaccount for the reduction in pipeline capacity due to the presence of thesesmaterials, an efficiency factor E is generally used as a multiplying factor.
A pipeline with E>0.9 is usually considered “clean.”
16
Type of gas Liquid content (Gal/MMSCF) E Dry gas 0.1 0.92Casing-head gas 7.2 0.77Gas and condensate 800 0.60
For high liquid contents two-phase flow conditions exist. Pipeline efficiencycan no longer represent the complex flow behaviour, and different equationsthat account for two-phase flow must be used.
TheThe factorfactor ((11/f/fMM))00..55 isis knownknown asas thethe transmissiontransmission factorfactor..
Single-phase vertical gas flow
4
, 29
,cos ,02 2
2
====++PTD
TZPqv
ZRT
PdLdz
Dg
dLvfdz
g
gdP
sci
scscg
ic
M
c π
γρθ
ρ
The effect of kinetic energy change is negligible because the variation intubing diameter is insignificant in most gas wells. With no shaft work deviceinstalled along the tubing string, the first law of thermodynamics yields thefollowing mechanical balance equation:
17
,08
cos29
2
252
22
=
++ dL
P
ZT
TDg
PQf
g
g
P
dPZRT
scic
scscM
cg πθ
γ
( )2/sExpPP hfwf =
( )sExpPP
dLg
g
P
dPTRZ
hfwf
cg
22
,0cos29
=
=
+ θγ
ZT
Ls
g θγ cos0375.0=In field units (qsc in MSCFD)
ZTRg
gLs
c
g θγ cos58=
Static bottom-hole pressure (q=0)
θ
z
29x229x2/(144xR)
Average ZT Method
( ) ( )[ ]θcos
11067.65
222422 scM
hfwfd
TZQsExpfsExpPP
−×+=
−
( ) ( )[ ]θπ
πθ
γ
cos
18
,08
cos29
252
222222
2
22
252
22
scic
scscMhfwf
scic
scscM
cg
TDg
TZPQsExpfsExpPP
dLP
TZ
TDg
PQf
g
g
P
dPTRZ
−+=
=
++
Ls
g θγ cos0375.0=
In field units (qsc in MSCFD, p in psia, d in inches, and L in ft)
ZTRg
gLs
c
g θγ cos58=
8(1000/24/3600)2(12)5(14.7/520π)2/32=6.67X10-4
18
( )θcos5
i
hfwfd
sExpPP +=ZT
s =
The Darcy–Wiesbach (Moody) friction factor fM can be found in theconventional manner for a given tubing diameter, wall roughness, andReynolds number. However, if one assumes fully turbulent flow, which is thecase for most gas wells, then a simple empirical relation may be used fortypical tubing strings (Katz and Lee 1990):
0.244 0.164
0.01750 0.01603 for 4.277 in., for 4.277 in.
M i M i
i i
f d f dd d
= ≤ = >
( )
2
1Guo (2001):
1.74 2log 2 /M
i
fdε
=
−
Cullender and Smith Method
,08
cos29
2
252
22
=
++ dL
P
ZT
TDg
PQf
g
g
P
dPZRT
scic
scscM
cg πθ
γ
R
LdP
TDg
PQf
P
ZT
g
g
ZT
P
g
p
p
scic
scscM
c
wf
hf
γ
πθ
29
8cos
252
222=
+
∫
wfpP
ZT∫
This Equation can be rearranged as:
In field units (qsc in MMSCFD)
pwf
∫
I
19
2 2
5
18.75
0.001cos 0.6666hf
g
p M sc
i
ZT dP Lf QZT
DP
γ
θ
=
+
∫ LdPI g
phf
γ75.18=∫
In the form of numerical integration:
or
( )( ) ( )( )L
IIppIIppdPI g
mfwfmfwfhfmfhfmf
p
p
wf
hf
γ75.1822
=+−
++−
=∫
where pmf is the pressure at the mid-depth. The Ihf , Imf , and Iwf are integrant Is
evaluated at phf , pmf , and pwf ,respectively.
28.97/R/144/0.001
Cullender and Smith method (cont.)
( )( )2
75.18
2
LIIpp ghfmfhfmf γ=
+− ( )( )2
75.18
2
LIIpp gmfwfmfwf γ=
+−
( )( ) ( )( )L
IIppIIppdPI g
mfwfmfwfhfmfhfmf
p
p
wf
hf
γ75.1822
=+−
++−
=∫
Assuming the first and second terms in the right-hand side each representshalf of the integration, that is,
and
20
22=
22=
( )hfmf
g
hfmfII
Lpp
++=
γ75.18
and
The following expressions are obtained:
and ( )mfwf
g
mfwfII
Lpp
++=
γ75.18
Because Imf is a function of pressure pmf itself, an iterative process is requiredto find pmf . Once pmf is computed, pwf can be obtained.
Gas flow through an annulus
2
and 2
5
22
1
2
1
2
=∆⇒=
=∆
Dg
qLfKp
D
qKu
Dg
uLfp
c
scFf
sc
c
Ff
ρρ
1 scqKu =
Although gas wells are generally produced through tubing, some wells may beproduced through the casing-tubing annulus. For flow in a pipe, we have:
For the case of annular flow, where the outside diameter of the tubing is dto and theinside diameter of the casing is dci, the velocity is related to diameter as follows:
21
22
1
toci
sc
dd
qKu
−=
( ) ( ) ( ) ( ) ( )tocitoci
sc
tocitocic
scF
tocic
scFf
dddd
qK
ddddg
qLfK
Dddg
qLfKp
−−=
−−=
−=∆
222
2
2
222
22
1
22
22
1 22 ρρ
Thus, the friction term becomes:
Thus, for the case of annular flow, d5 in the vertical flow equations must bereplaced by:
( ) ( ) ( ) ( )23222
tocitocitocitoci dddddddd +−=−−
where K1 and K2 are proportionality constants
Gas flow over hilly terrain
Transmission lines often deviate considerably from the horizontal depending onfield topography.In some cases, gas wells may also exhibit sections of different slope, such asthe directionally drilled wells from offshore platforms.
∆z
22
∆z
inlet outlet
Three approaches are available to obtain gas flow profile over hilly terrain
1. Static correction2. Flow correction3. General method
Static Correction
( )2/exp'spp oo =
ZT
zs
g ∆=
γ0375.0
To account for the differences in elevation between the inlet and outlet, the simplest way is to modify the outlet pressure for the pressure exerted by a static gas column.
If pi is the inlet pressure, and po the outlet pressure, then, the outlet pressure must be corrected as follows:
where s is given by
23
Note that ∆z is positive for uphill flowpositive for uphill flow and negative for downhill flownegative for downhill flow
∆z
inlet outlet
The flowline shown is equivalent to a horizontal flow line with an upstream inlet pressure of pi, and a downstream pressure of equal to p0exp(s/2) . This correction can be incorporated in to any horizontal flow correlation to find the flow rate.
( )TLZ
depp
p
Tq
avg
s
oi
sc
scsc
γ
3/1622
5027.13−
=
Flow CorrectionA more rigorous correction for the flow profile accounts for inclined flow in different sections of the pipe line.
[ ] [ ]sd
TZLqefep
d
TZqefepp
i
gsc
s
Ms
hf
i
sc
s
Ms
hfwf 5
25
2
5
222422
1105272.2
cos
11067.6 γ
θ
−×+=
−×+=
−−
ZT
z
ZT
Ls
gg ∆==
γθγ 0375.0cos0375.0
Rearranging this equation, and replacing the pressure with the inlet and
qsc is in MSCFD, d in inch, and L in ft.
24
( )
( ) ( )
( )[ ]
( )5.0
5225.0
522
5.0563538.5
173.14/105272.2
520/
−
=
−
−
×=
− TZLf
depp
p
T
TZLef
sdepp
p
Tq
egM
i
s
oi
sc
sc
g
s
M
i
s
oi
sc
scsc
γγ
[ ] 0 ,11
1
1 ≠−∑
=∑=
−
= ses
LeL
N
i
s
i
i
s
ei
i
j
j
Rearranging this equation, and replacing the pressure with the inlet and outlet pressures pi and po:
Where is the effective length for each single section of a flow line[ ]
Ls
eL
s
e
1−=
For the general case of non-uniform slope the profile is divided into a number of sections, the effective length is then:
Temperature profile in flowing gas systemsHorizontal pipelinesHorizontal pipelines
Assuming steady-state (Papay (1970):
( ) [ ]( )
( )( )
mkC
czczC
CCC
xCCC
C
xCC
xCC
CCCCCCCCTxT
pVVpLV
CC
CC
s
2
111
322
31
2
54
/
21
/
13225124
/
)1(
5))(/()(/32
32
=
−+=
+
++
+−
+
+−+=
Factors affecting the gas temperature in a pipeline are Heat exchange with thesurroundings, Joule-Thomson effect, velocity and elevation changes, phasechange, and friction.
25
( )[ ]
( )( ) ( )L
vvcc
L
ppzzC
Tm
DkLghv
L
vvQ
L
zzczcz
L
ppC
LcczzC
mkC
dVpVdLpLVV
oVVdVpVVdLpLV
pVpLVV
12
2
21125
111221
1121
4
123
2
/1
/))((
/
−−
−−=
−+−
+−
+−+−
=
−−=
=
µµ
πµµ
zV and zL are vapour and liquid mole fractions, respectively, µ (ft2-F/lbf)Joule-Thomson coefficient, p = pressure lbf/ft
2, L (ft) pipeline length, v (ft/s)velocity, cp (Btu/lb-F) specific heat, m(lb/s) mass flow rate, Q (Btu/lb)phase transition heat, k (Btu/ft-sec-F) thermal conductivity, g (32.17 ft/s2)gravitational acceleration, h(ft) elevation, Do(ft) pipe outside diameter, Ts
(°F), temperature of the soil or surrounding
Temperature profile in flowing gas systems (cont.)
( ) ( ) ( ) ( ) ( ) ( ) ( )
pV
velocity
Kx
pV
gravity
Kx
pV
KxdLKx
ss
mc
kK
L
xvve
KL
vvv
KLc
vve
KLc
ghe
KL
ppeTTTxT
=
−+−
−−
−−−−−
−−−+= −−
−
−−
44444444 344444444 2144 344 21444 3444 2144 344 21
12121
12
ThomsonJoule
21
ferheat trans Soil
1 111µ
( ) ( ) KxeTTTxT
−−+=
In the pervious equation, it was assumed that pressure, flow rate, and phase-transition are linear functions of distance from the inlet. This equation is, therefore,accurate for short line segments. For the case where phase changes can beneglected (single-phase flow), we have:
26
( ) ( ) Kx
ss eTTTxT−−+= 1
Ignoring JT effect, velocity and elevation changes
Flowing temperature in wells (Ramey, 1962)Flowing temperature in wells (Ramey, 1962)
( ) ( )[ ]Kx
T eKxGTxT−− −−−= 11
1
x = distance from the bottom hole or point of fluid entry, ftT = Temperature in °FT1 = temperature at point of fluid entryGT =geothermal gradient °F/ft
Gas gathering and transportNatural gas produced from several wells in a given area is collected and broughtto the field separation and processing facilities via a system of pipes known asgathering system. Processed or partially processed gas is then sent to the trunklines that transport gas to consumers. Gas is often distributed via pipelines grids.
Well centre Radial
well head
header
27
Trunk line wellhead
header
central gathering
station
flowline
Series PipelinesConsider three pipelines A, B and C, connected in series as shown below:
=−
3/16
2
2
2
2
15027.13 A
Aavg
sc
scABC
d
TLZ
T
pqpp
γ
2γ
++
=−
3/163/163/16
2
2
4
2
15027.13 C
C
B
C
A
C
sc
scABCavg
d
L
d
L
d
L
T
pqTZpp γ
1 2 3 4A B C
B CA
28
=−
3/16
2
2
3
2
25027.13 B
Bavg
sc
scABC
d
TLZ
T
pqpp
γ
=−
3/16
2
2
4
2
35027.13 C
Cavg
sc
scABC
d
TLZ
T
pqpp
γ( )
++
−
=
3/16
2
3/16
2
3/16
1
2
4
2
15027.13
d
L
d
L
d
LTZ
pp
p
Tq
CBAavg
sc
scABC
γ
Capacity of an equivalent single-diameter (d) pipeline is expressed as:
( )TLZ
dpp
p
Tq
avgsc
sc
γ
3/162
4
2
15027.13
−
=
++
=
3/163/163/16
3/16
C
C
B
B
A
A
ABC
d
L
d
L
d
L
d
L
q
q
Series Pipelines (cont.)
( )5
2
2
2
2
1
52
2
2
1
635382.5635382.5
A
AAavg
sc
sc
A
A
avgsc
sc
d
LfTZ
T
pqpp
fL
d
TZ
pp
p
Tq γ
γ
=−⇒
−
=
2
Lfpq
Series pipelines
qA= qB= qc= qt
∆pA ≠ ∆pB ≠ ∆pc∆pt= ∆pA+ ∆pB+ ∆pC
Equivalent length
1 2 3 4A B C
B CA
29
5
2
3
2
2635382.5 B
BBavg
sc
sc
d
LfTZ
T
pqpp γ
=−
5
2
2
4
2
3635382.5 C
CCavg
sc
sc
d
LfTZ
T
pqpp γ
=−
++
=−
555
2
2
4
2
1635382.5 C
CC
B
BB
A
AAavg
sc
sc
d
Lf
d
Lf
d
LfTZ
T
pqpp γ
( )
+
+
−
=
AC
CAC
AB
BABA
A
Aavg
sc
sc
fd
fdL
fd
fdLL
d
fTZ
pp
p
Tq
5
5
5
5
5
2
4
2
1635382.5
γ
( )
e
A
Aavg
sc
sc
Ld
fTZ
pp
p
Tq
5
2
4
2
1635382.5
γ
−
=
+
+=
AC
CAC
AB
BABAe
fd
fdL
fd
fdLLL
5
5
5
5
=
=
AC
CACeC
AB
BABeB
fd
fdLL
fd
fdLL
5
5
5
5
,
Parallel Pipelines
( )TLZ
dpp
p
Tq
avg
A
sc
scA
γ
3/162
2
2
15027.13−
=
( )TLZ
dpp
p
Tq
avg
B
sc
scB
γ
3/162
2
2
15027.13−
=
Consider three pipelines A, B and C, connected in parallel as shown below:
p1p2
( )−
A
B
C
30
( ) ( )2 2
1 2 16/3 16/3 16/331.5027 sc
ABC A B C A B C
sc g av
p pTq q q q d d d
p Z TLγ
− = + + = + +
( )3/16
3/163/163/16
d
ddd
q
q CBAABC++
=
( )TLZ
dpp
p
Tq
avg
C
sc
scC
γ
3/162
2
2
15027.13−
=
Capacity of an equivalent single-diameter (d) pipeline is expressed as:
( )TLZ
dpp
p
Tq
avgsc
sc
γ
3/162
4
2
15027.13
−
=
Parallel Pipelines (cont.)
( )AA
A
avgsc
scA
Lf
d
TZ
pp
p
Tq
52
2
2
1635382.5γ
−
=
( )BB
B
avgsc
scB
Lf
d
TZ
pp
p
Tq
52
2
2
1635382.5γ
−
=
qA+ qB+ qc= qt
∆pA = ∆pB = ∆pc
Equivalent lengthp1
p2
A
B
C
31
( )ee
e
avgsc
sct
Lf
d
TZ
pp
p
Tq
52
2
2
1635382.5γ
−
=
( )CC
C
avgsc
scC
Lf
d
TZ
pp
p
Tq
52
2
2
1635382.5γ
−
=
2
5
5
5
5
5
5
55551
++
=⇒++=
CC
e
e
C
BB
e
e
B
AA
e
e
A
e
CC
C
BB
B
AA
A
ee
e
Lf
f
d
d
Lf
f
d
d
Lf
f
d
dL
Lf
d
Lf
d
Lf
d
Lf
d
Looped PipelinesConsider a three-segment looped gas pipeline depicted below:
( ) ( )3/163/162
3
2
15027.13BA
Aavgsc
scBAC dd
TLZ
pp
p
Tqqq +
−
=+=
γ ( )
2
23/163/16
2
3
2
15027.13
+=−
sc
scC
BA
Aavg
T
pq
dd
TLZpp
γ
p1 p2
qA
qB
qCp3
L
LC
32
2
3/16
2
2
2
35027.13
=−
sc
scC
C
Cavg
T
pq
d
TLZpp
γ
( )( )
( )
+
+
−
=⇒
+
+
=−
3/1623/163/16
2
2
2
1
3/1623/163/16
2
2
2
2
1
5027.13
5027.13
C
C
BA
Aavg
sc
scC
C
C
BA
A
sc
scCavg
d
L
dd
LTZ
pp
p
Tq
d
L
dd
L
T
pqTZpp
γ
γ
Aavgsc TLZp γ
Applying the WeymouthWeymouth equation to the third segment (with diameter dC) yields
Adding these equations gives:
( ) + scBA dd
Looped Pipelines (cont.)
( )
−=
3/16
2
2
2
15027.13
d
LTZ
pp
p
Tq
avg
sc
scS
γ
Capacity of a single-diameter (d) pipeline is expressed as
Single
Looped
p1 p2
qA
qB
qCp3
L
LC
33
( ) 3/162
3/163/16
3/16
//
/
CCBAAS
L
dLddL
dL
q
q
++=
( )
( )
+
+
−
=
3/1623/163/16
2
2
2
15027.13
C
C
BA
Aavg
sc
scL
d
L
dd
LTZ
pp
p
Tq
γ
Looped Pipelines (cont.)
% Incre
ase in R
ate
34Katz et al.
% Incre
ase in R
ate
% Line Parallel
Blowdown and Purge
−=
u
dav
p
pTZtdV 102428.0 2
There are instances when it is necessary to blow down and purge a gas line.
For subcritical flow (pu/pd>0.2) (Izawa,1966)
For subcritical flow (pu/pd<0.2) (Izawa,1966)
35
T
tpdV av
24524.0=
V = Gas volume, MSCFd = pipe inner diameter, in.pav = average pressure, psigt = time, min.T = Temperature, °Rpu, pd = upstream and downstream pressures, respectively, psig
Pressure Testing
ip
Ldt
2
min
0.3=
Pressure testing of a pipeline is commonly done for the detection of leaks.The following relationship has been given for estimating the minimumvalue of this testing time necessary (Campbell, 1984):
where
36
ipwheretmin =minimum required shut-in time for the line, hrd =pipe diameter, inchL= length of the pipe section that is being tested for leaks, milespi = initial test pressure, psig.
d
tpp i
949max =∆
A flow line that has been shut-in for at least tmin hours is considered tohave no leaks ifif thethe pressurepressure lossloss isis lessless thanthan thethe ∆∆ppmaxmax (psi) givenbelow:
Optimum Pipe DiameterCompression costsCompression costs
The following equation can be obtained for the annual compression coststhe annual compression costs:
WhereCcomp: compression cost in $/year per foot of pipe lengthq : gas flow rate, ft3/sec
37
q : gas flow rate, ft3/secρ: the gas density, lbm/ft3
µ: gas viscosity, cpCe : cost of electrical energy, $/kwhHy : hours of operation per yeard : pipe diameter, in.E : compressor efficiency, fraction B* : a constant independent of pipe diameter d, accounting for all other energy looses in the flowing systemLfp : frictional loss due to pipe fittings and bends, expressed as equivalent fractional loss in a straight pipe
Optimum Pipe Diameter (cont.)
For most types of pipe, the purchase cost per foot of pipe is related to thepipe diameter as follows:
Cpipe = Cpdn
WhereCpipe : purchase cost of new pipe of diameter d inches per foot of pipe length, $/ftCp: a constant equal to the purchase price per foot for a 1-in. diameter pipe, $/ftd: pipe diameter, in.
Fixed Costs for Piping SystemFixed Costs for Piping System
38
d: pipe diameter, in.
WhereRp: ratio of total costs for fittings and installation to the purchase cost for new pipeCFp: annual fixed charges, including maintenance, expressed as a fraction of the initial cost for a completely installed pipe
The annual cost for an installed piping system can be expressed as (Petersand Timmerhaus, 1980):
Cpipe= (l+Rp)CpdnCFp
Optimum Pipe Diameter (cont.)
The total annual cost CT for the compressor and piping system can be obtained by
Differentiating with respect to diameter d, setting the resultant expression tozero, and solving for d gives the optimum pipe diameter, dop in inches, asfollows:
OptimumOptimum EconomicEconomic PipePipe DiameterDiameter
39
op
follows:
The value of n for steel pipes is about 1.0 for d < 1 in., and about 1.5 ford ≥1 in. Thus, for the commonly used greater than 1-in. diameter pipes,( Peters and Timmerhaus,1980):
Optimum Pipe Diameter (cont.)
Assuming typical numeric values for the terms involved.Using:Ce= $0.055/kwh, LfP = 0.35, Hy = 8,760 hrs/year, Rp=1.4, Cp: $0.45/ft for 1 - indiameter pipe, E = 0.50, CFp =0.20, and neglecting the viscosity term which isclose to unity for most cases for the small exponent involved, Peters andTimmerhaus (1980) presented the following equation for the optimum pipediameter:
40
diameter:
where m : mass flow rate, lbm/hrρ: fluid density, lbm/ft3
This equation is quite good for short pipe lengths and where the pressuredrop in the pipe length is not large.
Multiphase Flow RegimesIn addition to oil, almost all oil wellsproduce a certain amount of water, gas,and sometimes sand. These wells arecalled multiphase-oil wells.
41Fundamentals Of Gas Solids/Liquids Separation
Muller Environmental Design , Inc.
Liquid Holdup
In multiphase flow, the amount of the pipe occupied by a phase isoften different from its proportion of the total volumetric flow rate.
The liquid ‘‘holdup’’ is defined as:
whereV
Vy L
L =
42
yL = liquid holdup, fractionVL = volume of liquid phase in the pipe segment, ft3
V = volume of the pipe segment, ft3
Liquid holdup depends on flow regime, fluid properties, and pipe sizeand configuration. Its value can be quantitatively determined onlythrough experimental measurements.
Tubing Performance Models
HomogeneousHomogeneous modelsmodels treat multiphase as a homogeneous mixture anddo not consider the effects of liquidliquid holdupholdup (no(no--slipslip assumption)assumption)..Therefore, these models are less accurate and are usually calibrated withlocal operating conditions in field applications. The major advantage ofthese models comes from their mechanistic nature. They can handle gas-oil-water three-phase and gas-oil-water-sand four-phase systems. It iseasy to code these mechanistic models in computer programs.
43
easy to code these mechanistic models in computer programs.
SeparatedSeparated--flowflow modelsmodels are more realistic than the homogeneous-flowmodels. They are usually given in the form of empirical correlations. Theeffects of liquid holdupholdup (slip)(slip) and flow regime are considered. The majordisadvantage of the separated flow models is that it is difficult to codethem in computer programs because most correlations are presented ingraphic form.
where
Homogeneous-Flow Models
144
hkp
∆
+=∆
ρρ
Poettmann and Carpenter (1952), Cicchitti (1960), Dukler et al. (1964), Guo and Ghalambor (2005)
510
22
2
104137.7 D
Mqfk oF
×=
PoettmannPoettmann and Carpenterand Carpenter
where∆p= pressure increment, psi/ft
= average mixture density (specific weight), lb/ft3=(ρ1+ ρ2)/2∆h= depth increment, ftf2F = Fanning friction factor for two-phase flowqo = oil production rate, STB/dayM = total mass associated with 1 stb of oil, lbmass/STBD = tubing inner diameter, ftρ1= mixture density at top of tubing segment, lb/ft3
ρ2 = mixture density at bottom of segment, lb/ft3
44
ρ
Iterations are required to solve for pressure.
Poettmann–Carpenter Model
( )
( ) ( )
−++
++==
0.1420
7.14 615.5
17.50
ZT
pRGORBWORB
GORWOR
V
M
swo
gairwo
m
γργγρ
2048.1
00091.0
0125.0
10
10
18
=
t
API
gs
pR γ
The mixture density at a given point can be calculated based on mass flow rateand volume flow rate:
M = total mass associated with 1 stb of oil, lbmass/STB γo = oil specific gravity, 1 for freshwater
45
00091.01018
t
2.15.0
25.100012.09759.0
+
+= tRB
o
g
soγ
γ
o
WOR = producing water–oil ratio, bbl/STBγw = water-specific gravity, 1 for freshwaterGOR = producing gas–oil ratio, SCF/STBγair = density of air, lbm=ft3
γg = gas-specific gravity, 1 for airBo = formation volume factor of oil, bbl/STBBw = formation volume factor of water, bbl/STBRs= solution gas–oil ratio, SCF/STBp = in situ pressure, psiaT = in situ temperature, oRz = gas compressibility factor at p and T.t = in situ temperature in oF.
( )
( )D
MqvD
f
o
vD
F
5
log5.2442.1
2
104737.1
10
−
−
×=
=
ρ
ρ
Poettmann–Carpenter Model (cont.)
Because the Poettmann–Carpenter model takes a finite-difference form, this model is accurate for only short-depthincremental “h”. For deep wells, this model should be used ina piecewise manner to get accurate results (i.e., the tubingstring should be ‘‘broken’’ into small segments and the model
46
string should be ‘‘broken’’ into small segments and the modelis applied to each segment).
Guo–Ghalambor model (2005)
( ) ( )( )
( )211
2
2
2
00678.0
07.4
61.561.5
07.4
4.623503500765.0
cos144
tan144
tan144
144ln
2
21144
A
qTc
qT
qqqb
qT
qqqqa
LedaN
Mp
N
Mp
N
bMNc
bM
NMp
NMpbMppb
gav
gav
swo
gav
sswwoogg
hf
hf
hf
=
++=
+++=
+=
+−
+−+
−++
++−+− −−
γγγγ
θ
whereA = cross-sectional area of conduit, ft2
DH = hydraulic diameter, ftfM = Darcy–Wiesbach friction factor (Moody factor)=4 fFg = gravitational acceleration, 32:17 ft/s2
47
Iterations are required to solve for pressure
( )
( )
( )
( )( )22
2
2
cos
cos
cos
2
61.561.500166.0
00678.0
ed
ecN
ed
adeM
gD
fe
qqqA
d
Ac
H
M
swo
+=
+=
=
++=
=
θ
θ
θ
L = conduit length, ftp = pressure, psiaphf = wellhead flowing pressure, psiaγg = gas production rate, SCF/dqo = oil production rate, bbl/dqs =sand production rate, ft3/dayqw= water production rate, bbl/dTav = average temperature, oR
γ g = specific gravity of gas, air = 1
γ o = specific gravity of produced oil, water = 1
γ s = specific gravity of produced solid, water = 1
γ w = specific gravity of produced water, water = 1
Separated-Flow ModelsLockhart and Martinelli correlation (1949), the Duns and Ros correlation (1963),and the HagedornHagedorn andand BrownBrown methodmethod (1965).
HagedornHagedorn––Brown modelBrown model
zg
u
D
Mf
dz
dP
c
mtF
∆
∆+
×+=
210413.7144
2
510
2
ρρ
ρ
Mt = M = total mass associated with 1 STB of oil, lbm/STB
48
Mt = M = total mass associated with 1 STB of oil, lbm/STB
=in situ average density, lbm/ft3
um = uSL + usg, mixture velocity, ft/sρL = liquid density, lbm=ft3
ρG = in situ gas density, lbm=ft3
uSL = superficial velocity of liquid phase, ft/suSG = superficial velocity of gas phase, ft/sD = pipe inner diameter , ft
( ) GLLL yy ρρρ −+= 1
The superficialsuperficial velocityvelocity of a given phase is defined as the volumetricflow rate of the phase divided by the pipe cross-sectional area for flow.
Liquid Holdup CalculationsvL
4
4
Liquid velocity number, N :
1.938
Gas velocity number, :
1.938
Pipe dimeter number, :
L
L
vL SL
vG
vG SG
D
N u
N
N u
N
ρ
σ
ρ
σ
ρ
=
=
whereD = conduit inner diameter, ft σ = liquid–gas interfacial tension, dyne/cmµL = liquid viscosity, cpµG = gas viscosity, cp
49
3
14
120.872D
Liquid viscosity number :
0.15726
L
L
D
L
L L
N
N
N
ρ
σ
ρ σµ
=
=
µG = gas viscosity, cp
( )
( )[ ]3log
12195.054785.055100.015841.069851.2
10
1
4
1
3
1
2
11
+=
−+−+−=
=
L
Y
L
NX
XXXXY
CN
Liquid Holdup Calculations (cont.)
( )[ ] ( )[ ] ( )[ ] ( )[ ]
( )[ ] ( )
1 01.0for
3.1161746.2225325.123282176.491163.0 ,01.0for
, ,3log
6log401.06log29598.06log63295.06log61777.010307.0
38.0
3
4
3
3
3
2
3314.2
38.0
3
1.0575.0
1.0
21
4
2
3
2
2
22
LvG
D
LvG
DavG
LvLL
L
NNX
XXXXN
NNX
NpN
CNpNXNX
XXXXy
=≤=
+−+−=>=
=+=
+−+++−++−=
ψ
ψ
ψ
50
( ) ,102.2
1 01.0for
1
2
Re
14.23
LL y
G
y
L
t
LL
D
D
mN
yy
NX
−
−×=
=
=≤=
µµ
ψψ
ψ
where mt is mass flow rate in lbm/day
pa = atmospheric pressurep is the pressure at location where pressure gradient is to be calculated
where ρ is in lbm/day, v in ft/s, viscosity in cp, and D in ft.
For single phase flow:
1488Re
vDρ
µ=
Bubble-Flow Regime
−==<
D
uL
u
uL m
B
m
SGGBG
2
2218.0071.1 and , where λλ
The modified HagedornHagedorn andand BrownBrown methodmethod uses the Griffith correlation for thebubble-flow regime. The bubble-flow regime has been observed to exist when
which is valid for LB ≥0.13
u in ft/s and D in ft.
When the LB value is less than 0.13, LB = 0.13 should be used.
51
L
L
s
sG
s
m
s
mL
D
mN
u
u
u
u
u
uy
µ
2
Re
2
102.2 ,411
2
11
−×=
−
+−+−==
where mL is mass flow rate of liquid only in lbm/STB. The liquid holdup in Griffith correlation is given by the following expression:
LL
LF
yD
mf
dz
dP
ρρ
2510
2
10413.7144
×+=
where us = 0.8 ft/s.
Flow Regime Maps
52
Flow regime map for vertical flow Flow regime map for air-water system inhorizontal pipe
Approximate methods for two- phase flow systems
Flow streams with a GLR greater than10000 SCF/STB may be assumed tobe single-phase gas. This iscommonly the case for retrograde andwet gas reservoirs. The small liquidcontent of the gas can be accountedfor by modifying the properties thatare affected by the presence of liquid.These properties include MW, gravity,and compressibility factor. If the two-
53
oog
ogg
wMR
R
/132800
458
γ
γγγ
+
+=
and compressibility factor. If the two-phase Z-factor is essentially the sameas the single-phase Z- factor, themixture exists as a single-phase.
9.5
6084
03.1
29.44
−=
−=
APIM
o
oo
γ
γ
Pressure traverse for multiphase flow
For flow conditions that cannot be approximated as single-phase gas, it isnecessary to use more complex procedures.
Pressure traverse curves for horizontal gasPressure traverse curves for horizontal gas--liquid flowliquid flow
Developed using Eaton et al.’s correlation give a satisfactory results except forlow rates and low LGR. These curves were prepared for water, they can beused for oil, provided the free-gas/oil ratio is used for the G/L parameter asfollows:
54
follows:
1. Select the applicable curve for the given flowline size, flow rate, and GLR.2. On the pressure axis, locate the known pressure, go vertically down to the
applicable GLR, and read off the length on the length axis.3. Correct this length for the pipeline length by: adding the pipeline length to
the length determined in Step 2, if the known pressure is the outletpressure; or subtracting the pipeline length from the length in Step 2, if theknown pressure is the inlet pressure.
4. The unknown pressure is the pressure corresponding to the correctedlength determined in step 3.
1
23
6
55
45
56
57
Pressure traverse curves for vertical gas-liquid flow
1. Select the applicable curve for the given tubing size, flow rate, and GLR.
2. On the pressure axis, locate the known pressure, go vertically down to the applicable GLR, and read off the length on the length axis.
3. Correct this length by: adding the well depth to the
58
3. Correct this length by: adding the well depth to the length determined in Step 2, if the known pressure is the surface pressure; or subtracting the pipeline length from the length in Step 2, if the known pressure is the inlet pressure.
4. The unknown pressure is the pressure corresponding to the corrected length determined in step 3.
59
60
61
62
63
64
65
66
Handling Directional WellsFor directional wells with deviations from the vertical less than 1515--2020 degreedegree, thetrue vertical depth can be used to ascertain the pressure traverse. Thisapproximation, however, is invalid for deviation exceeding 20 degree, because adirectional well has aa greatergreater lengthlength thanthan aa verticalvertical wellwell for the same depth,resulting in a greater frictional loss. Also, liquidliquid holdhold upup differs and may begreater for inclined than for vertical flow.
An approximate answer can be obtained using the vertical and horizontal flowcurves as follows:
67
Flow over inclined or hilly terrainFlanigan (1958) presented a method for inclined flow.
where HF : elevation component of the total pressure drop, and fraction
And vg is given by:
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