3

Surface Production Operations

1

Surface Production OperationsHassan Hassanzadeh

EN [email protected]

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.

Aymen.r

Highlight

Aymen.r

Highlight

Aymen.r

Highlight

Aymen.r

Highlight

Aymen.r

Highlight

Aymen.r

Highlight

Aymen.r

Highlight

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


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


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


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):


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

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.

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:

68

3

Documents

Transcript of 3