P613_05A_Lec_06_Rate_Relations_(050302)

7/25/2019 P613_05A_Lec_06_Rate_Relations_(050302)

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PETE Slide 1Semi-Analytical Rate Relationsfor Oil and Gas Flow

T.A. Blasingame, Texas A&M U.Department of Petroleum Engineering

Texas A&M UniversityCollege Station, T !!"#$%$'

(.)!)."#*.++)+ t%-lasingametamu.e/u

Petroleum Engineering '$

Natural Gas Engineering

Texas A&M University

0e1ture 2'3

Semi-Analytical Rate Relations

for Oil and Gas Flow


2/46


4ate 4elations for 5il an/ 6as 7lo8

9istori1al Perspe1tives:Ba1;pressure: eyper-oli1, an/ >armoni1?.Derivation of Arps@ exponential /e1line relation.ali/ation of Arps@ >yper-oli1 /e1line relation.

Spe1ialie/ 6as 7lo8 4elations37et;ovi1> 6as 7lo8 4elation.

Ansa>%Bu-a%no8les 6as 7lo8 4elations.Spe1ialie/ 5il 7lo8 4elations37et;ovi1> 5il 7lo8 4elation.

nflo8 Performan1e 4elations =IPR?3Early 8or; =for rationale?.

5il IPRan/ Solution%6as Drive IPR.6as Con/ensate IPR.


3/46


6as ell Delivera-ility3

T>e original 8ell /elivera-ilityrelation 8as completely empiri-

cal=/erive/ from o-servations?,

an/ is given as3

!is relations!ip is rigorous for

low pressure gas reser"oirs,

=nF for laminar flo8?.

7rom3 #ac$-Pressure %ata on Natural-Gas &ells and !eir Application toProduction Practices 4a8lins an/S1>ell>ar/t =USBM Monograp>, )$*?.

9istory3 %eli"era'ility()#ac$pressure) E*uation

nwf

ppCgq )( 22=


4/46


Diffusivity E


5/46


:Dry 6as: PT Properties3 = g,vs.p?

Basis for t>e :pressure%s


6/46


:Dry 6as: PT Properties3 = g,vs.p?

Con1ept3 7 = g,? F 1onstant, T9EH p+%varia-le vali/.

= g,? 1onstant forpG+222 psia.

Even 8it> numeri1al solutions,p+formulation 8oul/ not -e appropriate.

dp

z

pp

pp

zp

gbasep

gpg

n

=

9istory3 Gasp+ondition= g,"s.p F+22 Deg 7?


7/46


Arps@ =Empiri1al? 4ate 4elations3Exponential /e1line 1ase =1onservative?.9armoni1 /e1line 1ase =li-eral?.9yper-oli1 /e1line 1ase =everyt>ing in -et8een?.

7et;ovi1> =4a/ial 7lo8? De1line Type Curve3Exponential, >yper-oli1, >armoni1 /e1line 1ases.

Derivation of t>e Arps@ Exponential 4ate 4elation3Com-ination of li


8/46


7lo8rate%Time 4elations3)exp( tDqq

ii =

bi

i

tbD

qq

/1)(1+

=

)(1 tD

qq

i

i

+

=

E+ponential3 ='F2?

/yper'olic3=2G'G?

/armonic3 ='F?

Cumulative Pro/u1tion%Time 4elations3

)]exp([1 tDD

qN i

i

ip =

])(1[1)(1

/11 b

ii

i

p tbDDb

q

N

+=

)ln(1 tDD

qN i

i

ip +=

E+ponential3 ='F2?

/yper'olic3=2G'G?

/armonic3 ='F?

Arps 4elations3 Summary =I+?


9/46


7lo8rate%Cumulative Pro/u1tion 4elations3

E+ponential3 ='F2?

/yper'olic3=2G'G?

/armonic3 ='F?

)()1(1 pbi

ib NNq

Dbq

=

= p

i

ii N

q

Dqq exp

pNiDiqq =

b

i

bi

p qDb

qNN

= 1

)1()(or

Plot of3 *versus Np

Plot of3 log=*? versus Np

Plot of3 log=N%Np? versus log=*?

Arps 4elations3 Summary =+I+?


10/46


a. Semilog :4ate%Time: Plot3 Barnett 6as

7iel/.

-. Cartesian :4ate%Cumulative: Plot3

Barnett 6as 7iel/ =Hort> Texas?.

1. 0og%0og :=G%Gp?%4ate: Plot3 Barnett 6as

7iel/ =Hort> Texas?.

a.

-.

1.

pNiDiqq =

b

i

bi

p qDb

qNN

= 1

)1()(

)exp( tDqq ii =

bi

i

tbD

qq

/1)(1+

=

bp

b

i

bi NNDb

qq

= 1

11

1

)()1(

=Exponential?

=Exponential?

=9yper-oli1?

=9yper-oli1?

=9yper-oli1?

Arps 4elations3 E+ample =I+?


11/46


)exp( tDqq ii = bi

itbD

qq /1)(1+

==Exponential? =9yper-oli1?

Arps 4elations3 E+ample =+I+?

E0RAnalysis3 Barnett 7iel/ =Hort> Texas =USA?? Semilog :4ate%Time: Plot3 Barnett 6as 7iel/. Hote /ata s1atter an/ apparent fit of >yper-oli1 fun1tion.


12/46


a. Semilog :4ate%Time: Plot3 SPE "#+"!

East Texas 6as ell .

-. Cartesian :4ate%Cumulative: Plot3 SPE

"#+"! East Texas 6as ell .

1. 0og%0og :=G%Gp?%4ate: Plot3 SPE "#+"!

East Texas 6as ell .

a.

-.

1.

pNiDiqq =

b

i

bi

p qDb

qNN

= 1

)1()(

)exp( tDqq ii =

bi

i

tbD

qq

/1)(1+

=

bp

b

i

bi NNDb

qq

= 1

11

1

)()1(

=Exponential?

=Exponential?

=9yper-oli1?

=9yper-oli1?

=9yper-oli1?

Arps 4elations3 E+ample + =I+?


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)exp( tDqq ii = bi

i

tbD

q

q /1)(1+==Exponential? =9yper-oli1?

Arps 4elations3 E+ample + =+I+?

E0RAnalysis3 SPE "#+!" ell =East Texas =USA?? Com-ination :4ate%Time: an/ :Pressure%Time: plot. Hote pressure -uil/up =use/ to 1>e1; 8it> PA?.


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7et;ovi1> :Empiri1al: De1line Type Curve3 1og-log )type cur"e) for t!e Arps )decline cur"es)=7et;ovi1>, )!$?. nitially /esigne/ as a grap>i1al solution of t>e Arps@ relations.

7et;ovi1> De1line Type Curve3 Empirical


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7rom3 SPE 2#'+) 7et;ovi1> =)!$?.

7rom3 SPE 2#'+) 7et;ovi1> =)!$?.

:Analyti1al: 4ate De1lineCurves3

Data from van Ever/ingen an/9urst =)#)?, replotte/ as a rate/e1line plot =7et;ovi1>, )!$?.

T>is loo;s promising -ut t>is isgoing to -e one really -ig :type1urve.:

>at 1an 8e /oJ Try to 1ollapse

all of t>e tren/s to a single tren//uring -oun/ary%/ominate/ flo8=7et;ovi1>, )!$?.

)Analytical) stems are anot!ername for transient flow 'e!a"iorw!ic! can yield estimates of

reser"oir flow properties.

Analyti1al Type Curves3 4a/ial 7lo8


16/46


7et;ovi1> :Analyti1al: De1line Type Curve3 =1onstantpwf?

0og%log :type 1urve: for transient flo8 -e>avior =7et;ovi1>, )!$?. 7irst :tie: -et8een pressure transient an/ pro/u1tion /ata analysis.

7et;ovi1> De1line Type Curve3Analytical


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7et;ovi1> :Composite: De1line Type Curve3 Assumes 1onstant -ottom>ole pressure pro/u1tion. 4a/ial flo8 in a finite ra/ial reservoir system =single 8ell?.

7et;ovi1> De1line Type Curve3 omposite


18/46


5il Material Balan1e 4elation3

5il Pseu/ostea/y%State 7lo8 4elation3

Steps3. Differentiate -ot> relations 8it> respe1t to time.

+. AssumepwfF 1onstant =eliminates d=pwf?Idtterm?.

$. E


19/46


ali/ation3Arps2 /yper'olic %ecline ase

a. 9yper-oli1 flo8rate relations for t>e 1aseof 1onstant pressure pro/u1tion from a

solution gas /rive reservoir =Cama1>o

an/ 4ag>avan =)")??.

-. 9yper-oli1 /e1line type 1urve 8it> /atasimulation performan1e /ata superimpos%

e/ =Cama1>o an/ 4ag>avan =)")??.

=Details of /erivation are omitte/, see paper

SPE )22), Cama1>o an/ 4ag>avan =)")??.


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Spe1ialie/ 6as 7lo8 4elations

7et;ovi1> 6as 7lo8 4elation =poor approximation?34ate%time.

C>ara1teristi1 -e>avior plot.

4esults from no8les%Ansa>%Bu-a 8or;34ate%time.4ate%1umulative.


21/46


6as Material Balan1e 4elation3 =,F K =i/eal gasJ??

6as Pseu/ostea/y%State 7lo8 4elation3 =7et;ovi1>?

7inal 4esult3 =7et;ovi1>?

1)( === ipii zzGGppp

7et;ovi1> 6as 7lo8 4elation3 Poor Appro+imation

n

wf

pp

g

C

g

q )( 22 =

0)(

122

)12(1

1 =

+

= wfp

nn

tG

giqn

giq

gq


22/46


7et;ovi1> :Analyti1al: 6as De1line Type Curve3 =pwfF 2?

C>eate/ =,F? ... t>is is not a vali/ solution =7et;ovi1>, )!$?.6oo/ intentions ... 8ante/ to /evelop a :simple: gas solution.

7et;ovi1> De1line Type Curve3 Gas


23/46


no8les 6as 4ate%Time 4elation

+

+

+

==1

)exp(1

11

)exp(

1

11

)1(

2

2

2

DdwD

wD

wD

DdwD

wD

wD

wD

wD

gi

ggDd

tpp

p

tp

p

p

pp

qqq

t

G

zp

zp

qt

ii

wfwf

gi

Dd

/

/1

2

2

=

=

ii

wfwf

wDzp

zpp

/

/

:no8les: rate%time relation for gas flo83Mo/els /e1line of gas flo8rate versus time.

Better representation of rate%time -e>avior t>an t>e :Arps:>yper-oli1 /e1line relations.

Assumptions3olumetri1, /ry gas reservoir.

pi G '222 psia.

Constant -ottom>ole flo8ing pressure.


24/46


:no8les: relations for gas flo83*g Gp follo8s


25/46


Simplifie/ 6as 7lo83 3alidation of 4nowles E*s.

a. Simulate/ Performan1e Case3 *gversus t=piF *222 psia,pwfF222 psia, G*uadF#.+2 BSC7?.

*gvs. t an/ Gpvs. t3Base plots L verify mo/els -y

Ansa>, et. al

Comparative tren/s of 2.)*gi, *gian/ .*gi . Comparison applie/

to all analysis plots.ery goo/ mat1> on -ot> plots,

a11ura1y verifies mo/el.

-. Simulate/ Performan1e Case3 Gpversus t=piF *222 psia,pwfF222 psia, G*uadF#.+2 BSC7?.


26/46


Simplifie/ 6as 7lo83 3alidation of #u'a E*.

:no8les%Bu-a: relations for gas flo83Simulate/ performan1e 1ase3 *g%Gp=es 8ell 8it>


27/46


Spe1ialie/ 5il 7lo8 4elations

7et;ovi1> 5il 7lo8 4elation34ate%time =De1line Type Curve Analysis?.

Delivera-ility =so1>ronal Testing of 5il ells?.


28/46


5il Material Balan1e 4elation3 =p56 formulation7?

5il Pseu/ostea/y%State 7lo8 4elation3 =7et;ovi1>?

7inal 4esult3 =7et;ovi1>?

pii NN

ppp2

22 )()()( =

7et;ovi1> 5il 7lo8 4elation3 =Appro+imation?

n

wf

pp

ip

p

oiJ

oq )( 2

2

=

0)(12

2

11

1

=

+

+

= wfpn

tNoiqoi

q

oq


29/46


7et;ovi1> :Analyti1al: 5il De1line Type Curve3 =pwfF 2?

C>eate/ =pressure-s*uared material 'alance relation8? ... t>is is not a

vali/ solution =7et;ovi1>, )!$?.

7et;ovi1> De1line Type Curve3 Solution Gas %ri"e


30/46


5il :Ba1;pressure: 4elation3 7et;ovi1> =I+?

a. Delivera-ility =:-a1;pressure:? plot

/evelope/ for ell +I#%+ prior to matrixa1i/iing treatment. =7et;ovi1> SPE

22#*+) =)!$?N?.

-. Delivera-ility =:-a1;pressure:? plot

/evelope/ for ell +I#%+ after matrixa1i/iing treatment. Hote mu1> >ig>er

flo8rate performan1e an/ apparent non%

linear =i.e., non%laminar? flo8 -e>avior

=7et;ovi1> SPE 22#*+) =)!$?N?.


31/46


5il :Ba1;pressure: 4elation3 7et;ovi1> =+I+?

a. Comparison of simulate/ an/ pre/i1te/IPR-e>aviors for solution%gas%/rive

1ase =ogel SPE 22#!' =)'"?N?.

-. Delivera-ility =:-a1;pressure:? plot/evelope/ using ogel /ata. Proof of

1on1ept for :-a1;pressure: flo8 relation

=7et;ovi1> SPE 22#*+) =)!$?N?.


32/46


nflo8 Performan1e 4elations =IPR?

Early 8or; =for rationale?5il IPRan/ Solution%6as Drive IPRogel IPR8or; =for familiarity 8it> approa1>?5t>er IPR8or; =for referen1eIorientation?

6as Con/ensate IPR7evang an/ >itson 8or; =for referen1e?


33/46


9istory 0essons Early Performance Relations

Early O6as Delivera-ility Plot,:

note t>e straig>t%line tren/s for

t>e /ata =1ir1a )$*?.

Early O6as IPR Plot,: note

t>e ip

-et8een 8ell>ea/ pressure

an/ flo8rate =1ir1a )$*?.

ell /elivera-ility analysis3 =after Rawlins and Sc!ell!ardt? T>ese plots represent t>e earliest attempts to avior an/ to

pre/i1t future performan1e.


34/46


6as ell Delivera-ility3

T>e original 8ell /elivera-ility

relation 8as /erive/ fromo-servations3

!e )inflow performance relation-

s!ip) =or IPR?for t!is case is3=assuming nF?

7rom3 #ac$-Pressure %ata on Natural-Gas &ells and !eir Application toProduction Practices 4a8lins an/S1>ell>ar/t =USBM Monograp>, )$*?.

9istory 0essons )#ac$pressure) E*uation

)( 22

wfppCgq =

nwf

ppCgq )(22=

0)()(2 ==

wfppCmax,gq

21

=p

wfp

max,gq

gq


35/46


9istory 0essons IPR %e"elopments(orrelations

Early :nflo8 Plot,: an attempt

to 1orrelate 8ell rate an/ pres%

sure -e>avior an/ to esta%

-lis> t>e maximum flo8rate,

=after 6il-ert =)*#??.

nflo8 Performan1e 4elations>ip =IPR?3Correlate performan1e, estimate maximum flo8rate. n/ivi/ual p>ases reavior

=after ogel =)'"??.

1

=p

p

qq wfmaxo,o

2

1

=p

p

qq wf

maxg,

g

2

0!80!21

=p

p

p

p

qq wfwfmaxo,o


36/46


Solution%6as Drive Systems 3ogel IPR

IPR-e>avior is /epen/ent on

t>e depletion stage =i.e., t>e

level of reservoir /epletion?.

Ho single 1orrelation of IPR

-e>avior is possi-le.

ogel IPRCorrelation3 Solution 6as%Drive Be>aviorDerive/ as a statisti1al 1orrelation from simulation 1ases.

Ho :t>eoreti1al: -asis ntuitive 1orrelation =*oma+an/pa"g?.

T>e ogel IPR 1orrelation an/

its variations are 8ell esta-lis>%

e/ as t>e primary performan1e

pre/i1tion relations for pro/u1%

tion engineering appli1ations.

T>e original 1orrelation is /e%

rive/ from reservoir simulation.

20!80!21

=p

p

p

pq

q wfwfmaxo,o

ogel Correlation3 =Statisti1al?


37/46


Solution%6as Drive Systems Ot!er Approac!es

5t>er IPRCorrelations3 7et;ovi1>3 Derive/ assuming linear mo-ility%pressure relations>ip.4i1>ar/son, et al.3 Empiri1al, generalie/ 1orrelation.

2

)(11

=p

p

p

pq

q wfx

wfx

maxx,x

7et;ovi1> IPR3 =Semi%Empiri1al?n

maxo,o

p

wfp

qq

=2

1

4i1>ar/son, et al.IPR3 =Empiri1al?

=+F p>ase =e.g., oil, gas, 8ater??


38/46


Solution%6as Drive Systems 5t>er Approa1>es

5t>er IPRCorrelations3iggins, et al.3 Use/ a polynomial expansion of t>e mo-ility fun1tion in

or/er to yiel/ a semi%rigorous IPRformulation.

Coeffi1ients =a, a+? are /etermine/ -ase/ on t>e mo-ility fun1tion

an/ its /erivatives ta;en at t>e average reservoir pressure.

iggins, et al.IPR3 =Semi%4igorous?

!!!13

3

2

21 ++++=

p

pa

p

pa

p

paq

q wfwfwfmaxo,o


39/46


Solution%6as Drive Systems 5t>er Approa1>es

5t>er IPRCorrelations3 strong fun1tion of pressure an/ saturation.

Semi%rigorous IPRformulation =/erive/ for t>e solution%gas 1ase? >ast>e same form of t>e 4i1>ar/son, et al.IPR =8>i1> is empiri1al?.

dpoBo

okp

basepnpo

koBoppop )(

=

Pseu/opressure 7ormulation Q 5il P>ase

pbapfpoBo

ok 2)( +==

Mo-ility 7un1tion

2

)(11

=p

p

p

pq

q wfo

wfo

maxo,o


40/46


6as Con/ensate Systems Pseudopressure

dpoBo

oksoBo

okp

wfps

w!rer

kgq

3/4)ln("

141!21

++

=

T>reeflow regions 8ere

1>ara1terie/3

4egion Main 1ause ofpro/u1tivity loss, oil an/ gas flo8simultaneously.

4egion + T8o p>ases 1oexist,-ut only gas is mo-ile.

4egion $ single%p>ase gas.

7evang an/ >itson Correlation3 6as Con/ensate systemsPressure an/ saturation fun1tions nee/ to -e ;no8 in a/van1e

654, PT properties an/ relative permea-ilities.


41/46


Mo/el%Base/ Performan1e Stu/y34a/ial, fully 1ompositional, single 8ell simulation mo/el

ParametersIfun1tions use/ in simulation3 4eservoir Temperature3 F +$2, +'2, $22 Deg 7 Criti1al 5il Saturation3 SocF 2, 2., 2.$

4esi/ual 6as Saturation3 SgrF 2, 2.*, 2.*

4elative Permea-ility3 ! sets of $ro%$rg/ata

7lui/ Samples3 # synt>eti1 1ases, + fiel/ samples

Assumptions use/ in simulation3 nterfa1ial tension effe1ts are negle1te/ Hon%Dar1y flo8 effe1ts are negle1te/ Capillary pressure effe1ts are negle1te/ 4efine/ simulation gri/ in t>e near%8ell region S;in effe1t is negle1te/

6ravity an/ 1omposition gra/ients are negle1te/Simulations -egun at t>e /e8 point pressureCorrelation of gas an/ gas%1on/ensate performan1e using 4i1>ar/son

IPRmo/el.

6as Con/ensate IPR %el astillo +22$=A90?


42/46


Gas ondensate IPR rends=ondensate?

Con/ensate IPRCorrelations =gas 1on/ensate reservoirs?All eig>t /epletion stages regresse/ simultaneously.Ex1ellent 1orrelation all stages.

Base

IPRplot =1on/ensate?

Case ' =gas 1on/ensate sys%

tem?.

Dimensionless

IPRplot

=1on/ensate?

Case ' =gas

1on/ensate system?

P4 Curves % Con/ensate Pro/u1tion

=Case&'?

2

&222

+222

$222

#222

*222

'222

2 +22 #22 '22 "22

*o , STBID

pwf,psia

HpIH F 2.&"R

HpIH F 2.$'R

HpIH F &.!)R

HpIH F $.*"R

HpIH F *.$!R

HpIH F !.&*R

HpIH F ".)#R

HpIH F &2.!$R

0egen/

Hormalie/ 5il 7lo8rate

=Case'?

2

2.

2.+

2.$

2.#

2.*

2.'

2.!

2."2.)

2 2. 2.+ 2.$ 2.# 2.* 2.' 2.! 2." 2.)

p wf(p 'ar

*o

(*o.ma+

HpIH F 2.&"RHpIH F 2.$'RHpIH F &.!)RHpIH F $.*"RHpIH F *.$!RHpIH F !.&*RHpIH F ".)#RHpIH F &2.!$RP4 Mo/el

0egen/

G =G ?


43/46


6as IPRCorrelations =gas 1on/ensate reservoirs?All eig>t /epletion stages regresse/ simultaneously.Ex1ellent 1orrelation even 8>en t>ere is a more pronoun1e/ 1urve

overlap =gas?.

Base

IPRplot =gas?Case

' =gas 1on/ensate system?.

Dimensionless

IPRplot =gas?

Case ' =gas 1on/ensate

system?.

P4 Curves % 6as Pro/u1tion

=Case&'?

2

&222

+222

$222

#222

*222

'222

2 &222 +222 $222 #222

*g, MSC7ID

pwf,psia

6pI6 F 2.2)R

6pI6 F 2.#!R

6pI6 F 2.)*R

6pI6 F #.!*R

6pI6 F ).*R

6pI6 F +$.!#R

6pI6 F #!.#"R

6pI6 F ''.#"R

0egen/

Hormalie/ 6as 7lo8rate

=Case&'?

2

2.

2.+

2.$

2.#

2.*

2.'

2.!2."

2.)

2 2. 2.+ 2.$ 2.# 2.* 2.' 2.! 2." 2.)

p wf(p 'ar

*g

(*g.ma+

6pI6 F 2.2)R6pI6 F 2.#!R6pI6 F 2.)*R6pI6 F #.!*R6pI6 F ).*R6pI6 F +$.!#R6pI6 F #!.#"R6pI6 F ''.#"RP4 Mo/el

0egen/

Gas ondensate IPR rends=Gas?

G d Diff i IPR T /


44/46



=Case&'?

2

&222

+222

$222

#222

*222

'222

2 +22 #22 '22 "22

*o , STBID

pwf,psia

HpIH F 2.&"R

HpIH F 2.$'R

HpIH F &.!)R

HpIH F $.*"R

HpIH F *.$!R

HpIH F !.&*R

HpIH F ".)#R

HpIH F &2.!$R

0egen/


=Case&?

2

&222

+222

$222

#222

*222

'222

2 &22 +22 $22 #22

*o , STBID

pwf,psia

HpIH F 2.#$R

HpIH F 2."'R

HpIH F #.+)R

HpIH F ".*)R

HpIH F &+.""R

HpIH F &!.&!R

HpIH F +&.#'R

HpIH F +*.!'R

0egen/

Con/ensate IPRS>ape =gas 1on/ensate reservoirs?4emar;a-le /ifferen1e in s>ape -et8een a very ri1> gas 1on/ensate

system an/ a lean one.

Base


Case ' =ery ri1> gas1on/ensate system?.

Base


Case =0ean gas 1on/ensatesystem?.

Gas ondensate Differen1e in IPRTren/s

Gas ondensate IPR Parameter = or ?


45/46


Con/ensate or gas IPRparameter =gas 1on/ensate reservoirs? 0o8 oor gvalues IPRmore 1on1ave.Exa1t value of not 1ru1ial similar 1urves for /ifferent oor gvalues.

0egen/

Dimensional P4 1urves

5il or 6as flo8rate

pwf,p

s

a

F 2.&* F 2.&" F 2.+)

F 2.#)

F 2.** F 2.'"

Dimensionless P4 1urves

2

2.&

2.+

2.$

2.#

2.*

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46/46

T.A. Blasingame, Texas A&M U.Department of Petroleum Engineering

Texas A&M UniversityCollege Station, T !!"#$%$'

(.)!)."#*.++)+ t%-lasingametamu.e/u


Natural Gas Engineering

Texas A&M University

0e1ture 2'3

Semi-Analytical Rate Relations

for Oil and Gas Flow=End of 1ecture?

Documents

P613_05A_Lec_06_Rate_Relations_(050302)