H dale beggs production optimization using nodabookfi

Production
Optimization
TM
Using Nodal
Analysis

H. Dale Beggs


Production
Optimization
Using NODALTM Analysis
PHAM HỐNG

mí ANH

H. Dale Beggs

OGCI and Petroskills Publications
Tulsa, Oklahoma

I

I


Production
Optimization
Using NODALTM Analysis

COPYRlGHT 1991, 200~. 2003 by
OGCI, Inc., Petroskills, llC. and H. Dale Beggs
P. O. Box 35448
Tulsa, Oklahoma 74153·0-148
AH rights rescrved. No part of this text
may be r~produced Oc transcribed in any fonn
oc by any mcans withoUl ¡he written pennission
of OGCI and Pelroskills.

lts use in adult training programs is specifically
reserved for OGCI and Pmoskills.
Printed in the United States of Americ.
Libr.ry of Congress Catalog Card Number: 90·064081
International Standard Book Number: 0·930972·14·7
Second printing-Febru.ry, 1999
Third printing-November, 2002
Second Edition-May, 2003


Contents

1

Introduction
1
Systems Analysis Approach
Applications
7
Summary
7

References
7

2

1

Production Systems Analysis
2

Reservoir Performance
Introduction
9
Well Performance Equations
9
OarcyOs Law
9
Factors Affecting Productivity Index
15
Factors Affecting Inflow Performance
15
Orive Mechanisms
17
18
Oissolved Gas Orive
Gas Cap Orive
18
Water Orive
18
Combination Orive

19
HJ
Orawdown or Producing Rate
Zero Skin Factor' 19
Non-zero Skin Factor
20
Effect 01 Oepletion
20
IPR Behavior 01 Gas Wells
20
Predicting Present Time IPRs lor Oil Wells
21
Vogel Method
21
Application 01 Vogel Method-Zero Skin Factor
23
24
Saturated Reservoirs
UndersOaturated Reservoirs
24
Application 01 V0gel Method-Non-Zero Skin Factor (Standing Modification)
Undersaturated Reservoirs
29
Oetermining FE from Well Tests
29

9

'

26

1°


Fetkovich Method
30
31
Flow-After-Flow Testing
Isochronal Testing
31
Modified Isochronal Testing
32
Jones, Blount and Glaze Method
35
Constructing IPRs When No Stabilized Tests Are Available
37
IPR Construction for Special Cases
Horizontal Welis
37
Waterflood Welis
37
Stratified Formations
38
Static Reservoir Pressure Unknown
39
Predicting Future IPRs for 011 Welis
40
Standing Method
40

Fetkovich Method
42
Combining Vogel and Fetkovich
42
Predicting Present Time IPRs for Gas Welis
43
Use of the Back Pressure Equation
43
Jones, Blount and Glaze Method
45
46
Predicting Future IPRs for Gas Wells
Weli Completlon Effects
47
48
Open Hole Completions
Perforated Completions
48
Perforated, Gravel-Packed Completions
53
Innow Performance Summary
54
Oil Welis ·54
Gas Welis
54
References
55

3

36

Flow in Pipes and Restrictions
Introduction
57
Basic Equations and Concepts
58
The General Energy Equatlon
58
Single-Phase Flow
62
64
Two-Phase Flow
Two-Phase Flow Variables
64
Liquid Holdup
64
No-Slip Liquid Holdup
65
Denslty
65
65
Velocity
Viscosity
66
SUrface Tension
66
Modification of the Pressure Gradient Equation for Two-Phase Flow
66
Elevation Change

Friction Component
67
Acceleration Component
67
Two-Phase Flow Patterns
67
Pressure Traverse Calculation
67
Procedure When Temperature Distribution is Unknown
69
Fluid Property Calculations
72
Fluid Density
75

vi

57

66


Gas
75
Oil
75
Waler
75
76
Fluid Velocity

Gas
76
Oil
76
Water
76
Empirical Fluid Property Correlalions
76
Gas Compressibility Factor
77
Salution or Dissolved Gas
78
Formation Volume Factor
79
Gas
79
79
Oil
Water
79
Isothermal Compressibilily
79
80
Viscosily
Oil
80
Waler
80
80
Gas

Interfacial Tension
81
Gas/Oil Inlerfacial Tension
81
GasN'laler Inlerfacial Tension
81
Predicling Flowing Temperatures
81
Flowing Temperature in Wells
82
82
Flowing Temperature in Pipelines
Well Flow CarrelaClons
83
84
PoeHmann a"d Carpenler ~lethod
85
Hagedorn ane Brown Method
86
Duns and Ros Melhod
Orkiszewski 1'.lethod
86
87
Bubble Flow
Slug Flow
87
Transition Flow
87
Misl Flow
87

Aziz, Govier and Fogarasi Melhod
87
88
Chierici, Clucci and Sclocchi Method
Beggs and Brill Method
88
MONA, Asheim Method
90
Hasan and Kabir Method
90
Flow in Annuli
90
Hydraulic Radius Cancept
90
Cornish Methad
91
Evaluation al Correlations Using Field Data
91
Elfects 01 Variables on Well Performance
93
93
Liquid FlolV Rate
Gas/Liquid Ralio
93
94
Waler/OiI Ratio or Water Cut
Liquid Viscosity
95
Tubing Diameler and Slippage
95

96
Flow in Gas Wells
97
Flaw in Direclional Wells
Use 01 Prepared Pressure Traverse Curves
98

di


Preparation of Pressure Traverse Curves
Generalized Curves
98
Application of Traverse Curves
98
Pipeline Flow Correlations
104
Horizontal Flow Pattern Prediction
108
Eaton, et al., Method
109
Dukler, et al., Method
110
Seggs and Srill Method
111
Flanigan Method for Hilly Terrain
112
Hybrid Model
114
MONA, Asheim Method

114
Evaluation of Pipe Flow Correlations
114
Effects of Variables on Pipeline Performance
Liquid Flow Rate
116
Gas/Uquid Ratio
116
Water Cut
117
Liquid Viscosity
117
Pipe Oiameter
117
Single-phase Gas Flow
117
Use of Prepared Pressure Traverse Curves
Parallel or Looped Pipelines
122
Pressure Orop Through Restrictions
123
Surface Chokes
123
Gas Flow
123
Two-Phase Flow
124
Subsurface Safety Valves (SSSVs)
127
Gas Flow

127
Two-Phase Flow
127
Valves and Pipe Fittings
128
Eroslonal Velocity
129
References
129

4

116

118

Total System Analysis
Introduction
133
Tubing Size Selection
135
Flowline Size Effect
136
Effect of Stimulation
139
Systems Analysis for Wells with Restrictions
Surface Chokes
141
Subsurface Safety Valves
143

Evaluating Completion Effects
143
Nodal Analysis of Injection Wells
146
Effect of Oepletion
148
Relating Performance to Time
150
Analyzing Multiwell Systems
151

5

98

133

141

Artificial Lift Design
Introduction
155
. Continuous Flow Gas Uf!

155
155

l'iii
'.

Well Performance
156
Valve Spacing
160
Gas Uf! Valve Performance
165
Otis Design Procedure
167
Submersible Pump Selection
174
Sucker Rod or Beam Pumping
177
Hydraulic Pumping
183
Summary
183
References
185

Nomenclature

187

Appendix A

191

Two-phase Flow Correlation Examples Hagedorn and Brown Method

197

Appendix B
Pressure Traverse Curves

191

197


Production Systems
Analysis

1

INTRODUCTION
.-\ny procluctioll \Vell i~ drillcd :lnd completcd lo mQVC
Ol" g"l~ fnJl1l irs original iocatioll in the rescrvoir

(;-:(' oil
¡~...

¡he stock tank or sales line. ~foYel1lcnr or {rampart of
!luids rcquircs ~ncrgy to t)V('rcol11c friction losscs
::1 Ihe syslcm and lO !in (he products to lhe surfJcc. The
(uids mus! travel Ihrough rhe rescrvoir and lhe piping

¡:-,¡;';C

~~ ~tem

and ultinltHe]y 110\\1 into J scparator for gas-liquid
Thc produclion system can be relati"cly simr!c or can ínelude many components in which energy al'
fíessurc losscs occur. Far cxample. íl diagram of:1 COI11rkx production systcm, which ¡Ilustrares a numbcr of
~('rJration.

l:-;e componcnls in which prcssure losses OCCUf, is shown
ioFig.I·1.
Thc prcssurc drop in thc fotal syslcm al any lime will
~ the iñitial nllid prc!'surc minus Ihe final nuid pres~:Jre, pI{ ~ P...",. This pressure drnp is the slIm of the
rressure drops occurring in all ol' lhe componcnts of the
~~ stem. Since the pressure drop through any component
yaries with producing rate, the producing rate will be
cQntroJled by Ihe components selecled. The selcction and
~izing of the individual components is very important,
r:Jt because of Ihe intcraction í.lmong the components, a
ch.:lIlge in the prcssurc drop in one Il1l~e Oowing !luid is compressiblc. amI. thercfore. the
r~~ssurc drop in a pal1icuJar componcpt dcpcnds not only
en the now rate through the component, bUI also on the
J\'aage pressure lhat exists in lhe componcnt.
Thc final dcsign of a production syslel1l cunnot be sep2í3tcd into rcservoir performance amI piping .systell1

perfonnance and handled independenlly. The amollnl
ol' oil and gas no~ing inlo the \\'ell fram the reservoir
depen,-!s on the pressure drap in the piping sY:'lem .
and the pressurc drap in the piping system dq:-ends
on the amount of fluid tlowing through il. Ther-=,iore,
the t:nlire production system mus[ be analyzcd J:; a
unir.

The production rale ar delivcrability of a \\'ell can aften
be sevcrely restricted by the perfonnancc of on!y one
component in the system. If the effcct of cach component on Ihe tot<'l.1 system perfannance can be isoIalcd. the
systern performance can be optimized in the rnost economical way. Past experience has sho\\'n thtH Jarge
amounts of moncy have been wasled on stimulating lhe
[onnation when the we¡¡-s producing capacily \\'3:' acrually bcing restricted because the mbing or Oowline was
[,-X, smal!. Another cX3mplc of errors in completiC'n design is to install tubing that is too largc. This often happen, on wells Ihat are expected to produce at high rales.
lt will be shown thar this practicc not only was[es Illoney
on oversized equipmcnt. bUI tubing that is too large ('an
actually reduce the rate at which a well will no\\'. This
can cause the wcll to load up with liquids and dic. which
ncccssitatcs lhe early ínst.tllation of artifici.tl lift equip~
menl or comprcssion.
A mcthou for analyzing a wel1. which will alk'!\\' dck'mlination of the producillg c;:IracJty for any C'llmbin:ltion of componcnts, is desnibed in lhe following st::ction. This melhod may be uscu (o delcnninc locafit1rts of
c.\(:essive Oow rcsislance or prcs~urc drop in any p~lrt of
the systcm. Thc crfcct of changing th<' total wcll performance can be easily dclcrmincd,

I


Prodllctioll Oplimi=alioll Usillg Nodal Allalysis

2

f<-A PB =(P wh -i'sep)4 rr===::::::;;--~SALES UNE
PWhO lAPB=(Posc-Psep)r'
GAS

T2

;~~;~~-

~p~se~p~S~EP~A~R~A"iO~R~~~
lIQUID

Pose

Posv

STOCK
TANK

AP4=(PUSV-r-uov'P_' pI,

I

t.p,
t.P2

AP7=
Pwl-Pwil

AP3
PUA-POA
AP4 = Pusv-Posv
A Ps
Pwh -PDse
c> P6 = POSC-Psep
c> P7

Pwf-Pwh
c> PB = Pwh-Psep

BorrOMHOLE
RESTRICTION
AP3=
(PUA -POA)

PR-Pwfs
Pwfs-Pwl

Á

LOSS IN POROUS MEDIUM
LOSS ACROSS COMPLETION
"
"RESTRICTION
"
SAFETY VALVE
SURFACE CHOKE
IN
FLOWlINE
TOTAL LOSS IN TUBING
" FLOWlINE

~;""';;'7\,

t. P, =(Pw's - Pw,)--1

I

....------1

A P, =(PFi - Pw's)

Fig. 1-1. Possible pressure losses in complete system.

SYSTEMS ANALYSIS APPROACH

The systems analysis approach, oflen caHcd NGDAL'"
Analysis, '" has becn applied for many years to analyze
the perfonnance of systems eomposed of interacting
eomponents. Electrical circuits. complex pipeline networks and centrifuga! pumping systems are a11 analyzed
using mis method. lts applicadon to well producing syslems was firsl proposed by Gilbert' in 1954 and discussed by Nind' in 1964 and Brown' in 1978.
The procedure consists of selecting a division poiot or
node in lhe well and dividing the syslem al lhis point.
The locations of lhe most commoruy used nodes are shown
in Fig. 1-2.
AH of the components upstream of (he nade comprise
(he inflow section. while the ou(f1ow section consists af
all of lbe componenLS downstream of the node. A relationship between fiow cate and pressure drop must be
availabJe foc each component in the system. The flow
rale lhrough lhe system can be delermined once the following requirements are satisfied:

1. Flow inlo lhe node equals flow out nf lhe node.
2. Ooly one pressure can exist at a nade.
·"NODAL Analysis" is a uademark oC Flopetrol JohnSlon, a di8
'IisiDn oC Schlumberger Technology Corporation, and is proteeted by
. U.S. Palent #4,442,710.

At a particular time in the Efe af (he well. there are

always [wo pressures !.ha( remain flxed and are not fun.:tioos of ilow rateo Ofie of these pressure~ is (he aver.lge
reservoir pressure PRI and (he other is the s)lstem audet
pressure. The outlet pressure is usually (he separaror
pressure Pup' bU[ if the well is cOnlrolled by a surface
choke the fixed oullet pressure may be the wellhead
pressure P...IJ.
Once the nade is selected. the nade pressllre is calculated from both dírections starting at [he fixed pressures.
inj10w fo the node:

fiR - t:.p (upstream componen!s)

= P"..J<

Outflow from the lIode:
PUP

+ l1p

(downstream eomponents) ::::: P""dc

The pressure drop, IIp, in any component ,varies wilh
flow rale, q. Therefore. a pJot of nade pressure versus
flow rate will produce two curves, the intersection of
which will give the canditions satisfying requiremenlS 1
and 2, given previously. The procedure is illustrated
graphically in Fig. 1-3.
The effect of a change in any of the eomponents can
be analyzed by recalculating the nade pressure versus
flow rate using the new characteristics of the componenl

3

PrOdllcliOIl Syslems Allalysis

NODE
1

2
3
4
5
6
7
B
1A
1B

LOCATION
SEPARATOR
SURFACE CHOKE
WELLHEAD
SAFETY VALVE
RESTRICnON
PWF
PWFS

PR
GAS SALES
STOCKTANK

,e,g. 1-2. Localion 01 various nades.
rhat

waS

changed. Ir u changc was made in


I ,~(l(}H'

ro

Hode:

(oll1poncnt. lhe Gutllo\\' curve wil1 rCI1l3in unchanged.

Ho"ve ....er. if cithcr curve is chang:cd, lhe intersection \ViII
Pt: shifteu. and a IlCW !lo\\' capadty and node pressure
\\'¡II cxist. Thc CtlfVCS \ViII also be shifted if eithcr of the
fixcd prcssures is changcd. which may occur with depletion or a chunge in separatioll conditions.
The procedurc can be furthcr illustrnted by considcring the simple producing systcm shown in Fig. 1-4 3nd
~elccting

the wellhcad as ¡he nade.

Qutj70w from nade:

The effect on the flow capxity of changing. the tubing

size is illustrated in Fig. 1-:'. and thc crfce! l.."'If <1 dwnge
in flowline size is shown in Fig. 1-6.
The effcct of increasing: thc tubing sizc. :.1:' long as the

I•

g

Oulflow from
node

~

w

OC

::>

(f)

~
oc

:

o...
W

1

,

Inllow lo node

-----~:~-

L:
°z

VERTICAL :-::;
INClINED· .'9ING

I

1Syslem Flc-...
I CapacUy

FLOW RATE, q -

,C¡g. 1-3. Oeterminalion

o( ffow capacity.

.:::'g. 1-4. Simple producing

s.'stem


Productioa Optimizalioa Using Nodal Analysis

4

OUtflOIV from node:
Pup

Pwh

q-=.---=J

L

Fig. 1-5. Ellecl 01 lub{ng size.

tubing i5 not too large, is to give a higha node or wellhead pressure ror a gi\'cn flow rate, bec3use the prc.ssure
drop in [he tubíng wiB be decreased. This shifts the ioflow curve upward and lhe intcrsection te the right.
A lar'2.er tlowline will reduce {he pressure drop in (he
nowline~. shifting the outtlow down and (he ¡ntceseerian
to the right. The erfeer of a change in any component in
{he system can be isolarcd in lhis manoa. Also, {he ef~
feet of declining reseryoir pressure or changing separator
prcssure can be determined.
A more frequently used analysis procedure is to select
(he node betwecn (he reservoir and me piping system.
lllis is labeled as paint 6 on Fig. 1-2, and lhe nade pres5Ufe is P..f' Selecting the node at Ihis point esséntially
divides lhe well ioto a reservoir dominated eomponent
aod a pipiog system dominated eomponenL The inflow
and outf1ow expressions for the simple system will theo

be:

Inflon'

10

nade:

+

!J.Pllwli.<

+

:'J.p,",;., = P-t

The effect of a change in tubing size 00 the total systcm producing eapacity when P..f ¡s the node pressure is
illustrated in Fig. lo?~
A producing systcm may be optimized by selceting {he
eombination of component characteristics that will give
the maximum production rate foc {he lowest cost. Although the overall pressure drop available for a system,
PR - PUP' might be fixed at a particular time, the producing capacity of rhe system depends 00 where the
pressure drops occur. If too mueh pressure drop occurs
in one component or module, there may be insufficiem
pressure drop remaining for efficicnt perfornlance of the
other modules. This is illustrated in Fig. 1-8 for a system
in which (he tubing is too slllal!. Even [hough the reservoir may be capnbJe of praducing a large amounl of
fluid, if too much pressure drop occurs in (he tubing, the
well perfonnance suffers_ For this eype of \Vell eompletion, it is obvious (ha¡ improving the reservoir performance by stimulation \\'ould be a waste of effort unkss
larger tubing were installed.
A case in which the well performance is controlkd by
the inflow is shown in Fig.·1-9. In rhis case, the exccssive pressure drop could be caused by fOITn

or inadequate perforarions. II is obvious from tlle plor
[hat improving the performance of the piping system or
outflo\V or placing lhe weH on artitlcial !ift would be
fruitless u~Iess the inflo...... performance were a150 irnproved.
An increase in produclion rate achieved by íncreasing
tubing size is iIlustrated in Fig. 1-7. Howevcr, if lubing
is too large, the velocity of (he fluid moving up the tubing may be too low lo effectively ¡ift the ¡iquicls lo the
surface. This could be caused by either large tubing or
10w production rates. This phenomenon will be dis-

Pwf

q-

Fig. 1-6. Ellecl oOlowline size.

Lt

q_-_-

Fig. 1-7. Elfect 01 lubing size..

_


5

ProdUCliol1 Syslellls Allalv.sis

,

I

,I
I

Pwl

,
I
I

Im¡
I

I ,~

m

__

:~:

mm
q

Fig. 1-10 Efleel 01 lubing size.
Fig. 1-8. \.'Iell res/rieled by plping sys/em.

cusscd in dctail in Chaptn J. Thc nuid \'eloeity is the
produClion rate divided by lhe area of lile lubing. A qual~

itative cxample of selccting lhe optimu11l tubing size ror

a well

th~H

is producing both gas
in Fig. 1-10 and 1-11.
As tubing sizc is increased. ¡he frictiL'1l losses dccrt:
innow. Hthe wcll hegins loading with liquid ami th~ rlo\\' bccoll1cS
inlamittent nr unst'lhle. As lhe liquid k\'el in lhe \\'el1
huilds Ihe \\"ell will c"cntllally die. Fig. 1·11 illustratcs

Ihis graphically.
Once a \\'ell lhat is producing liqllids ail10g wiril [he
ga;; rcaches Ihe stagc in \\'hich it \Vil! nl~ longer tlow
naturdis('usscd cxlcnsivcly in Chapter 5, bu! :ln example of
dctermining (he optimulll gas injcction rale for a wcll on
gns lirt is illustratcd in Fig. 1-12 nnd 1-13. The pu~ose
of injccling gas inlo the tubing is to dccrea:,e the dcnsity
nI' the tl(lwing gas-liquid mi.\tllTc amI. ¡hercfore. dc~

crease the required tlawing ballom hale pressure. Hawever, as the gns rate is incrcas.ed, lhe nuid \'clocity and,
lherefore, th~ !'rictian lasses alsa ¡nerease. A paint \ViII
cvcntually be reached ~uch that the frictian losscs ¡ncrense more lhan the density or hydrostatic losscs dccrease with an ¡ncrease in gas rate. This can be dctcrmincd using NODAL n, Analysis as ilIustrared in Fig.
1-12.

A plot of liquid productil11l rale vcrSU5 gas injection
rate can the inllow anJ oUIOo\\, cur\'es for various injection rates.
A plot of (his dat~l is shown in Fig:, 1-1 J. This mClhod
can also h~ uscd lo allo('n[c ¡he injection ga~ J\'íúlahle
among scYt~ral wells in a field producing by gas lin.
In recent years. it has becn found th3t Jn inadcquntc
numbcr of perforations can be very dctrimenta! to the
performance of SOIllC wells. lf Ihe boltom hl)le flowing
pressurc is selecled as the node pressure, rhe inrlow can
be broken clown iota pressure drop through the rack and
prcssurc drop through the perfarations. The inllaw i:md
outtlow cxprcssions would lhen consist of:

PR ---------------------------------------

1

I

UNSTABLE

q

REGION

I

p

l----------t!

d FOR MAXII.IUM q~

sep I

q--~

Fip. 1-9 l'lelf rcsfricled by inf!ow.

d~-

Fig. 1-11. Finding optimum tucing size.


6

ProducliO/l Optimiza/ion Using Nada/ Ana/ysis

lnflow

ExcessivQ GlA

/

P

N2

"

,1"

t

Oullbw

A

> N1

P.1

NJ

N,

Pwl

> N2

GlR

\

/"
l'l/Iow

Fig. 1-14. Elleel o/ per/orating densily en inflow.

Fig. 1-12. Efleel 01 gas rale on outflow.

IllfloU' lO /Jode:

OU{/7ow ¡ro", Ilode:

Since ¡he perforation pressure drop is a function of the
number of perforations open, as well as prodüction rate,
a different ¡nnow curve would exisl [or each perforating
density. This is illuslraled qualitatively in Fig. 1-14.
As [he numba of perforations is increased. a point
will eventually be rcached such (hat the perforation pressure drop is ncgligible, and, therefore, a further ¡necease
in perforating density would be useless. A pIar of the
production rate resulting from various perforating densities, that ¡s, lhe intersection af the various inflow curves

L:~·
t
I

q L __

:~I Avail~ble
!
I
l·

l. Determine which eomponents in (he system can be
changed. Changes are limitcd in some cases by previous decisions. For example, once a cert:lin hale sizc
is drilled, the casing size and, thcrclorc, (he tubing
size is limited.

2. Select ane component lo be optinlized.
3. Select lhe node location thar will bes! emphasize (he
effeet of the change in lhe selectcd componcnt. This
is nol critica1 because lhe same ovcrall resuh will be
predicted regardless of the node local ion.
4. Dcvelop expressions ror the infiow and oulOow.
5. Obtain required data to calculare prcssure drop versus
rate ror aH [he components. This may require more
dara than is available, which may necessitate per-

=---c:-__..

I
I
I
I

I
I

with the autflo\\' curve, i5 shown in Fig. 1-15. M~thods
ror calculating perforation pressure drop ar~ discusscd in
ehapler 3.
A suggested procedure for applying NüDAL '" Analysis is given as follows:

T
q
Gas Voluml!'

Economlc optlmumqlnJ

i

q

lo'.

Fig. 1-13. Efleet 01 gas injeclion rate en liquid rate.

N~mber 01 p~rlorations

--'to

Fig. 1-15. Elfect 01 perlori!ting density on rate.


Producriol1 S.1'",rems Ana~)'sis

7

the
13. Analyzing a multiwell producing SystClll.

6.

Oelenninc lhe cITect of ehanging (he characteristics of
¡he selcctcd eomponent by plotting innow versus out~
now alld reading lhe intcrscctioll.

I..!. Relating ficld pcrfonl1ancc to time.

7.

Repeallhc proccdurc ror ench component tlwt is to be
optimizcd.
111. APPLlCATIONS

The nodal systcms analysis approach may be uscd to
proccdure Cwells. ifthe ef(cet ofllle arlificiallift lllClhod 011 the pres5ure cnn be cxprcssed as a fUllelion of Ilo", ratc. The proccdure can also be appl:cd to the analysis of injection
\vell performance by applications is givell as follo\Ys:

l.

Selccting tubing sizc.

i

Sckcling nO\vlinc sizc.

_'o

Gran'l pack designo

..1..

Surrat'c chokc sizing.

"

Sub~urrat'c ~arely

('.

..\nalyzillg:ln

\"I\'e siling.

C\i~lillg

syslclll fl)r abnonnal

lll)\\'

rc~trictil)!l~.

IV. SUMMARY

Thc nodal systcms analysis approach is a \'cry ncxihlc
melhod that can be uscd to impro\'c lhe perfonmlllcc of
many "'el\ systems. To apply Ihe systcl11s analysis proccdure to a \Vell, it is nrCeS:':i3ry to be ablc to calculnlc thc
prcssure drop tl13t wiJl (lCcur in al1 the sy:-Iem CL.nlponcl1ts listed in Fig. 1-1. These. prcssure drops depcnd
not only on now mte. bUI on the sizc and other charaCleristics of Ihe eomponents. Un1css accuratc Illcthods
can be fOUlld to cnlculalC thcse prcs:-ure drops. lhe systems analysis can produce crrOllcous rcslllts.
The following sections in this book prcscnt Ihe l<1tcsl
and Illost accurate 111cthods for calculating lhe rci::lliollship bctwcen now rnte ane! pressure drop fol' <111 Ihe COI11p0nelHs. This rcquircs a thorough rcvicw of rescr\'oir ell~

ginecring concepts to dClcrmine rescn'oir intlow pcrf(lrl11~nCe; an underslanding of Illultiph~sc nO\\' in pipes
1(" ca1culatc tubing and nowlillc pcrlorm'ance: procedures
Il) dct12'rminc the performance 01' pcrlorgri.l\·CI~rflCk t:olllplcriol1s. ami damelgcd or :,til1l11lal12'd
\Yell:,: :llld <111 undcrs(andillg ofartil1ciallil'¡":'Y.:;(L'llls.
Once procedures are presclllcd to an:llY7c ('~h:h C0111pl1!lelH :'eparalely. Ihe sys!e1l1S allillY'sis :lpprl)adl will be
applicd lo many dilTcren\ \\'ells (o dCl11l)J1stratc Ihe proccdurc:, to optimize "'cl! performance,

.\nillt'iallirt designo

V. REFERENCES

:\

\\'cll stilllulation c\·:llllatioll.

9.

Delcrmining Ihe clrcet ofcomprcssil)il on gas \\·cl1 pcr(onn:1I)cc.

10..-\naIY7.ing cllccts

l)r

pcrli.mlling dCJ1::;ity.

11. Predicting lhc c1Tccl
paclli .'
l~.

01'

dt'plctioll

l)ll

producing ca-

AJloc<1ting injeclion gas among ga:- hit wells.

1. Gilbcrt, W. E.: "Flowing and G,b-Lirt \Vell
Performance," API Drill. Prod. i'raclicc. \95-J..
:\ind. T. E. W.: Pril/ciples (?( Gil IIdl ProdUcliol1.
'IcGraw-f1ilJ. 196~ .
.. Bro\\'n, K. E. tl!ld Beggs. H. D.: rhe Ti:cllllology (~(
.irr(/icia{ L~¡; '\/er¡'od\. Vol. 1, Pelln \\'ell Publ. Co..
Tuls<1. Okl<1homa. 197fl.



Reservoir Performance

2

1. INTRODUCTION
Oile of thc m(l~l illlJ10rlrllll componenl:, i'n llll' 1\11;11 \\"cll

s\"~;('m is Ihe rc~\?r\"oir. Ulllc:,s tlCCllr;J{C pn'dil'tl\Il\:' l':1I1

\0 \\ lut will I1tH\ ¡nltl ¡he lltlrt'llt11t' 1"("111 ¡!lL'
Ihe pcrl'llrll1¡lllCC (Ir {he Systclll (';1(lfln! l·,' ,111;1h ;:;:J. As disCllS:,cd in lhe prC\"i0US s,xlinll. IJlI" ,>1 111\..'
1~\~'J pn:ssu["c:-,. ¡JI :lllY lil1lc in ¡he 1ire l,ftlll' 11",\'1\,'11. IS
111;: J\'cragL' rcscr\ nir prcssurc Ji". T11I..' 1111\\" l111t' 1111..'
\\(':', dcpcnds 011 lhe dra\\"(1L1\\"Il or prl..'Ssun.: 111111' 111 lh\.'
rc~:rq)ir. />R -/\". The rclati\lIlship b\'I\\",11 1!l1\\"
rí1¡~ ~\Ild prcssurc drop occurring in Ihe ptlrtlll" 111,\1111111
cm be vcry cOlllpkx and dcpcnds 011 parallll'lL'IS :--11,-11 as
rl~~"~ propcrtics. !luid propcrtics. !lo\\' !"l',t:.illll'. I1l1hl :,;~Ill­
r~I1:\ms in IhE: rack. comprcssibility 01' Ihe 1l11WIII!: llllllb.
ft1¡-::wtion damll1.:.::nanisl11. It al:'(l dcpcnds on lhe rcs('r\'t1lr prl':,~lll\' ¡Ise1;' and, dcpcnding 011 lhe dri"c Illccll;1lli~l1" ,hl~ 1\1:1)'
dcccasc \\'¡Ol limc nI' cUll1Ulalivc produClitlll.
Thc rcsen'oir c\llllponcnl will ah\"ay:-: \w ;111 111\'''\l~';1I11
cllmpOncnl. That is. it will hnrdly cver he 11I;1~·lh·,ll. lo
~l,:'Jc('t PR as the Ilodc prcssurc, although Ihe ~:llhll:ll'e
t;l' :,l:1dc

rC':';:;-\L,ir.

prc":,surc Pub is sOllletimcs selccted. This \\'ill ISI1!:lh' [he
ctl'c,,:ts of ihc prcsslIrc drop gr3\"el pack.
The no\\' frolll thc reselToir inlO the \\"l'1I 11:\:-: bl'1..'1l
C;1:)('l! "inflo\\' performance" by Gilb('rl ' alld ;\ phI! t.lr
pr\'..lucing ¡'clL'd ílll "inflo\\" performallce rc!aliollsl1ip" (}I 11'1' 11~'s
:,lh'·.ild nol b(' cOllfuscd wilh Ihe ínllo\\" [I};l 11l11k :1:' dlSl'\l>~('d in Charter l. The in!low to l!le. node l':lll llh'llId~
fhe !low Ihrollgh oll1el" eomponellls. dC¡ll'ndill!~ \'11 lhe

ll'~'.ltiOIl of the IlDlk selccted.
!;] this chílplcr Ihe \\'el! pcrform:111cc eqll:llillll...; \\ dI be

prc:,('nted for various r~ser\"oir types and drive I1lcehanisll1:'. Thcse equtl\iOll5 \\"ill pcnnit Ihe calculation of J.Pt
fJll:f.~ or. if Ihero:;- is negligiblc prcssure 105s across
the ~·~'ltlrlctioll. ÓI'I = P.I? -1'". where Pu'- is the ll(l\\'ing
\\·c¡;·:'~'rc_pressurc. The I.:'ffl.:'c\:' ~1( ch:lllging conditit111s (In
the ::..:'cllfncY 01' Ihe cqualil1 n:, ",jI[ be disclIsscd :llld
clllr::-i~'alll1cthOlls lo (\lrrc'(l fN failurc nI' {he lhenry \\-il1
be ;':-~'sl'[lIcd. \lclhods rOl' prt'Jieling IPRs rnr l1lllh Ihe
IWC5;,';lt 01' reCl1s>;,'~i. l30th oil tlnd g:lS rcsl'f\l)ir pl'rformancc \\"ill he
¡m::'(':l1cd.
Fi¡ully, mcthods for obtaining the nccessary roc.k and
fluid rropcrties rOl' use in the equatiolls \\'ill be ollllincí.l,
ane. ¡he accuracy of the data \\-ill be disCllsscd.

: : ro: -

11. WELL PERFORMANCE EQUATIONS
T\' ..:::tlctdate the prcssure tiror occurring in a ITsc[\·oir.
all ;,'..:juation that cxpresscs the cncrgy al' prcssurc losscs
du\:' w viscous shcar or frictit'!n forces as a funclion al'
ve!L'..::ily al' f10\V rate is requircd. :\1though lhe form ofthc
cqu~liol\ can be quite difierenl for various typcs offluids,
the [,Js.ic cqllation on which all of lhc variolls form5 are
ba::~d is Darcy's law.
A. Darcy's Law

Ir. 1856, whilc pcrforming ('.\pcrill1~nts for {he dL'sig.n

of s.md filler beds for \\'alcr purifi(',ltion, Ileury Dan:y
pmrl':,ed an cquatioll rclating arparcll! lluid vclocily lo
prc:,:,urc drop aeross Ihe filter bed . .\ithough the c.\pcril1l~I1IS \\'crc performcd with tlll\\' only in the dO\\"ll\\"ard
veni.:al dircction, the c.\prcssi011 is abo val id ror horizoll-


PJVdUClioll Oplimizalioll UsiJlg Nudul A¡;_I~rsi.\

JO
tal flow, which is of mast ¡nterest in lhe pctroleum industry.
It should al so be noted that Darcy's experiments involved only one Huid, water. and that the sand [¡{ter was
completcly saturated with the water. Thercfore, no effccts of fluid properties oc saturation were involved.

Darcy's sand tilters wece of constan! cross-sectional
arca, so lhe cquation did nol account roc changes in vclo~¡ly with localion. Writtcn in diffcrentiai form, Darcy's
law is:
kdp

v=--

(2-1)

!l ti,

=_ kA dI'

(2-2)

!l el,

wherc

k
v

q
A
P
dpld,

=

Units

Variable

Symbol

Darey

Field

cc/sec

bbl'day

Flow rate

q

PermeabHity

k

darcys

md

Area

A

cm'

ft'
psi

Pressure

p

almo

Viscosity

~

cp

cp

Lenglh

L

cm

ft

Thc gcol1lctry of ¡he linear systclll is illllstr[\ted in Fig.
l.
It can be obsaycd from Eguatían 2~J lhat a plot on
c3rt~sian coordillJleS of p vs. L \ViII pr.oduce a srr¡:üglli
line of constanl slop~, -qW'kA. TIJat is, the variarion al
pressure with distance is linear.
If the nowing tluid is compressiblc, lhe in-siru 00\\
rate is a fUl1ctiol1 01' pressure. Using the t:1et lhal lhe mas~
now rate pq mus! be constant and ~xpressing the
dClIsity in tcnns of prcssure, temperatun: and ga:i specinc gravity, it can be showll thut Equation 2-3 be-comcs'
1~

or in tcrms oh'olumctric now rate q

q = vA

TABLE 2-1
Units for Darcy's Law

pc.:-rrncability oflhe paraus medium,
appararcnt fluid vclocity,

volurnctric flow rale,
afta open lo now,
fluid viscosity, and
pressure gradient in lhe direclion of flow

,

,

JI, - Pi

8.93ZT¡tL
kA
'/."

(2·:' .

(llL'gati\'c).

J. Linear Flol\"
For linear now. that is [or constant area flO\y, the cquation may be intcgratcd to gívc the pressure drop occurring
Qver some Icngth L:

l'
T
ft
L
k
A

P' kdp
= -qJ1
- fL el,

J

J.l

PI

kA o

PI

qp JL elr
-dp=-kA o

ep,

ti,
md,

fe,
scf / day

If it is assum~d that k, ~ and q are independent of pressure, or that they can be cvaluated at lhe average pressure
in the system, the equation bccomes

JP'

pSIJ.

°R.

(2·3)

FOl" high-velocity f10w in which turbulcnce or 0011Darcy Oow can exist, Darcy's la\\' must be moditied lo
accaunl for the extra pressure drop cJused by the tUfbulcnc~. Applying [he turbulencc corrccti,on to Equatians 2-3 and 2-5 givcs:
Oil Flow

Integration givcs:

-q¡.¡

p,-Pt =--L
kA

(2-4)
(2-6)

or

q

e

CkA(p,- 1',)

J1L

where is a unit conversioo factor. The correet value for
is 1.0 for Darcy Units and 1.127 X 10-3 for Field Units
(See Table 2"1).

e

where

1',
1',
!lo

Bo
L

upstream pressure, psia,

do\,·mstream pressurc, psia,
oil viscosity, cp,
üil fonnation volume f"ctor, bbl/STB,
Length of flow path, n,


11

J. Radial FIOlI'
Darcy's law

q~

----L---~

F"f;. 2-1. Geometry for linear tlo','/.

lío
A

pcnncability to oil. md,

arC
Po
~

(1"

l;.'

=-:

c~n be us.cd to calculatc the no\\" into a
\,"eH whcrc lhe fluid is convcrging radially into a rclati,"cly small hale. In this cnsc, Ihe arca open lo now is nat
constnnt amI mllst ¡herefare be included in lhe integrntion
01' Equation 2-2. Rdcrring lo Ihe no\\" geomclry iIIustratcd in Figure 2-2. the cr05s-scclion~1 arc~ apcn lo lhe now
~t any radius is A = 2Itr".
.-\150, dcfining the changc in prcssure wilh localion to
be negalivc \Vith rcspcct lo lhe dircction of no\\". dl'ldr
becomcs -dpldr. ~Iaking thcsc substitutions in Equatioll
2-2 gives:

n=.
q:::o

oi! dcnsity. IbnlitV.
\"Clocily coclliciClll. n-l. and
oil llo\\' raleo STB day.

2¡rrhkd[J

'

~

FIfJ\\"

.:, Oil Flan: \\·hcl1 applying the Darcy cqualion lO !lo","
,,1jl in a IT~Cf\"ojL il is a~sumcd that lhe oil is only
:-li-;¡Hiy comprcssiblc. lile small, changc in q wilh prcs~ur~ i::; halllllcd with lhe oil fOrlllJ::ion "olume factor B", so
{h~t Ihe Ilo\\" rate can be cxprcs5cd in surnlce llr stock
ud: '"olullle:-i. For oil Oo\\', EqUJlion 2-9 b,,:coll1c::;:
\)f

:

'2

I't -P:::::

+

S.l)3Z~1..LT
Cjg

kg.1

(2-7)

1.247 xl ,)-J('PZTLy " ,
~

r

q:

el B = 2rrrhk,,_i dp
~t,,; .Ir
,1

g'IZ dL'\i'llillll l;h.::~'r 1.,'\'1111:llct! ;11

"

g;l:, gr:n'ity (~lir;=; : J.
gil:' 111..1\\ rate ¡It :,:. - psiga:, \'iscosity a( T. ¡J. ep.
J'('nllcabi~ity 10 ~.1_';. md, lhl\\' arca, n2

k

2rrh J "" --"- el"
. p"B"

\n cstimatc l~lr Ihe \'c!oci¡~ ClldliciCll! [3 Clll be ob-

fr'.
md. and a and b

Formation Type

Consolidtlled
Uncorlsolidaled

approximalcd ¡'rom:

3fC

a

b
10 10

1.2

1.J7 X 107 ,

0.55

2.329

X

\llhough lincar !lo\\' rafe!y L1ccurs in tl n,::sL'r\-t1ir, Ihese
.:Jtinlls ",ill be llscd blter 1\1 .:,\!culalc lhe pre~:\urc drop
",'-,)~~ ;1 gr¡¡n~l pack COlllplcli,,111. Ihal i:-;. ófJ = / 1 ,,; - pur.

. d,-

=.r"1~'r

1';-11 )

\\"h.. . ll inlegraling: Ihi~ cqutllioll. it i:-; u~llally assulllcd that
lh,:- prcssurc rUIlClioll . .tfl'J oc:: k P.,B,. is indcp('ndcnt ('Ir
rr~~~lIrc or lhílt il ":-tln be c\';lll1J~cd al avcrngc prc""urc in

";:d rrom:

k

I
I

T. .!'.

J1l)\\ing [l'mpl.'ra:::r;.'. ~ 1?

P

12-9)

dI'

~'':

F;), 2-2" Rad;aJ (fow sys{em.


ProdllClioll 0plimizotioJl UsiJlg Noda/ .-lll,:~rsis

12
Ih~ w~lI's

drainuge volumc. This is ne..:essary beca use no

which gives upon inlcgration:

simple ílllalytical equation for this tenn as a functiol1 oC
pressllr~ can be forll1ulm~d. Utilizing this assumption and
inlcgrating Equation 2-11 Qyer the drainagc radius of the
wcll gin~$:
2¡r.k.,h(p, - 1'''1)

llaBlJ

11'1 (1~

(2-12)

/1"11')

2

qsc =

kg

where

1'.'

p,,,.
r

"

h

i,,!lo\\' ml<, STB / day
cn~ctivc oil pcnneabiliIY. md,
rcservoir lhickncss, n,

prcssure al r = 1"1" psia,
wdlborc tlowing pressur~ al r
wdl's drainage radius, fL

pR
Puf
~l.!:

=

Z
T

r,.., psia

\Vcllbore rJdíuS, n,
oil \'iscosiry. cp, and

oil fonll:Hiúl1 volulllc faclI..)f. bbl/STB.
Equatian 2-13

applie~

foe stcady-sr:.:.¡t

703XIO-6k"h(p~ - 1',:,)
_
'

(2-17)

gas tlow rate, Mscl'd,
penncabililY lo gas, md,
n::scn'oir thickness, n,
. averagc res,,:rvoir prcssurc. !)sia
wcllbor~ t10Willg prcssur~. psia,
gas viscosiry al T, Ji =.5 (PR + PI')' cp
gas compressibility factor at T, p ,

rescrvoir lcmperalurc, o R,
drainuge radius, n, and
wcllbore mdius, ft.

c. Resermir Pressure Profile. The behavior ol' th~ presin lhe rcscn'oir as a fUllction ol' radius can be ,:m3Iyzed by plouing pressurc verSus radius as predicleJ b~
Equatiol1 }-14. Assuming a fixed average rescn·oir pressurc Px al r:= 0.472 r~ and solving for pressure. Equarion
2-14 gives:
SlIfC

(PI' =

constant),

laminar tlo\\' 01' a \\'dl in lhe ccnter of:.I circular drainngc
arca. It is mOfe useful if exprcsscd in I~nns of averag~
rescr\'oir prcssur~ PRo and for pseudo-s:r~3dy s.tate or stabilized !lo\\' (PR - Puf:= constant) as:
O,OOIOSk)~PR - 1'",)

t2·16)

.1'('

where
qsc

..

k /1'

JI g 1

IlgLTln (.472 1;/1;,,)
(2-13)

/¡

=

Modifying Equalion 2-16 for stabilizcd no\\", a\"t~rag~
reservoir presslLfc, defining Psc = 14.7 psia and T,•. =.
520 °R givcs un cquation for gas inf10w rate in ficld units.

For fidd units, Equation 2-12 bccomes:

i/"
k

qS(}lgZTpsc(lnJ~/r,,)

2

Po: - P"f

(2-14)

1l0B" ~1 (.4721~ / r,)

l~-IS\

whcrc

average pr~$sure in Ihe drJinage volume of
th~ well.
The other tcnns are the same as rhose defined for
Equation 2-13.

'PR

b, Gas floa: To inh:grate Equation 2-9 for flow of
gases, the faet that pq is constant is used along with the
gas cquation of state
.
pM

p~ ZRT

(2-15)

A plOl of prcssure versus radius for typical wcll con-

ditiOllS, Figurc 2-3, shows thc large increasc in pr~ssurc
gradient t10wl the pressllre drop around the wellborc is even more
severe.
Examination of Equation 2-18 revea(s lhat a plOI of p
versus In r will result in a straight hne of constant slope

m,
where

or

111

141.2qu~UB(l

(2-19)

k)l

This type oC plol is illuSlratcd in Figure 2-4_ !I should
be emphasizcd that the slope remains constant only if all
oC the [erms on Ihe righl-hand side oC Equation 2-19 remain constan!. A different slope and, therefore, a differ-


Resen'uir Pc/formollce

/3

versus qf) on Cartcsian caardinatcs rcsu'¡is in a straighl

line having" slopc of -IIJ and an ¡ntercepl of
al qo = O.

'"I!

p

r_\'I---c_~

(2·33)

-::cr - __

~
I

.~_7_2_r_e-,

Fig. 2-3. Reservo;r pressure profile.
en!

PR

"aloe of P\l/' \Vould be obtaincd for each flo\V rate

Ir cond;tiolls are such tha[ J is constant ",illt drawdO\vl1, once a valuc of J is obtaincd from one production
lest or calclIlated using Eqllation 2-20, it may be used to
predict inflow performance ror other cOllditions.
Examp/e 2-1:
A wel1 that is producing from a reservoir having an
average pressure of 2085 psig produced al arate of
282 STB/day when bottomhole flowing pressure was
1765 psig.

q".

A. similar anaiysis of Equation 2-17 for gas flo\V rc\'('als thnt a 1'101 of P~ \'crsm'" r rcsu\ts in a strnight lille

Calculate:

1. The productivily index J.
2. The producing rate ir Pwf is decreased lo 1485
psig.
3. The bollomhole pressure necessary lo oblain an
inflow 01 400 STB/day.
4. The inflow rate if p\~J is reduced to zero, Le.,
Absolute Open FlolV polenlial (AOF) or

of :-.Iope:

3. Producrid(1' hule.\' Conccll/
The rclnlionship bctWCCll m:ll inflo\\' ralc and rrcssure

dLl\\-clo\\'11 hns o((cI1 been t'xprcsscd in Ihe form of a
r,.u)ucr;l·il.l· ¡lIdex./,

qo(max)'

So/ution:
J =

00070:'k"h
~L"B"

, ~-2())

1. J

111 (.--F~J:.I r".)

PR - P.,_

282
2085 -1765

=0.88 STB/day-psl.

2. qo =JÍPR - p",) = 0.88(2085 -1485) =528
STB/day

Thc inflc'l\\ equn(;on for oil !lo\\' cnll l!len be \\Tilten as

qo =JCPR-P,,/)

0,
=-_-=

(2-21)

=PR -o,1J =2085-400/0.88 =1630psig
4. qo(mo,¡ =JIPR • O) = 0.8812085) =1835 STB/day

3. P",

.1 = -=--,-
(~-12)

PR - Pul

$0lving rOl' p,,(in

tel'IllS

('Irq" rcv(,
Tlle predictiolls llliJdc in "Examplc 2-\ are val id only if
J rcmains constan!. This implies tll
,
¡
r

tion./{p) = k,/p"B" remains constant, which ;s scldol1l the
case, as will be discussed further in following sections.
Tlle productivity index can also be exprcssed as:

01

/

J

Pwfl

0.0070817

(Pn -1"'l)ln (.472r,. r,)
I

sr,

I"if

k"

--dp
¡I j3"

(2·2-l)

',>"1

lr~/

I

L--------~cc!

'In ce: ~~'

In r....

The producli\'ity index concept could gas \\'ell inno,," r~rrOrJllallc(' by dcfining n gas productiv-ity indcx as
(2-25)

Fig 2-4. Semi-Iog plot of pressure

VS.

radius.

or


Prodm.:t;oJl Optinú::otiolJ Using Nodo!

14

Jg

and

703xlO-6 kgh
(2-26)

J.lgZTln

703xlO- kg/~ ¡i~ - P ~/)
6

(.472I;II~.)

A plOI of p~\fversus liJe would not be linear 011 Cartesian
coordinalcs. A more commoll procedure roc gas well
anaiysis \ViII be discussed in following scctions.

(2-28)

q" = ~JT [In (.472rJ r,,) +S']

The ski n faclor S' inciudcs the crrccts uf both turbukncc ami actual fanuncion damagc as:

S' ~ S + 0'1
4. Permeability Altera/ion Clnd Turbulem:e
Darcy's law was bascd 011 the assumptions that permeability 10 (he nowing fluid was con~tant in lhe enlice
drainage area ol' the well and that only laminar flow existcd. The effcctivc pcrmeability lo oil is lhe product of
lhe fclativc permcability to oil and lhe absolute penne~
ability al' lhe reser\'oir, thal ¡s,
k.. . =k k ro

Thc absohuc pcrmeabilily k, can be either increased
around [he wcllbofe by wcll stimulation oc decreased by
fonnation damage. such as clay swelling oc pore plugging. This would changc Ihe slope of lhe pressure profile
out to the radius lO which the pcrmeability was altered.
This is illuSlrated in Figure 2-5.
Figure 2-5 illustrales that foe a CDIlSt.1nt flo\\' rate, less
prcssure drawdown ",ould be cequired if the weJl had
bccn slimulatcd and mor~ drawdowll would be requircd
for a damaged wel!. The bottomhole tlowing pressure
required for 110 changc in permcability is labelcd P~lf'
II is oftcn impossibk 10 delennine either the aItered
mdius r. , 01" lhe alten.::d permeability k... In this. case it is
assumed Ihat the prcssure change due 10 the altered permeability oceurs al the wellbore in Ihe fonn of a skin
cffee\. The skin effecl is defincd as a d!mensionless quantity and can be includcd in Equ3tions 2-14 and 2- t 7
as:

(2-29)

whcre
ski n factor due lo pcrmcability change,

S


turbulence cocfficicnt

D

The tcrm S wil! be positivc ror damage, ncgati"c for
improvemcnt, Dr zero for no changc in pcrmeability. The
turbulcnce cocfficient. D, will be citller púsitivc 01' zera.
Thc effccts oC S' on lhe pressure profik for an oil rescrvoir are illllstratcd in Figure 2-6. Althollgh a sudden
large prcssurc drop cDuld occur al lhe wcllbore as indicated for a posilive 5', if, ror cxamplc, a small l1umber
of pcrforations are open, it would be physically impossible for a pressure ¡necease lo occur as illustrated for a
negative 5/, The actual siluarían is illu~tratcd in Figure
2-5.
Equl.ltiDns 2-27 and 2-28 are cDl11l11only lIscd to describe psC'udostcady s{ate Oow in a circular drainagc area.
Ir rhe drainage radíus is not circular, [hen Ihe use 01'
Equations 2 27 and 2-28 rnay !cad to appreciablc errors.
Odch 2 dc\'elopcd (h~ following equations to describe
pseudostcady ::italc: t10w in a noneircular Jrainilgc ar~a.
w

0.00708 kjl (P R - P ,,¡ )

J.l,B, (In (472 x) +S')

0.00708k)I(PR - p,,¡)

q,

J.luSu [In (.4721; 11;,.) + S']

(2-27)

J = -::-q-"",-p, - p,¡

J.luSo (In (.472x) + S')

.... ~'.!~i ~!..]

1
P

"¡II
.

Pwl

/

P;'"

__-i'--k.

>ti

\ '..

P

<,
-~s'>o

Pwl(+}

PwI

k.

In Cw

In (.472 r.)

In ra

Inr-Fig. 2-5. Effects of allered permeability.

In (.472 r.)

In rw

In r - -

Fig. 2-6. Effacts of skin factors.

•
Reservoir Pef/Onnance

15

where x is givcn in Figurc 2-7 for diffcrcnt dr<'linagc arca
shapcs and well locations.
The l11<'lnd turbulcncc. dcfined as:

be decreascd around the wellborc. This siluation can
occur cven though PR may be wcll aboye 1"'.
As pressure deplction in the reservoir occurs. PR will
likely drop bclow Ph and free gas \ViII exisr !hraughout
the reservoir.

(2-30)

can be ca1cul141.2qfl~lvB{lS'

(2-31 )

k(lh
:\ value for S' can be oblained from allalysis ofvarious
types of prcssurc transíent tests.

B. Factors Affecting Productivity Index
The cxpress ion for the prodllctivity inc;?x for Lln oil

wel!. including ski n cffcc!, can be writtcll 35:
J~

O.00708kh

(PR - Puf )[ln(.-I72r/ r,) +Sl

Si', --(1'
k. I
1'",

~ ,B"

(2-32)

From this exprcssion, il can be observcd ¡h~t.l willllot
be constant uJlless lhe flressur~ fUllction is ír:dcpcndcnt of
rrc5surc. In arri\'ing ti! Eqllation 2-14. it 'Xas assul11ed
Ih:Jt kili \Vas ('(lllstant ami that ~I" tlnd B/I c(lu1d be evalu.-\}50. (he dcpcndcncc of S' on l/" wns negj;?cted. In lhis
:'t'(lioll, sorne al' lhe f~1Clors that can cause J to chang.: are
di:='cl1sscd. The bcha\·ior of 11ll' B" amI k,lJ . 'xith prcssurc
and nuid salur
J. PITase Be}¡ul'ior in Reserl'Oirs
.-\ thorough discussion of lhe pllase beha\ ior of oil resc'fyoirs \ViII no! be giycn herc. bu! may be f..1tllHI in books
\.111 tluid prop~nics'~'uch as ~IcCain3 and Ai":lYX. Bass and
\\·hiting.-t _.
Thc conccpl of bubblcpClint prcssurc and dewpoitH
pressure wil! be rcvie\Vcd bcctlusc of lhe importance of

gas saturation on Ihe rchltiyt: pcrmc(lbility to oil. A typical pressurc·lcmperalure phase diagram f('lf nn oil res(,Iyoir is sho",n in Figure 2-8. Thc liquid. gas and tworhase regions arc 5hoWI1, <'lnd lhe bubblepoinl pressurc is
indicated as the prcssurc al \\'hich free ga~ first forms in
Ihe reservoir as rrcssurc is redllced at COll5tanl rescrvoir
lemperaturc.
Thc rcscrvoir nuid dcpictcd in figure 2-8 is aboye
lhe buhblcpaint pressurc p". at initial l"Csc,,·oir pressurc
1'_:;; élnd. thcrefore, no free gas \yould.exisl anywhcre in
lh;: rcscrvoir. Ho\\'c"er, if lhe prcssurc nI an)' paint in lhe
rcs.en·oir drops below I'h , free gas \ViII form ami km ",ril'
~(' relluced. Thcrcforc, if n \\'cll is produad al rC\.luircs Ihat 1',,/ be Icss than I'h' km and, tL'rcforc,.J will

l. ReJative Permeability Bellm"ior
As free gas forms in lhe pores of a Tesen·oir rack, the
abilily uf lhe ¡iquid phase lo no\\' is decreased. Evon
though Ihe gas saturalío n may not be great enough to
allow gas to Oow, (he space occupied by the gas rcduees
the effective flow area for the liquids. The behavior ofthe
rclative pcrrncability to oil as a function ofliquid saturalion is ShO\Vl1 in Figure 2-9. The relativc pemleability is
delined as the ratio of effectiye pcnncability to a particu·
lar fluid to the absolute pcrmeability of the rock, km =
k" k. The absolutc permeability. k, is the permeability to a
tluid when the nuid completely satura tes the roe k and is
independent ofthe fluid as long as the fluid is :\ewtonian.
The rclative pcrmeabiliry to g3.$ wil1 be dCCrC3$ed if liquid saturation develops in a gtls reservo ir. eilher as a
result of retrograde condensalÍon or water ii.-¡mlatioll in
¡he pores.

., Oi¡ l'i.'icosi~\· Bellador
The viscosity of oil 531uriltl~J w¡lh gas at c('n~tallt tcmJ:'c-r.Hurc will d('clTase ;1$ Pl\.·~~urc is decl'ca~;?¿ from illi~

¡i::JI prcssurc lo bllbbkpoinl r~~ssur~. 8cI0\\-?, the vis("osilY \ViII increase as gas cor~~('s out uf SOlu!l011. lcaving
lh? hcavicr molccllles in th(' :i(Juid pllasc. Fi~urc 2-10
iJlu5tmles qllalilalively the bchavior of 11" \-('[sus prcs5Ufe al constant tempcraturc. Equations for cJlculating
bcha\·ior of vi$cosity with pressllrc ch:mgc \ViII be presented in Chapter 3.
-l. Oi/ Forma/ion Vo/ume FacTo" Behm'ior

:\5 pressurc is decrcascd (ln a liquid. the liquid will
expJnd. Whcn the bubblcpl1int pressure \.lf .111 oi! is
reachcd, gas coming out ol' sC'lution \ViII cam~ the oil lo
shrink. The behavior of 8 0 \'ersus ]J al C011:'13nl tcmpcrature js shoWIl grapllically in Figure 2-11.
The oil fonnalion voJume factor is dcfincd 35:
B
11

Vollll11e of oil plus ils dissoh'cd gas

al

p. T

Volull1e of oil
e, Factors Affecting

Infiow Performance

The lnno\V Pcrfonnnncc Rdationship (IPR) for a \\'ell
is the rclalionship bet,,"ccn now rate into Ihe wcllbore
and H'ellbore llowing prcssmc Pllf' The IPR is illuslrated

graphically by plolting]J",.vcr511s q. (flhe IPR can be rep.resentcd by a constant productivily indcx J. lh~ pl01 \vill
be linear and th~ slope of the line will be -1 J. wilh inlcr-


ProdllclivlI Optimiza/ion Using :\fui/u! Al1í1~D'¡S

16

SYSTEM

X

O
D
O

--

X

SYSTEM

I

re
rw

0.966A'"

EJ~}

-

rw

2

0.571 A'"
rw

t--j~j 1

1.44 A'·'
rw

2

0.565 A'"
fw

t-~~+

2.206 A'I'

rw

2

--

.-

0.604A'"
rw

8
-ti!
--

f --

-!.----JI

1.925 A'h.

rw

4

0.61 A

ll1

L-f-~--II

fw

6.59A"
fw

4

9.36A"

0.67BA'"
fw

I[~
"3" ¡---'

E-~--~--JI

fw

4

.. 0.66B A'"
fw

[~}
2

[

.-~

--

•

1.36B A'"
fw

1

1.724 A'"

D

~--

-

fw

[.:f:::+

1

4

2

2.066 A'"

I

•
5

rn
m

1.794 A'h
fw

1

4.072A'"

H+8
2

fw

1

0.884 A'I'

1

~¡;,!,
- r - r - -I ~II

fw

I

, . : 1 I

fw

9.523 A'"
fw

2

1.485 A'"
rw
.

Fig. 2-7. Factors for differenl shapes and wel1 posiUons in a drainage 8(ea. 2

!2.

10.·135 A'I'
rw