00077549

Copyright 2002, Society of Petroleum Engineers Inc. This paper was prepared for presentation at the SPE Annual Technical Conference and Exhibition held in San Antonio, Texas, 29 September–2 October 2002. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract

This paper shows how to forecast liquid condensate and water production from gas and gas-condensate wells. The methodology is based on the Whitson and Fevang’s5 method of handling fluid properties using two-phase pseudopressure function that not only takes into account the pressure dependence of the fluid properties but also retrograde behavior (phase change) of light hydrocarbons in the reservoir. Tedious mathematical treatment indicates that the oil phase production can also be predicted using the surface gas production in case of two-phase, and gas and water production data in case of three-phase producing wells.

To establish Inflow Performance, one needs relative permeability data along with PVT data. In this study, we have used pressure transient methods to obtain the effective permeability as a function of pressure. Once the correlation between the effective permeability and pressure is obtained, it is used in the pesudopressure integral. Sapphire well test simulator was used to simulate pressure test data and PVT data for condensate wells. The effective permeability correlation was developed from the well test analysis. This was done by simulating 2-phase and 3-phase gas-condensate reservoirs with reservoir pressure equal to P*.

The main source of the liquid condensate is considered the 1st region, closest to the wellbore, which is spread areally between the wellbore and the pressure P*, in the reservoir. Flow in this region is two phase in nature. Some of the liquid phase may go into the gas phase due to retrograde conditions if exist. Region-2 consists of liquid and the gas phase but it is assumed that the liquid condensate is immobile since the liquid saturation has not builtup enough to initiate flow. The

pressure range in this region is from Pd to P*. The water production, however, is the contribution of the entire reservoir since no phase change is expected in the aqueous phase. More importantly, water phase is considered as the most reliable phase in gas condensate systems from engineering point of view, since water properties hardly change with pressure in the reservoir.

SPE 77549

Forecasting Liquid Condensate and Water Production In Two-Phase And Three-Phase Gas Condensate Systems S.A. Jokhio, D. Tiab /University of Oklahoma, and F. Escobar/Universidad Surcolombiana

Two examples are solved with simulated data to show the use of the technique developed. Introduction

Downhole liquid production in gas wells is a global problem. Indian Basin, New Mexico, wells have an average water production of 1,500 B/D. If not unloaded these well will kill themselves with brine they produce. Downhole produced liquid managemnet can be an expensive business too. There are many issues that have to be adressed before putting an artificial lift system in such wells. The first and foremost is to forecast the liquid production..

Retrograde gas-condensate systems like volumetric gas wells, always have liquid condensate and water production at certain stage of depletion or may be at the start of the production on very first day. Such liquid production may reduce the gas deliverability, the main production of such wells. When combined with water production it may even kill the wells. Therefore, it is in the interest of the operator to forecast the liquid production so that the well can be unloaded continuously or intermittently. Several three-phase and two phase IPRs are vailable in the literature but they do not completely represent the gas condensate systems. Water phase, however, may be predictable with such co-relations. The main basis of this paper is the definition of two-phase pseudopressure and the distribution of the reservoir in three distinct regions as given by Whitson and Fevang5, as visualized in Fig. 2, 3, and 4. For convenience, those three regions are described here: 1) Region-1: around the wellbore where both oil and the gas phases are flowing if the system is 2-phase. All the three phases are mobile in case of 3-phase system. Its external boundary is the distance around the wellbore at which reservoir pressure is equal to P*, the pressure at which liquid begins to move. 2) Region-2: the region between the pressure P*, and the dew point pressure,

2 S. JOKHIO, D. TIAB, AND F. ESCOBAR SPE 77549

Pd, at which gas phase begins to liquefy. In 2-phase systems only gas phase is mobile. However, gas and water both phases are mobile if the system is 3-phase. 3) Region-3: this region is the farthest region where only gas-phase is flowing in 2-phase system and both water and gas are mobile if system is 3-phase. The total pseudo pressure function is obtained by integrating the fluid properties at the pressure range or reservoir pressure and wellbore-flowing-pressure. The effective permeability as a function pressure was obtained from the well test data in such systems for this study. The complete discussion of the permeability as a function of pressure is left for other publication. Literature Review

The depletion of gas-condensate reservoirs has been the topic of continuous research. Quantitative two-phase flow in reservoirs was first studied by Muskat and Evinger14. They were the first researchers who indicated that the curvature in IPR curve of solution gas drive reservoirs is due to the decreasing relative permeability of oil phase with depletion. Simple correlation for productivity estimations (J = ∆P/q) was being used until 1968 for solution gas reservoirs too. Vogel1, 1968, first published IPR for solution-gas reservoirs, which handles the two-phase flow of oil and gas. His work is mainly based on Weller’s2 approximations which did not require assumption of constant GOR. Instead, he assumed that the de-saturation of the oil phase at a given moment in depletion is constant everywhere in the reservoir. Vogel1 using Weller’s concepts was able to generate family of IPR curves in terms of only two parameters, flow rate and BHP, which revolutionized the art of well performance forecasting.

Recently Raghavan and Jones4 discuss the issues in predicting production performance of condensate systems in vertical wells. Fevang and Whitson5 model the Gas-Condensate well deliverability and by keeping the track of saturation with pressure and relative permeability. Two-Phase Systems Modeling Liquid Condensate Production

The main source of the liquid condensate production is the Region-1. In two-phase systems (oil and gas), total liquid hydrocarbon production is the some of the liquid condensate production and the vapor content in the gas phase. Mathematically,

ofree,gooT Rqqq += (1)

( )oToT Pm.Cq ∆= (2)

( ) ∫

µ+

µ=∆

r

wf

P

Po

gdgd

rg

oo

ro dpR.Bk.k

.Bk.kPm (3)

For vertical wells

+−

=

aw

e S75.0rrLn

h.00708.0C (4)

For horizontal wells

+−+

=

aHw

2/1S75.0LnC

rALn

b.00708.0C (5)

Water Production ww mP.Cq ∆= (6)

( ) ∫

µ

=∆r

wf

P

P ww

rww dp

.Bk.kPm (7)

Producing Gas Oil Ratio (Rp) As the pressure drops below the dew point, producing

gas oil ratio GOR, increases monotonically12, i.e., a one-to-one relationship exists between the producing gas oil ratio and the pressure as shown in Fig.6. It dives as the P* approaches and liquid becomes mobile. However, it stabilizes as effective liquid permeability stabilizes. By Definition

ofree,gfree,o

sfree,Ofree,g

OT

gTP Rqq

Rqqqq

R++

== (8)

µ+

µ

µ

+

µ==

ogg

rg

oo

ro

Soo

ro

gg

rg

oT

gTP

RBk

BkC

RBk

Bk

C

qq

R (9)

On simplification

( )Pogg

oo

ro

rgsP RR1

BB

kk

RR −

µµ

+= (10)

( ) ( )P

BB

kk

R1

BB

kk

R

PR

gg

oo

ro

rgo

gg

oo

ro

rgs

P

µµ

+

µµ

+

= (11)

Solving for krg/kro results, ( )

( )

µµ

−−=

oo

gg

Po

sP

ro

rg

BB

RR1RR

kk

(12)

Solving for gas and oil effective permeability, results ( )( )

{ }

µ

µ−

−==oo

rogg

Po

sPrgg B

kkBRR1RRkkk (13)

( )( )

{ }

µµ

−−==

gg

rgoo

sP

Poroo B

kkBRRRR1kkk (14)

Producing Oil-Water Ratio (Rpow) To get insight in the production phenomenon, let us derive the water oil ratio and see how it behaves and what kind of information can be derived from it.

FORECASTING LIQUID CONDENSATE AND WATER PRODUCTION IN SPE 77549 TWO-PHASE AND THREE-PHASE GAS CONDENSATE SYSTEMS 3

w

free,Oofree,g

w

oPow q

qRqqqR

+== (15)

µ

µ

+

µ==

ww

rw

oo

roo

gg

rg

w

oPow

Bk.kC

Bk.kR

Bk.k

C

qqR (16)

Substituting the oil effective permeability from Eq. 14 and simplifynig, results

+

−−

µµ

== o

sp

po

gg

ww

rw

rg

w

oPow R

RRRR1

BB

k.kk.k

qqR (17)

Solving for water and gas effective permeability respectively:

µµ

+

−−

=Pow

rg

gg

wwo

sp

porw R

k.kBBR

RRRR1

k.k (18)

( )

+

−−

µµ

=

osp

porw

ww

ggPowrg

RRRRR11k.k

BB

Rk.k (19)

It is important to note that even the producing oil-water ratio can be derived from the free gas production and the water production data, Eq.17. The base fluid in gas condensate wells is the gas phase. Oil production may be hard to measure downhole. Therefore every parameter is expressed in terms of gas properties and water properties. Producing Gas-Water Ratio (Rpgw)(Oil Phase Absent) Similarly

w

sgwwfree,g

w

gPgw q

Rqqqq

R+

== (20)

Where Rsgw is the solution gas-water ratio expressed as SCF /STB.

µ

µ

+

µ==

ww

rw

sgwww

rw

gg

rg

w

gTPgw

Bk.kC

RB

k.kB

k.kC

qq

R (21)

sgwgg

ww

rw

rg

w

gTPgw R

BB

k.kk.k

qq

R +

µµ

== (22)

Solving for water and gas effective permeability respectively:

( )

µµ

−=

gg

ww

gwsPgw

rgrw B

BRR

k.kk.k (23)

( )( )

µµ

−=ww

ggrwgwsPgwrg B

Bk.kRRk.k (24)

Above equations indicate that only one phase effective permeability is required to know all other phase effective permeabilities provided their production is known.

Modeling Relative and Effective Permeability as a Function of Pressure: Vertical Wells (Pressure Drawdown tests)

The effective oil, gas, and water permeability during pressure transient period can be expressed as follows9:

( )

∂∂

µ−==

tlnPh

Bq6.70kkk

wf

oofree,oroo (25)

( ) SP

wf

free,grgg

tlnmPh

q6.70kkk

∂∂

−= (26)

( )

∂∂

µ−==

tlnPh

Bq6.70kkk

wf

wwfree,wrww (27)

Above equations are valid for a fully developed semi-log portion (straight line). Several algorithms are available in literature for estimationg the log derivative of the pressure recorded during a pressure test. Pressure Buildup

∆∆+∂

∂

µ−==

ttt

ln

Ph

Bq6.70kkk

ws

oooroo (28)

Similarly

SP

ws

free,grgg

tttln

mPh

q6.70kkk

∆∆+∂

∂

−== (29)

∆∆+∂

∂

µ−==

tttln

Ph

Bq6.70kkk

ws

wwwrww (30)

To be more accuarte following equation can be used:

( )

SPtigi

tg

ws

free,grgg

ctctt

lnd

dmPh

q6.70kkk

µ∆µ∆+

−== (31)


Several gas well tests were simulated in order to establish relationship between pressure and effective permeability for gas wells. Modeling Two-Phase Pseudopressure Function for Gas-Condensate Fluids

The absolute permeability is usually determined from well test analysis. For multiphase flow conditions effective permeability is required. Thus from the theory of well testing during pressure transient period for tD > 50, when the pressure wave has crossed the wellbore and skin effects (During a semi-log straight line period)

( ) SP

free,grg

tlnmP

h

q70.6kk

∂∂

= (26)

Eq. 26 is expressed in absolute value. mP can be replaced by flowing and shut-in conditions depending on the well test being Drawdown or Buildup. Horner time and adjusted time can also be used. Equations 25, 26, and 27 indicate that the effective permeability is inversely proportional to the derivative of the pressure with natural lograthim of time. On semi-log plot of time versus pseudopressure, the rate of change of pseudopressure is just the slope of the straight line, mgSP. Thus Eq.30 results a relationship of the effective permeability with the prerssure at certain level of depletion in time. A carefully designed pressure test with this additional purpose in mind can provide the average effective permeability over a long range of pressure that can be used for psedosteady state over a long period of time. Oklahoma Corporation Commission requires every well to be tested every year. Thus, fortunately, the value of effective permeability can be updated every year for every well. Relative permeability curves, if available, can also be used to evaluate the two-phase integral.

Solution gas oil ratio is function of API gravity of condensate, gas specific gravity, the bubble point pressure of the condensate, and reservoir temperature. Most of the gas reservoirs produce much over the bubble point pressure of the condensate and free gas gravity and API gravity are constant values. Thus for the resevoir pressure above the bubble point pressure of condensate and well producing at the wellbore flowing pressure within the test pressure range in the Region-1, the oil phase peudopressure (Eq. 3) can be written as: Oil Phase (Region-1)

( ) ∫

µ+

µ=∆

*P

Po

gdgd

rg

oo

roo

wf

dpR.Bk.k

.Bk.kPm (32)

Substituting Eq.14 and 13 in above equation and simplifying results pseudopressure function in terms of more reliable phase, gas phase effective permeability.

( ) ( )∫

µ

+

−−

=∆*P

P gdgd

rgo

sp

poo

wf

dp.Bk.k

RRRRR1

Pm (33)

( )gdgd

osp

pog,o .B

1RRRRR1

Mµ

+

−−

= (34)

( ) ∫

−−

µ=∆

*

wf

P

P po

so

oo

roo dp

RR1RR1

.Bk.kPm

(35)

−−

µ=

po

so

oooo, RR1

RR1.B

1M (36)

Now substituting Equation 19 in Eq. 32, and simplifying yields pseudopressure function in terms of second reliable phase, water phase.

( ) ∫

µ

=∆*P

P ww

rwpowo

wf

dp.B

k.kRPm (37)

Equations 33, 35 and 37 indicate that the oil production can also be predicted from water and gas production. Since water phase itself is the most reliable phase (water properties rarely change with pressure and temperature), therefore, it is not necessary to express water phase in terms of oil phase and gas phase terms, Eq.7.

Gas Phase Produced gas at the surface is combination of free gas and dissolved gas in oil. The total pseudopressure thus can be written as5:

( ) ∫

µ+

µ=

r

wf

P

P gg

rgs

oo

roT dp

.Bk.k

R.Bk.kPm (38)

Region-1 (Inner wellbore region)

( ) ∫

µ+

µ=

*

wf

P

P gg

rgs

oo

ro1 dp

.Bk.k

R.Bk.kPm (39)

Region-2 (Region where liquid develops)

( ) ∫

µ=

d

*

P

P gg

rg2 dp

.Bk.k

Pm (40)

Region-3 (Only gas region)

( ) ∫

µ=∆

R

d

P

P ggwirg3 dp

.B1)S(kkPm (41)

Thus total ∆mPT is equal to ∆mP1+∆mP2+∆mP3 .

Region-1 Substituting Eq.14 and 15 in above equation

respectively, result the gas phase pseudopressure function in terms of gas and oil effective permeability.

( ) ( ) ( )

−−

µ= ∫

*P

P sp

SOP

gg

rgg1g,

wf

dpPRR

)RR1(R)B(

k.kPm

(42)


One can write

( ) ( )∫=*P

P1g,gegg1g,

wf

dp)P(MPkPm

(43)

( )( )PRR)B()RR1(RMspgg

SOP1g,g −µ

−= (44)

( ) ( )( )∫

−−

+µ

=P

P po

sps

oo

ro1o,g

wf

dpRR1RR

R.Bk.kPm (45)

Similarly

( ) [ ]∫=P

Po,geo1g

wf

dp1MkPm (46)

( )( )

−−

+µ

=po

sps

oo1o,g RR1

RRR

.B1M (47)

Water Phase (oil phase absent, pure gas reservoirs with water production) Substituting Eq.23 in Eq.7 and simplifying it results

( ) ∫

µ

=∆*P

P freeww

rww

wf

dp.B

k.kPm (7)

( ) ( )∫

µ−=∆

*P

P gg

rg

sgwpgww

wf

dp.Bk.k

RR1Pm (48)

Eq. 48 shows how water phase pseudopreessure can be expressed as a function of gas properties and producing gas water ratio. Three-Phase Systems: Producing Gas Oil Ratio (Rpgo). By definition

ofree,gfree,O

sgwwSfree,ofree,g

OT

gTP Rqq

RqRqqqq

R+

++== (49)

sgwoT

w

oT

Sfree,ofree,g

oT

gTP R

qq

qRqq

qq

R ++

==

sgwoT

w

ogg

rg

oo

ro

soo

ro

gg

rg

OT

gTPgo R

qq

RB

k.kB

k.kC

RB

k.kB

k.kC

qq

R +

µ+

µ

µ

+

µ== (50)

Last term on right hand side of above equation is total producing water oil ratio, Rpwo. On simplification

µ+

µ

µ

+

µ=−

ogg

rg

oo

ro

soo

ro

gg

rg

pwoPgo

RB

k.kB

k.k

RB

k.kB

k.k

RR (51)

Simplifying and solving for individual phase effective permeability, yields:

−

−

µ

=µ 1AR

RAB

kkB

kk

o

s

oo

ro

gg

rg (52)

−

−

µ=

µ s

o

gg

rg

oo

ro

RA1AR

Bkk

Bkk (53)

Where sgwpwopgo RRRA −= (54)

Producing Gas-Water Ratio (Rpgw3p) in Region-2 and Region-3

w

sgwwfree,g

w

gPgw q

Rqqqq

R+

== (55)

Where Rsgw is the solution gas-water ratio expressed as SCF /STB. For two phase systems Rsgw = 0.

µ

µ

++

µ==

ww

rw

sgwww

rw

gg

rg

w

gTPgw

Bk.kC

RB

k.kB

k.kC

qq

R (56)

Simplifying

sgwsoo

ww

rw

ro

gg

ww

rw

rg

w

gTPgw RR

BB

k.kk.k

BB

k.kk.k

qq

R +

µµ

+

µµ

==

(57) Solving for water and gas effective permeability respectively.

( )

µ−µ

=sgg

rg

gwsPgw

wwrw B

k.kRR

Bk.k (58)

( ) ( ggww

rwgwsPgwrg B

Bk.kRRk.k µ

µ

−= ) (59)

Pseudopressure Function for Three phase Systems (mP) Substituting Eq.52 in Eq.3 and simplifying it yields

the oil pseudopressure function in terms of gas phase and the oil phase properties. Oil Phase

( )( ) dP

1ARRAR

1B

k)p(m

o

so*P

P oo

o

wf

o,o

−

−+

µ

= ∫ (60)


We can write as

dPMk)P(m o,o

*P

Po

wf

o,o ∫=

( )( )

−

−+

µ

=1AR

RAR1B1M

o

so

ooo,o (61)

sgwpwopgo RRRA −= Now Substituting Eq. 63 in Eq. 3 gives:

( )( ) dP

RA1ARR

Bk

)P(ms

oo

*P

P gg

g

wf

g,o

−

−+

µ= ∫ (62)

We can write as

dPMk)P(m*P

Pgg

wf

g,o ∫=

( )( )

−

−+

µ=

s

oo

ggg,o RA

1ARRB1M (63)

Water Phase Equation 7 also applies for three phase systems. There will be no water vapor phasse in the gas phase since the saturation temperature, temperature at which vapor phase begins to condence, is very high at high pressure as indicated in Figures 8a and 8b. Establishing IPR Rawlins and Shellhardt13 equation now can be used to establish oil and water phase IPRs. Mathematically:

( )([ noo PmCq ∆= )]

] (64)

( )[ nww PmCq ∆= (65)

Effective Permeability Estimation Using Measured Surface Rate from Well Test Analysis in Two Phase Systems

In phase changing multiphase environment such as gas condensate systems, it is hard to measure the free rate at surface. The total rate is the combination of the free oil and gas flow and dissolved gas in oil and vapor phase in the gas phase. Thus a scheme is devised to get effective permeability using the surface measured rate from well test analysis instead of free rate. Pressure transient response in terms of pseudopressure can be represented as:

( ) ( )

+

−

φµ+

=−<

S8686.02275.3

rc)P(klog)tlog(

hq

6.162PmPm 2wt

emeas,g

wf*PP

(66)

Oil phase pseudopressure for Region-1 has been define by Eq.32 and 45. Eq. 66 with equation 43 can be expressed as:

( )

+−

φµ+

=∫

S8686.02275.3rc)P(klog)tlog(

hq

6.162dp)P(MPk 2wt

emeas,g

*P

P1g,geg

wf

(67) Taking derivative of above equation with respect to pressure and re-arranging, yields:

( ) dPdk

rc)P(k

434.0dtdP.t

hPkq

6.70)P(M eg2wt

2eg

1

eg

meas,g1g,g

φµ+

=

−−

(68)

Eq. 68 provides gas phase effective permeability as function of pressure from well test analysis in Region-2 in gas-condensate reservoirs.

Now with Eq. 46, Eq 66 can be similarly re-written as follows:

( ) dPdk

)P(k.krc434.0

dtdP

.thPk

q6.70)P(M eg

egeo2wt

1

eo

meas,g1o,g

φµ+

=

−

(69) Eq. 69 provides oil effective permeability as function of pressure with gas phase well test analysis. Effective Permeability Estimation Using Measured Surface Rate from Well Test Analysis in Three Phase Systems

In phase changing multiphase environment such as gas condensate systems, it is hard to measure the free rate at surface. The total rate is the combination of the free oil and gas flow and dissolved gas in oil and vapor phase in the gas phase. Thus a scheme is devised to get effective permeability using the surface measured rate from well test analysis instead of free rate. Pressure transient response in terms of pseudopressure can be represented as:

( ) ( )

+

−

φµ+

=−<

S8686.02275.3

rc)P(klog)tlog(

hq

6.162PmPm 2wt

eomeas,o

wf*PP

(70)

Oil phase pseudopressure for Region-1 has been define by Eq.60 and 62. Equation 68, with equation 60 can be written as follows:

+−

φµ+

=∫

S8686.02275.3rc

)P(klog)tlog(h

q6.162dPMk 2

wt

eomeas,o

o,o

*P

Po

wf

(71)

Taking derivative of above equation with respect to pressure and re-arranging, yields:

( )

φµ+

=

−

dPdkeo

rck434.0

dPdt

t1

h)P(kq

6.70PM 2wt

2eo

o

meas,oo,o (72)


Or one can write as follows:

( )dP

dkeorc

k434.0dtdP.t

h)P(kq

6.70PM 2wt

2eo

1

o

meas,oo,o

φµ+

=

−−

(73) Above equation provides the oil phase effective permeability as a function of pressure from the pressure test data.

Substituting Eq. 62 in Eq. 72, and simplifying similarly results:

( )dP

dkegrckk

434.0dtdP.t

h)P(kq

6.70PM 2wteoeg

1

g

meas,og,o

φµ+

=

−

(74) Eq. 74 provides gas effective permeability as a function of pressure. Concerns Following should be considered in modeling gas-condensate flow in porous media: 1. In Region-1 where gas is lean and all the liquid may have

dropped out, dry gas properties should be used. 2. If retrograde conditions are reached in Region-1, dry gas

properties may no longer be appropriate. 3. In Region-2 liquid dropout is in process. Gas is neither

lean nor rich but in between. Conclusions

1. Pseudopressure integral method has been successfully applied to retrograde gas-condensate systems to predict liquid production including water phase, in two and three phase systems.

2. Multiple equations that use oil, gas, and water physical properties and surface production data have been developed and can be used to predict any desired phase production.

3. Pressure transient data has been successfully used to establish relationship between effective permeability and the pressure, thereby, eliminating the need of core derived relative permeability as a function of saturation.

4. Well testing equations have been modified to estimate effective permeability using surface produced rates.

Nomenclature Bo = Oil FVF, RB/STB Bgd = Dry gas FVF CF/SCF CH = Geometric Factor for horizontal well kro = Oil relative permeability krg = Gas relative permeability L = Length of horizontal well qg = Gas Flow Rate, scf/D Rs = Solution GOR, SCF/STB Rsgw = Solution Gas water ratio, SCF/STB Rp = Producing GOR, SCG/STB (qg/qo)

Rpgw = Producing Gas water ratio, SCF/STB Rpow = Producing oil water ratio, STB/STB S = skin SR = Skin factor due to partial penetration m(P) = pseudo-pressure function, psia2/cp µo = Oil viscosity, cp µg = Gas viscosity, cp Subscripts d = Dewpoint g = gas o = Oil r = Relative w = Water sp-trans = Single phase from transient test 1 = Region-1 2 = Region-2 3 = Region-3 g1,g = Gas in Ragion-1 using gas effective permeability g1,o = Gas in Ragion-1 using oil effective permeability g1,w = Gas in Ragion-1 using water effective permeability o1,g = Oil in Ragion-1 using gas effective permeability o,1o = Oil in Ragion-1 using oil effective permeability o1,w = Oil in Ragion-1 using water effective permeability w1,g = Water in Ragion-1 using gas effective permeability w,1o = Water in Ragion-1 using oil effective permeability w1,w = Water in Ragion-1 using water effective

permeability References 1. Vogel, J.T.: “Inflow Performance Relationships for Solution-

Gas Drive Wells,” JPT Jan. 1968, (83-92). 2. Weller, W.T.: “ Reservoir Performance During Two Phase

Flow,” JPT Feb.1966 (240-245). 3. Fetkovich, M.D., Guerrero, E.T., Fetkovich, M.J., and Thomas,

L.K.: “Oil and Gas Relative Permeabilities Determined from Rate-Time Performance Data,” paper SPE 15431 presented at the 1986 SPE Annual Technical Confference and Exhibition, New Orleans, Oct. 5-8

4. Raghavan, R., Jones, J.R.: “Depletion Performance of Gas-Condensate Reservoirs”, JPT Aug. 1996

5. Fevang, O. and Whitson, C.H. “Modeling Gas-Condensate deliverability,”Paper SPE 30714 presented at the 1995 SPE Annual Technical Confference and Exhibition, Dallas, Oct. 22-25.

6. McCAin, W.D. Jr.: The Properties of Petroleum Reservoir Fluids, Second Edition, PennWell Publishing company.,

7. Gopal, V.N.: “Gas Z-Factor Equations Developed For Computer,” Oil and Gas Journal (Aug. 8, 1977) 58-60.

8. Standing, M.B. and Katz, D.L.: “Density Of Natural Gases,” Trans., AIME (1942), 146, 140-149.

9. Serra, K.V., Peres, M.M., and Reynolds,. A.C.: “Well-Test Analysis for Solution-Gas Drive Reservoirs: Part-1 Determination of Relative and Absolute Permeabilities” SPEFE June 1990, P-124-131.

10. Penuela, G. and Civan, F.: “Gas-Condensate Well Test Analysis With and Without Relative Permeability Curves”, SPE 63160.

11. Economides M.J. et al. “The Stimulation of a Tight, Very-High-Temperature Gas Condensate Well” SPEFE March 1989, 63-72.

12. Guehria, F.M.: “Inflow Performance Relationships for Gas-Condensates”, SPE 63158.


13. Rawlins, E.L. and Shellhardt, M.A.: “Backpressure Data on Natural Gas Wells and Their Application to Production Practices”, USBM (1935).

14. Evinger, H.H. and Muskat, M.: “Calculation of Theoretical Productivity Factors”, Trans., AIME (1942) 146, P126-139.

15. Cenggel, Y.A. and Boles, M.A.: Thermodynamics, McGraw-Hill Publishing Company, 1989.

16. Ghetto, G.D., Paone, F., and Villa, M.: “Reliability Analysis on PVT Correlations”, Paper SPE 28904.

Fig.1 Phase behavior of the condensate fluids.

P e

P d

P *

P w f

S w c

Fig.2. Three regions in a gas condensate reservoir with vertical well.

Pi Pd

P*

Pwf

Fig.3 Three regions indicating two-phase flow around the well, single-phase flow but with liquid buildup, and the free

gas flow in the farther region.

Pi Pd

P*

Pwf

Fig.4 Fluid and pressure distribution around the fully penetrating horizontal well.

Fig.5. Two-phase system with developing oil phase

8000

10000

12000

14000

16000

18000

20000

3800390040004100420043004400450046004700480049005000

Pressure[psia]

[scf

/STB

Pd = 5000 psi

P*

Rp

1/Ro

Fig.6 Determination of P*, pressure at which liquid is mobile, from pressure test data in a multiphase system.


Three Phase Systems

Figure 7. Thee-phase system with developing oil phase

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

320

340

360

32 82 132 182 232 282 332 382 432

Temperature [oF]

Pres

sure

[psi

a]

0

500

1000

1500

2000

2500

3000

3500

432 482 532 582 632 682 732

Temperature, oF

Pres

sure

, psi

a

Fig.8a and 8b. Saturation pressure of water vapor at various temperatures, Steam Tables15. Appendix-A Example-1: Two-Phase (Condensate and Gas) This example was simulated using Sapphire well test simulator. Well, reservoir and fluid data for Example-1 PI 5000 psia qc 100 STB/D Pd 4800 psia h 50 ft GOR 15000 scf/STB qg 2 MMscf/D T 200 oF rw 0.35 ft Gas SG 0.71 API 50

Procedure 1. Having calculated Bo, Bg, µo, µg, Rs, and Ro calculate

Mo,g. and Mo,o using Eq.34 and 36 for 2-phase and 44 and 47 for 3-phase.

For Ro use following correlation

ssso RRRxR 3815.42623.110706.466.11 39 −++−= −

2. Having calculated the Mo,g Mo,o terms, calculate the group (t.dp/dt) and get the inverse of it using following equation.

[ ]1i1i

1i1i

1i1i

1i

1i

i )tln()tln(

)tln()tln(

Pd)tln()tln(

Pd

dtPd.t

−+

−+

++

−

−

∆+∆

∆

∆

∆+∆

∆

∆

=

∆

3. Plot Mo,g Vs (t.dP/dt)-1 on a Cartesian graph. From

the point where wellbore storage and skin effects are over, calculate the derivative of the curve as follows m = (y2 -y1)/(x2-x1) and convert it into absolute value if necessary.

4. Effective permeability then is as follows:

=

hmq

6.70)P(k meas,oeo

For two phase systems, If gas phase pseudopressure is used then use qg,meas.

5. Using a good curve fitting software, get a correlation for the effective permeability such as at zero pressure ke = 0. Get gas, oil, and water effective permeability correlations.

6. Using these correlations, evaluate the pseudopressure integrals for oil using Eq. 33, or 35 for 2-phase and Eq 60 or 62 for 3-phase, depending on the gas or oil effective permeability determined. For water phase use Eq. 7.

7. Now convert the wellbore flowing pressure data into pseudopressure data using Eq. 33 or 35 depending on the effective permeability term for 2-phase and Eq.60 or 62 for 3-phase. Use any suitable numerical technique to evaluate the integral.

8. Plot flow rate Vs pseudopressure on a log-log plot and calculate slope n, and intercept, C.

9. Establish IPRs using Eq. 64 and 65.


3000

3500

4000

4500

5000

5500

0.01 0.1 1 10 100 1000

Time [hrs]

Pres

sure

[psi

a]

Semi-log Straight line

Fig.9. Transient Pressure semi-log plot.

0

0.2

0.4

0.6

0.8

1

1.2

4700 4750 4800 4850 4900 4950 5000

Pressure [psia]

Gas

Effe

ctiv

e Pe

rmea

bilit

y [m

d]

Pd and Phase Change Effects

Single Phase Region

Two Phase Region

Fig.10 Gas effective permeability as function of pressure.

Fig.11 Gas effective permeability as function of pressure, curve fit.

0

10

20

30

40

50

60

70

80

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

Pressure [psia]

m(P

) o,g

[106 p

sia2 /c

p]

Fig.12. Oil phase pseudopressure using gas effective

permeability. [Eq.33]

0

400

800

1200

1600

2000

2400

2800

3200

3600

4000

4400

4800

0 25 50 75 100 125 150 175 200 225

Condensate Flow Rate [STB/D]

Pres

sure

[psi

a]

Fig.13. Oil phase IPR against pressure with pseudoprerssure

using Eq. 33. [n =0.8 c = 0.0001 Assumed values] Example-2: Three-Phase Vertical Well This example was simulated with reservoir pressure just above the dew point pressure to simulate the Region-1, Region,2 and Region-3 together. But the pressure did not drop far below to see all the three regions altogether. The lowest pressure is 3500 psi. But the initial data is masked by the wellbore storage effects, Region-1, Pd < Pwf = 4800 psi, is well developed. After 100 hours we are in radial portion and in the Region-1. Thus using same procedure as in example 1, well performance is established.

Well and Fluid Data for Example-2

Pi 5,000 Psi h 100 ft GWR 10,000 CF/STB C 0.2 STB/Psi WGR 100 STB/MMscf S 3 SG 0.7 kh 100 md-ft Pd 5,000 psi k 1 md Tp 1,000 hrs qg 1 MMcf/D Cr 3.00E-06 Psi-1 qo 100 STB/D T 250 F qw 100 STB/D GOR 20,000 cf/STB API 50 rw 0.35 Ft


3000

3500

4000

4500

5000

5500

0.01 0.1 1 10 100 1000

Time, Hrs

Pres

sure

, psi

Start of Semi-log Staright Line

Fig.14 Semi-log plot of pressure test indicating start of semi-log straight line.

keg(P) = 3E-146P39.355

R2 = 0.9811

0.00000001

0.0000001

0.000001

0.00001

0.0001

3250 3300 3350 3400 3450 3500 3550 3600

Pressure[psia]

Keg

[md]

Fig.15. Gas phase effective permeability Integral. Example2)

keo(P) = 4E-144P39.175

R2 = 0.9821

0.000001

0.00001

0.0001

3250 3300 3350 3400 3450 3500 3550 3600 3650

Pressure, Psia

Keo

[md]

Fig.16. Oil effective permeability as a function of pressure.

Fig.17. Water effective permeability as a function of pressure.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000

Pressure, psiaFo

rmat

ion

Volu

me

Fact

or, R

B/S

TB

Oil

Water

Gasx1000

Fig.18. PVT data for example-2.

0

2000

4000

6000

8000

10000

12000

14000

0 1000 2000 3000 4000 5000 6000

Pressure, psia

m(P

) w, p

sia2 /c

p

Fig.19. Water phase pseudopressure.

0

5000

10000

15000

20000

25000

30000

135.4 135.6 135.8 136 136.2 136.4 136.6 136.8 137 137.2 137.4

Condensate Flow Rate, B/D

Oil

Phas

e Ps

eudo

pres

sure

, psi

a2 /cp

Fig.20. Condensate IPR [n = 0.8, C = 0.04]


0

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100Water Rate, B/D

Wel

lbo

re F

low

ing

Pre

ssur

e,

120P

Fig. 21. Water phase IPR [n = 1, C =0.5 Assumed values]

Table-1 PVT data for Example-1 P

psia Bo

bbl/stb µo cp

Bg bbl/scf

µg cp

Rso scf/bbl

Ro bbl/scf

Mo,g 2-phase

4107.709 1.836679 0.154726 0.000717 0.023184 1430.138 6.24E-05 6.817752 4112.585 1.837855 0.154585 0.000716 0.023204 1432.097 6.25E-05 6.816188 4117.385 1.839013 0.154447 0.000716 0.023224 1434.025 6.26E-05 6.814634 4122.117 1.840156 0.154311 0.000715 0.023244 1435.926 6.27E-05 6.813088 4126.78 1.841282 0.154177 0.000715 0.023264 1437.8 6.28E-05 6.811552 4131.378 1.842392 0.154045 0.000714 0.023283 1439.648 6.28E-05 6.810023 4135.913 1.843488 0.153916 0.000714 0.023302 1441.471 6.29E-05 6.808502 4140.387 1.844569 0.153788 0.000713 0.023321 1443.27 6.30E-05 6.806991 4144.804 1.845637 0.153663 0.000713 0.02334 1445.046 6.31E-05 6.805486 4149.165 1.846692 0.153539 0.000713 0.023358 1446.8 6.32E-05 6.803988 4153.892 1.847836 0.153405 0.000712 0.023378 1448.701 6.33E-05 6.802352 4159.119 1.849101 0.153257 0.000712 0.023401 1450.804 6.34E-05 6.800528 4164.894 1.8505 0.153094 0.000711 0.023425 1453.128 6.35E-05 6.798491 4171.287 1.852048 0.152914 0.00071 0.023452 1455.701 6.37E-05 6.796214 4178.323 1.853754 0.152716 0.00071 0.023482 1458.534 6.38E-05 6.793679 4186.051 1.855629 0.1525 0.000709 0.023515 1461.646 6.40E-05 6.79086 4194.523 1.857685 0.152264 0.000708 0.023552 1465.059 6.42E-05 6.787728 4203.785 1.859934 0.152007 0.000707 0.023591 1468.791 6.43E-05 6.784255 4213.887 1.862389 0.151727 0.000706 0.023635 1472.863 6.46E-05 6.780408 4224.915 1.865072 0.151424 0.000705 0.023683 1477.311 6.48E-05 6.776137 4236.872 1.867983 0.151097 0.000704 0.023734 1482.135 6.50E-05 6.771426 4249.788 1.87113 0.150745 0.000703 0.02379 1487.347 6.53E-05 6.76624 4263.682 1.87452 0.150369 0.000701 0.023851 1492.958 6.56E-05 6.760551 4278.632 1.878171 0.149966 0.0007 0.023916 1498.998 6.59E-05 6.754302 4294.574 1.882068 0.149541 0.000698 0.023987 1505.442 6.63E-05 6.747494 4311.489 1.886209 0.149092 0.000697 0.024061 1512.284 6.66E-05 6.740105 4329.395 1.890599 0.148621 0.000695 0.024141 1519.531 6.7E-05 6.732101 4348.189 1.895212 0.14813 0.000693 0.024224 1527.142 6.74E-05 6.7235 4367.77 1.900025 0.147624 0.000692 0.024312 1535.078 6.79E-05 6.71432 4388.102 1.90503 0.147102 0.00069 0.024404 1543.324 6.83E-05 6.704553 4408.956 1.910172 0.146572 0.000688 0.024498 1551.787 6.88E-05 6.69429


Authors

Sarfraz A. Jokhio Djebbar Tiab Freddy H. Escobar Dr. Sarfraz A. Jokhio is currently working as a postdoctoral fellow and lecturer at Mewbourne School of petroleum and geological engineering, university of Oklahoma. Previously, he worked as an Application Engineer with Woodgroup ESP, Inc. in Oklahoma City where he handled ESP applications, ESP assisted BHA, downhole liquid management in gas wells using ESPs, downhole water separation system application along with ESP, downhole ESP corrosion control, and highly viscous crude pumping issues. He has worked on the wells in Indian Basin, NM, Permian Basin (West Texas), Oklahoma, Venezuela, and offshore Sicily. He also worked with Dowell Schlumberger in Production Enhancement in Eastern NM for some time. Dr. Jokhio holds BS (MUET Jamshoro, Sindh, Pakistan, 89), Msc (1997), and Ph.D. (2001) from University of Oklahoma, all in Petroleum Engineering. Dr. Jokhio also worked as an assistant professor in Mehran University of Engineering and Technology, Jamshoro, Sindh, Paksitan where he taught various undergraduate petroleum courses. His main interests include Artificial Lift System Optimization, Oil and Gas Well Test Analysis, Gas-Condensate Reservoir Performance, Production System Optimization, and Petrophysics. Prof. Dr. Djebbar Tiab

Dr. Tiab is the Senior Professor of Petroleum Engineering at the University of Oklahoma. He received his B.Sc. (May 1974) and M.Sc. (May 1975) from the New Mexico Institute of Mining and Technology, and Ph.D. (July 1976) from the University of Oklahoma - all in Petroleum Engineering. He is the Program Director of “The University of Oklahoma Graduate Program in Petroleum Engineering in Algeria”. Before joining University of Oklahoma in 1977, he worked as an assistant professor at the New Mexico Institute of Mining and Technology.

Dr. Tiab is also the president of United Petroleum Technologies (UPTEC): 1980-84, 1989 - present. He is a member of the U.S. Research Council, Society of Petroleum Engineers (SPE), Core Analysis Society, Pi Epsilon Tau, Who is Who and American Men and Women of Science. He served as a technical editor of various SPE, Egyptian and U.A.E. journals. He is currently a member of the SPE Pressure Analysis Transaction Committee. Dr. Tiab is the author of over one hundred journals and conference technical papers in the area of pressure transient analysis, petrophysics, natural gas engineering, reservoir characterization, and reservoir engineering and injection processes. In 1975 (M.S. thesis) and 1976 (Ph.D. dissertation), Dr. Tiab introduced the pressure derivative technique which revolutionized the interpretation of pressure transient tests. He has two patents in the area of reservoir characterization (identification of flow units). He is the senior author of the textbook “PETROPHYSICS”, published by Gulf Publishing Company: 1st Edition in October 1996.

Dr. Tiab supervised 21 Ph.D. and 56 M.Sc. students at the University of Oklahoma. Most of his Ph.D. students are now professors at universities in the U.S.A., South America, Africa, Asia and the Middle East. He received the Outstanding Young Men of America Award (1983), the SUN Award for Education Achievement (1984), Kerr-McGee Distinguished Lecturer Award (1985), the College of Engineering Faculty Fellowship of Excellence (1986), the Halliburton Lectureship Award (1987-89), the UNOCAL Centennial Professorship (1995-98), and the P&GE Distinguished Professor (1999 – 2000). He also received the prestigious 1995 SPE Distinguished Achievement Award for Petroleum Engineering Faculty. Dr. Freddy H. Escobar Dr. Freddy H. Escobar holds a BS degree from the Universidad de America (Bogota-Colombia), M.Sc(1995 OU). and Ph.D. (2002, OU) from University of Oklahoma all in Petroleum Engineering. Currently, Dr. Escobar is a full-time professor in Universidad Surcolombiana (Neiva-Colombia) where he teaches Reservoir Engineering, Petroleoum Reservoir Development and Well Test Analysis. His main interests include well test analysis, reservoir characterization and simulation, numerical analysis and software development. He has authored and coauthored more than 30 publications related to Petroleum Engineering.

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