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Generalized Inflow

Performance

Relationships

for

Three Phase Flow

Michael Wiggins

SPE,

of Oklahoma

Summary

This paper presents generalized liquid inflow performance relationships (IPR's) for three-phase flow in bounded, homoge

neous reservoirs and new methods to predict present and future performance during boundary-dominated flow.

1

. . , , , (2)

Conclusions

1 Generalized three-phase IPR's that are suitable for use over a

wide range of reservoir properties have been presented.

2 ThegeneralizedIPR's have been verifiedby useof information

presented by Sukam0

4

and by comparison to Brown

3

and Sukar

no's4 three-phase methods.

3. For the first time, a method has been proposed for predicting

future performance during three-phase boundary-dominated flow

where the subscripts and

P

represent future and present conditions,

respectively. As the figures indicate, some variation exists between

the curves because

of

relative permeability and fluid property ef

fects. The curves suggest that care should be taken in estimating fu

ture performance over large stages

of

depletion because the error

may increase as prediction periods increase. Updating

of

initial fu-

ture performance estimates every 6 months to 1 year is recom

mended; this would progressively reduce the uncertainty in earlier

estimates as depletion occurs in the reservoir.

(4)

(3)

o.max

f

_ p r

f

) Pr

f

2

-

0.15 =

+

0.84 =

o m xp

Prp Prp

(

_ 2

w.max

f

Prf Prf

and

= 0.59 =

+

0.36 = ,

w m x

p

Prp Prp

Predicting Future

Performance

While IPR's yield estimatesof well performanceat the currentstage

of

reservoir depletion, there are times when the engineer wants

to

predict future performance. Relationships were developed to pre

dict future performance with the simulator results generated during

this study, The ratio

of

the maximum production rate to the current

maximum production rate was plotted against the ratio of the aver

age reservoir pressures. Figs. 3 and 4 present the results, which

were fit with a linear regression model.

The resulting relationships for predicting future maximum pro

duction rates are

To test theirreliability, the generalized IPR's were comparedwith

Brown

3

and Sukarno's4 three-phase IPR methods. Both methods

differ from the generalized three-phase IPR method presented be

cause they couple the water and oil rates. The proposed method as

sumes that each phase can be treated separately.

To evaluate the three methods, information generated by Sukar

n0

4

using a simulator and presented in his Tables 6-24 to 6-26 was

selected for comparison. This informationwasnot usedin the devel

opment ofthe proposed methodand shouldgive an unbiased indica

tion of the reliability of the proposed IPR's. All three methods pro

duce similar producing rate estimates, indicating that the

generalized three-phase IPR's yield suitable results. The maximum

difference between the simulator results and the generalized IPR is

3.98 for the oil phase and7.08 for the water phase. This analysis

shows that any

of

the three methods appear suitable for use during

boundary-dominated flow; however, the proposed method is much

simpler to usewithout yielding any degree of reliability. Because of

their simplicity, the generalized IPR's are recommended for use in

applications to field data.

Generalized

IPR s

Figs. 1and 2 present the simulator results for all cases studied along

with the resulting IPR equations. Overall, the average absolute error

was 4.39 for the oil IPR and 6.18 for the water IPR, indicating

that the generalized curves should be suitable for use over a wide

range

of

reservoir properties

if

the reservoir is producing under

boundary-dominated flow conditions.

The generalized IPR's are

= 1 0 52 (P : )-0.48

P.: )2

qo m x

Pr

Pr

2

and = 1 0 72 (P : )-0.28 P.: ) .

qw m x

Pr

Pr

Original SPE manuscript received for review March 21, 1993. Revised manuscript received

March 2, 1994. Paper accepted forpublicationMarch17, 1994. Paper SPE 25458) firstpres-

ented at the 1993 SPE Production Operations Symposium held in Oklahoma City. March

21-23.

Copyright 1994 Society of Petroleum Engineers

Development of Simulator Results

To develop generalized equations to predict inflow performance,

IPRcurveswere generated from simulatorresults for four basic sets

of relative permeability and fluid property data. Each data set was

used to generate simulator results from irreducible water saturation

to residual oil saturation (ROS). A total of 16 theoretical reservoirs

were examined from initial pressure to minimum flowing bottom

hole pressure in 91 simulator runs. Reservoir properties varied as

follows: porosity, 12 to 24 ; permeability, 1 to 100 md; height,

10 to 25 ft; temperature, 150 to 200F; initial pressure, 1,500 to

3,500psi; oilgravity, 15 to45 API; gas gravity, 0.6 to0.7; water sol

ids, 12 to 30 ; ROS, 5 to 45 ; irreducible water saturation,

10 to 50 ; critical gas saturation, 0 to 7.5 ; and drainage ra

dius, 506 to 1,085 ft.

Simulator results were obtained for a radial flow geometry and

constant oil rate production. The model grid was established geo

metrically so that each succeeding radius was 1 1 times larger than

the previous radius. The initial cellblock radius was 0.329 ft, with

a wellbore radius of 0.328 ft Refs. 1 and 2 give additional reservoir

property details and simulator parameters.

Introduction

IPR's are empirical relationships basedon linear regression analysis

of simulator results that cover a wide range of reservoir fluid and

rock properties. The IPR's developed are compared with other

three-phase methods and yield similar results for production-pres

sure behavior during boundary-dominated flow while being much

simpler to use.

The proposed IPR's were developed from analysis of multiphase

flow in bounded, homogeneous reservoirs without external influx

of fluids into the reservoir and apply to the boundary-dominated

flow regime. The relationships are limited by the assumptions that

(1) the reservoirs are initially at the bubblepoint, (2) no initial free

gas phase is present, (3) a mobile water phase is present for three

phase studies, (4) Darcy's law for multiphase flow applies, (5) iso

thermal conditions exist, (6) no reactions take place between reser

voir fluids and reservoir rock, (7) no gas solubility exists in the

water, (8) gravity effects are negligible, and (9) the wellbore is fully

penetrating.

SPE Reservoir Engineering, August 1994 181

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1.0

.8

SimulalorResu1Is

0.6

.4.2

----....---r--.---- r--

- r - - . - - - - r - - . - ~

0.0

0.6

0.2

0.8

0.4

1 . 0

1.0

.8

SimulalorResulIl

0.6

.4.2

o . o - - . - ~ - - - - ~ - - . - _ _ . - ~ _ _ -

0.0

0.6

0.2

0.8

0.4

1.0

_ _

::

pwf/pr

Fig. Comparison

of

simulator results with generalized oil

IPR.

pwf/pr

Fig.

2 Comparison

of simulator results with generalized water

IPR.

1.0 . . .

1 0 r ....

1.0.8

.6.4

.2

Simulalor

Resulrs

-

Proposed Relalion

y

=

0.59x +0.36x

2

0.0 'lC.::. . , .. . .-, . . . . ..--.---r--.---r-

.....

t

0.0

0.2

0.4

0.6

0.8

1.0

.8.6.4.2

y = 0.15x + 0.84x

2

Simulalor

Resu1Is

-

Proposed

Relation

0.0

. . .

-=:;.....;;;.....,r----.--,--

r . . . . .

...... .

0.0

0.4

0.2

0.8

0.6

pr,f/pr,p

Fig. 3 Comparison of simulator results with proposed method

fo r determining future oil-phase performance.

pr,f/pr,p

Fig.

4 Comparison

of simulator results with proposed method

for determining future water-phase performance.

SPEREConversion factor is exact.

Michael L Wiggins is anassistant professor of

petroleum and geological engineering at the U

of Oklahoma. He has industrial experience with

major and independent producers. He holds BS,

ME

and

PhD

degrees in petroleum engineering

from Texas A&M

U

Wi ggin s was a 1992-93

member of the Production Operations Sympo

sium program committee and has served as U

of OklahomaSPE Student Chapterfa culty advis

er since 1991.

51 Metric

Conversion

Factors

o

API 141.5/ 131.5+

0 API)

= glcm

3

bbl x 1.589

873 E-Ol

= m

3

ft x 3.048*

E-Ol

= m

O CF-32 /1.8

= c

galx3.7854l2 E 0 3 = m

3

md x 9.869 233 E - 04 =

Ilm

2

psi x 6.894 757 E

+

00 = kPa

Nomenclature

Pr

= average reservoir pressure, mlLt

2

,

psi

Pw f

= flowing wellbore pressure, mlLt

2

, psi

= oil production rate,

L3/

t

, BID

%,rnax

= maximum oil production rate,

L3/t

BID

qw

= water production rate,

Oft,

BID

qw,rnax = maximum water production rate, L3/

t

, BID

x

=

prjlprp

y = qrnax f/qrnax p

References

1.

Wiggins, M.L.:

Generalized Inflow

Performance

Relationships

for

Three-Phase

Flow,

paperSPE 25458 presented at the 1993 SPE Produc

tion Operations Symposium, Oklahoma City, March 21-23.

2. Wiggins,

M.L.:

Inflow Performance

of Oil

Wells

Producing

Water, PhD

dissertation, Texas A&M D., College

Station

1991).

3.

Brown,

K.E.: The Technology ofArtificial LiftMethods

PennWeli

Pub

lishing

Co., Tulsa (1984) 4, 18-35.

4.

Sukarno,

P.:

Inflow

Performance

Relationship Curvesin Two-Phase

and

Three-Phase Flow

Conditions, PhD dissertation, D. of Tulsa,

Tulsa

1986).

182

SPE

Reservoir

Engineering,

August

1994