a comprehensive comparative study on analytical pi-ipr correlations

8/11/2019 A Comprehensive Comparative Study on Analytical PI-IPR Correlations

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SPE 116580

A Comprehensive Comparative Study on Analytical PI/IPR CorrelationsSuk Kyoon Choi, SPE, The University of Texas at Austin, andLiang-Biao Ouyang, SPE, and Wann-Sheng (Bill)Huang, SPE, Chevron Energy Technology Company

Copyright 2008, Society of Petroleum Engineers

This paper was prepared for presentation at the 2008 SPE Annual Technical Conference and Exhibition held in Denver, Colorado, USA, 2124 September 2008.

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 have not beenreviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, itsofficers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission toreproduce in print is restricted to an abstract of not more t han 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

Inflow performance is one of the significant components to quantify the reservoirs capability to produce hydrocarbon.There are two commonly-used quantities to represent reservoir inflow performance: productivity index (PI) and inflow

performance relationship (IPR). Both relate fluid flow rate to pressure difference between bottomhole and reservoir. Much

effort has been made on developing the PI or IPR solutions suitable for specific circumstances since Darcy proposed the

simple and useful Darcys law in 1856. As a consequence, various correlations for PI or IPR calculation have been proposedfrom simple analytical solutions to rigorous numerical formulations in the literature.

As horizontal or multilateral wells have been occupying an ever-increasing share of hydrocarbon production since the

1980s, more accurate PI or IPR estimation has been emerging as an important issue in the petroleum industry.11 The

correlations become more and more complicated and rigorous in order to accurately describe inflow performance for

complex well geometries. They can provide a better prediction or estimation of inflow performance, though they would be

costly and computationally demanding. On the other side, researchers have been tried to simplify the complex solutions intoanalytical forms through extensive case studies. These provide a useful tool for the researchers and engineers to make quick

estimations although they are confined to limited conditions.

In this paper, analytical correlations of PI and IPR from a comprehensive literature survey are reviewed. They have beencategorized by well deviation, fluid phases and time dependence. The well deviation is divided into vertical (less than 15),

slanted (15 to 60), highly-deviated/horizontal (60 to 90), and multilateral wells; while fluid phases are categorized into

single oil, single gas, and oil flow in two phases. For time dependence, steady state and pseudo-steady state have beenconsidered, however the transient state has been excluded. In addition, case studies for the specific input parameters have

been conducted to show the effective range, trends, and limitations of correlations as well as to provide the selection

guideline for an appropriate estimation of inflow performance. All correlations of PI and IPR have been organized in the

table for quick reference.

IntroductionInflow performance serves as an important component with outflow performance to quantify hydrocarbon production

from a reservoir. Outflow performance reflects the flow capability in the pipelines and surface facilities from the bottom hole

to the surface storage tank; while inflow performance represents the reservoir capability which relates well production rate to

driving force, i.e., pressure difference between the outer boundary or average reservoir pressure and flowing bottom pressure.

Both performances are essential factors to generate a well deliverability curve which enables to predict an optimal wellproduction rate.

There are two different ways to express inflow performance inflow performance relationship (IPR) and productivity

index (PI). IPR presents the well production rate as a function of the flowing bottomhole pressure. The lower flowingbottomhole pressure provides a higher driving force in the reservoir, thereby resulting in a higher hydrocarbon production

potential. Theoretically, the zero flowing bottomhole pressure gives the maximum production potential in the reservoir

which is called Absolute Open Flow Potential (AOFP). The simplest and most widely used IPR is the straight-line IPR,implying that rate is directly proportional to pressure drawdown. The constant of proportionality is defined as the

Productivity Index (PI), another way to define inflow performance. One of the main objectives of production engineering

is to maximize PI which can be obtained by maximizing the flow rate for a given pressure drawdown or minimizing the

pressure drawdown for a given flow rate.


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2 SPE 116580

There are many PI or IPR correlations published in the literature, ranging from simple analytical correlations to rigorous

numerical solutions. Since the proposition of Darcys law in 1856, various theoretical and empirical correlations have been

presented for variable reservoir parameters (reservoir shape, area, and petrophycal parameters), fluid phases, well trajectory,and time dependence.

This study intends to review as many simple analytical solutions for PI or IPR as possible through a comprehensiveliterature survey, and perform case studies for the comparison of correlations in the same category in order to provide a

selection guideline. Under certain circumstances, it makes more sense to use simple analytical solutions to predict PI or IPR,

because they are much quicker and easier to use with acceptable accuracy when compared to reservoir simulation. Twentyeight (28) correlations have been found from this review and categorized according to well deviation, fluid phase, and timedependence.

Analytical Solutions for PI / IPRSummary for the Solutions

A total of 28 PI or IPR correlations reviewed through literature survey have been classified into three categories: fluid

phase, well deviation, and time dependence. In fluid phase category, analytical correlations are available only in single-phase(single oil and single gas), and oil flow in two phases in the solution-gas drive reservoir. Well deviation is divided into three

well geometries: vertical (less than 15); slanted (15 to 60); horizontal (60 to 90); and multilateral well. The solutions forsteady and pseudo-steady state are included in this paper. However, the transient state is excluded.

Table 1shows a summary for the analytical PI or IPR solutions. No analytical correlations were found in the literature

for multilateral well in single-phase gas reservoir and two-phase reservoir. In particular, there were also no analytical

correlations identified for slanted well geometry; instead, three correlations for deviation skin were applied to combine withany correlation in the vertical well geometry to calculate PI or IPR for slanted wells. Therefore, the skin correlations are

included for the slanted well section in this table except for the case of oil flow in two-phase reservoirs which has its owncorrelation for PI or IPR. All the correlations in this paper are displayed in the form of productivity index with oil-field units.

Literature Review

Single Oil Well in Vertical Geometry

In a vertical geometry with single-phase oil under steady state, Darcy proposed a constitutive equation that describes the

flow of a fluid through a porous medium. Darcys law can be expressed in various forms according to reservoir geometries.The following form represents the steady state radial flow in a circular drainage area with potential skin effect in the near

wellbore.11

( )141.2 ln /o

oe wf o o e w

q khJ

p p B r r s= =

+ (1)

whereJois productivity index; qois oil flow rate;pwfis bottom hole flowing pressure; peis external boundary pressure; kispermeability; his pay thickness; Bois oil formation volume factor; ois oil viscosity; reis external boundary radius; rw is

wellbore radius; andsis skin effect.The equation for pseudo-steady state with the same conditions can be obtained by simply replacing ln(re/rw) term in

equation (1) with ln(0.472re/rw), which is shown in equation (2):11

( )141.2 ln 0.472 /o

or wf o o e w

q khJ

p p B r r s= =

+ (2)

where pr is average reservoir pressure. Dietz8developed a series of shape factors to account for irregular drainage shape

and/or asymmetrical positioning of a well. The following model is a generalized form of equation (2) for any shape factor

proposed by Dietz:

2

1 4

141.2 ln2w

oo

r wf

o oA

q khJ

p pA

B sC r

= =

+

(3)

whereAis drainage area; is Eulers constant (1.78); and CAis shape factor.Two correlations to account for non-Darcy flow effects have been included in the category of the vertical geometry with

single-phase oil in pseudo-steady state. Note that non-Darcy flow occurs when the linear relationship between pressure drop

and rate (Darcys law) becomes invalid at high flow velocity. Jones20presented a pressure drawdown versus rate relationship

by employing the Forchheimer factor:

1oo

r wf o

qJ

p p A B q= =

+ (4)


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SPE 116580 3

where

ln 0.75

0.007078

eo o

w

rB s

rA

kh

+ = (4.1)

13 2

2 2

9.08 10

4

o o

p w

BB

h r

= (4.2)

whereis non-darcy factor; ois oil density; and hpis perforate interval.The following correlation represents an alternative form of the PI (in vertical geometry with single-phase oil in pseudo-

steady state), which employs the non-Darcy coefficientD.2 The additional term of Dq is added as an equivalent skin term

in equation (2) in order to consider the non-Darcy effect:

( )

0.00703

ln 0.472 /

oo

r wf o o e w o o

q khJ

p p B r r s D q= =

+ + (5)

whereDois non-darcy factor.

Another equation for irregular drainage area was proposed by Odeh22based on the work of Matthews and Russel15.

[ ]

37.08 10

ln 3 4

oo

r wf o o

q khJ

p p B X s

= =

+ (6)

where X is the correlation between rwandA(or re) for various drainage areas and well locations.22

Single Oil Well i n Slanted GeometryFor the slanted well, three correlations for slanted skin have been introduced in this paper because not even a single

analytical PI/IPR correlation for the slanted well has been found in the literature. They can be combined with any of the

correlations for vertical geometry with single phase (single oil and single gas) so as to provide the PI or IPR solutions for the

slanted well geometry. Cinco-Ley7proposed a simple correlation for slanted skin based on the study of unsteady state flowof a slightly compressible fluid. This correlation is valid for the well deviation angles between 0 to 75:

2.06 1.865' '

10log41 56 100

w w Dhs

= (7)

where ( )' 1tan tanvwh

k

k

=

(7.1)

v

h

w

Dk

k

r

hh = (7.2)

wheresis slanted skin; hDis reservoir dimensionless thickness; khis horizontal permeability; kvis vertical permeability; andis slant angle measures from the normal to the bedding planes.

Besson3proposed another slanted well skin correlation from the results of a semi-analytical simulator. In this model,

pay thickness (h) and horizontal well length (L) are used to represent slanted angle (). For isotropic reservoir and slantangles between 0to 90:

4ln ln

4

w

w

r h Lhs

L L r

= +

(8)

For anisotropic reservoir:

24 1ln ln

4 1 1/

w

w

r h Lhs

L L r

= + +

(9)

where h vk k= (9.1)

2

2 2 2

1 11

h

L

= +

(9.2)

whereLis well length.

Rogers and Economides23 derived an expression for a slanted skin starting from the previous comprehensive semi-

analytical PI models by Economides9:1.77 0.184

0.821

sin1.64 D

ani

hs

I

= forIani< 1;

5.87 0.152

0.964

sin2.48 D

ani

hs

I

= forIani1 (10)


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where haniv

kI

k= , h x yk k k= (10.1)

Dw

hh

r= (10.2)

where kxis permeability in x direction; and kyis permeability in y direction.

Single Oil Well i n Horizontal Geometry

For the horizontal geometry, Economides10presented an inflow performance relationship for the single-phase oil flow in

steady state that was modified from Joshis equation. This model requires the assumption that the drainage area is

ellipsoidal, with the large half-axis of the drainage ellipsoid, related to the length of the horizontal well:

( )

( )

22 / 2141.2 ln ln

/ 2 1

o ho

e wfani ani

o o

w ani

q k hJ

p pa a L I h I h

B sL L r I

= = + + + +

(11)

where haniV

kI

k= (11.1)

0.50.5

4

0.5 0.252 / 2

erLaL

= + +

for 0.92

eL r< (11.2)

Other similar solutions are available in the literature: Borisove4, Giger14, Renard and Dupuy16.

For Borisove:

( )0.007078 /

ln 4 ln2

h o ooo

e wf e

w

k h BqJ

p p r h h

L L r

= = +

(12)

For Giger:

( )

( )

( )

2

0.007078 /

1 1 / 2

ln ln/ 2 2

h o ooo

e wfe

e w

k h BqJ

p pL rL h

h L r r

= = +

+

(13)

For Renard and Dupuy:

( ) ( ) ( )1 '0.007078 1

cosh ln / 2

o ho

e wf o o w

q k hJ

p p B x h L h r

= = +

(14)

where '1

2w wr r

+= (14.1)

2x a L= for an ellipsoidal drainage area (14.2)

h vk k= (14.3)0.5

0.54

0.5 0.252 / 2

erLaL

= + +

(14.4)

For pseudo-steady state, equation (11) can be simply converted by adding ln(0.472) term at the place equivalent toskin.10

( )

( )

22/ 2

141.2 ln ln 0.75/ 2 1

o ho

r wfani ani

o o

w ani

q k hJ

p pa a L I h I h

B sL L r I

= = + + + +

(15)


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Another pseudo-steady state correlation proposed by Economides9is shown in equation (16). This model employs the

dimensionless pressure of a point source in a box shape reservoir with no-flow boundary conditions. Furthermore, this model

is applicable to estimate the inflow performance for multilateral wells with specific well configurations by using horizontalplane shape factors:

887.22 2

o xo

xr wfo o D

q kLJ

Lp p

B p sL

= =

+

(16)

where4 2

x A xD x

L C Lp s

h L = + (16.1)

ln2 6

x ew

h hs s

r L

= +

(16.2)

22 21 1

ln sin2 2

w w we

z z zhs

L h h h

=

(16.3)

3x y vk k k k = (16.4)

wherezwis distance of well from middle of reservoir.Babu and Odeh proposed the widely used correlation for horizontal well pseudo-steady state calculations. The reservoir

is assumed to be in a bounded rectangular shape with an arbitarily located horizontal well:19

( ) ( )

( )1

0.007078 2 /

ln / ln 0.75

e y v o ooo

r wf w H R

x k k BqJ

p p A r C s

= =

+ + (17)

Where 1 2 eA y h= (17.1)2

y

y

2 1ln 6.28 ln sin 180

k 3 2 2

20.5ln 1.088

k

oe v w w wH

e e

e v

y k y y zC

h y y h

y k

h

= +

(17.2)

wherexwis distance from the horizontal well mid-point closest boundary in the x direction;ywis distance from the horizontalwell to the closest boundary in the y direction;zwis vertical distance between the horizontal well and the bottom boundary;xeis reservoir half length in the direction parallel to the wellbore;yeis reservoir half length in the direction perpendicular to the

wellbore; andsRis skin factor due to partial penetration of the horizontal well in the areal plane19.

Single Oil Well i n Multi lateral Geometry

For steady state flow in a multilateral well, Borisove proposed models for calculating inflow performance of multilateral

wells with either plannar or stacked laterals.12 Planar laterals means the spokes of a wheel from a single spudding location.For plannar laterals:

( )0.007078 /

ln ln2

o ooo

e wf e

w

kh BqJ

p p r h hF

L nL r

= = +

(18)

For stacked laterals:

( )0.007078 /

ln ln2

o ooo

e wf e

w

kh BqJ

p p r h hF

L Lmn mr

= = +

(19)

whereFis 4, 2, 1.86, 1.78 for n = 1, 2, 3, 4, respectively; nis number of spokes (laterals); and mis number of elevations or

levels at which laterals are drilled. These models are confined to 1, 2, 3 or 4 laterals. As stated above, for pseudo-steady

state calculation, the correlation proposed by Economides9, equation (16), can be used.

Single Gas Well

The available analytical solutions for single-phase gas well are fewer than those for single-phase oil well. Aronofskyand Jenkins 1developed the solution for vertical single gas well in steady state from the solution of the differential equation

for gas flow through porous media by using non-Darcy factor:


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( )2 2 1424 ln /g

g

e wf g e w g

q khJ

p p ZT r r s Dq= =

+ + (20)

where qg is gas flow rate; g is average gas viscosity; Z is average z factor; and T is temperature. This model can be

converted to the pseudo-steady state correlation by simply replacing ln(re/rw) term with ln(0.472re/rw).

Jones20proposed a similar correlation to that of a single oil case that includes Forchheimer factor,:

2 21g

g

gr wf

qJA B qp p

= =+

(21)

where

1422 ln 0.75egw

rT Z s

rA

kh

+ = (21.1)

12

2

3.161 10 g

w

Z TB

h r

= (21.2)

Odeh 22also derived the similar pseudo-steady state flow equation with noncircular drainage area to his oil flow equation:

( )

6

2 2

703 10

ln 0.75

gg

r wf g g g

q khJ

p p TZ X s D q

= =

+ + (22)

Economides11suggested a steady state flow equation for horizontal wells analogous to equation (11):

( )

( )

2 222

/ 21424 ln ln

/ 2 1

g hg

e wfani ani

g g

w ani

q k hJ

p pa a L I h I h

ZT s DqL L r I

= = + + + + +

(23)

where

v

hani

k

kI = (23.1)

5.05.0

4

2/

25.05.0

2

++=

L

rLa e for er

L9.0

2

< (23.2)

This model can be converted to the pseudo-steady state correlation by replacing ln(re/rw) term with ln(0.472re/rw).

Two-Phase Oil Well

In two-phase flow, only the solutions for flowing oil wells in solution gas drive reservoir are available. Vogel26

proposed an empirical correlation for the inflow performance relationship through a number of history matching simulations:2

,max 1 0.2 0.8wf wf

o or r

p pq q

p p

=

(24)

where qo,maxis absolute open flow potential.

Fetkovich13introduced a similar correlation to improve the deficiency of Vogel correlation which is often not accordancewith field data:

2

,max 1 wf

o or

pq q

p

=

(25)

For the negative values of the ideal flowing bottomhole pressure, Vogel or Fetkovich correlations predict incorrect

behavior, resulting in wrong prediction of IPR. Harrison suggested a correlation that works for both positive or negative

values in place of Vogel and Fetkovich correlations:5

( )1.792 /,max 1.2 0.2

wf rp p

o oq q e =

(26)

These three correlations are applicable to the horizontal wells by employing productivity index of horizontal pseudo-

steady state calculation. Cheng6proposed a versatile correlation to calculate IPR for vertical, slanted, and horizontal wells,based on the results from NIPERs well simulators:


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SPE 116580 7

2

,max 0 1 2wf wf

o or r

p pq q a a a

p p

=

(27)

where a0~ a2are correlation constants for well deviation degree.6

Case StudiesCase Description

The case studies for PI and IPR have been performed to identify the effective range and the sensitivity for thecorrelations, and to provide some insights for the correlation selection. A total of 38 input parameters required for the case

studies were carefully chosen as default values from the literature and were summarized in Table 2. These default valueswere used in all the case studies unless specified otherwise.

For the PI correlations, one variable that would be considered the most important was chosen in each well-deviationcategory, and the case studies have been performed for that variable. Table 3shows the selected variables with the ranges

used in these studies. Table 4shows the detailed case descriptions with the corresponding variables and the figure locations.

Note that the correlations in two-phase category are excluded in these PI case studies because no PI is available due to

inconstant proportionality of the rate for pressure drawdown in the reservoir.For the IPR correlations, the IPR curves for all the correlations in the same category were generated and compared each

other. Table 5 shows the detailed case descriptions for the IPR studies. Considering the result similarities (that will beaddressed in the following section), the results of case studies are differently displayed between PI and IPR: the results of PI

studies follow the order of the well-deviation category; and the results of IPR studies are sorted in the order of the fluid phase

/ time dependence category. The corresponding figure locations for the IPR results are also included in Table5.

Study Results

Productivity Index

Figures 1 through14show the results for case studies in PI. For the vertical well, as shown in Figures 1 through4, the

PI value decreases logarithmically, as the ratio of external boundary radius (re) and well bore radius (rw) increases. This is

because the ratio (re/rw) is placed in natural log term at the denominator. The term ln(re/rw) indicates that the drainage area

assigned to a well has a relatively small impact on the production rate. In particular, it is observed that all the correlations inthe same category agree with one another very well as shown in Figures 2and 4.

Figures 5 through8show the comparison results for the slanted well. As mentioned in the previous section, the slantedskin was combined with the correlations in vertical well geometry. For simplicity, only the first correlation in vertical well

geometry was used for these case studies, because all the vertical correlations are in good agreement as shown in Figures 1

through4: Darcy in steady state single oil; Dake in pseudo-steady state single oil; and Aronofsky and Jenkins in steady state

and pseudo-steady state single gas. In the section below 75, all three skin correlations for deviation are in good agreement.Cinco-Ley and Besson give almost same skin values, while Rogers produces a little conservative result. Above 75, a big

difference is observed between Cinco-Ley and Rogers. In particular, Besson correlates h(pay thickness) andL(horizontal

well length) to calculate (Slanted angle). Therefore, skin value cannot be generated at 90 , because L(horizontal welllength) term should be infinite. Based on these observations, it is recommended that all analytical skin correlations for

slanted wells be used in the range between 0 and 75.Figures 9 through 12 represent the results for horizontal wells. Figure 9 displays the results for four different PI

correlations in single oil horizontal well in steady state. All the results are in good agreement within less than 5% deviation

range. Giger produces the most optimistic PI, followed by Borisove, Renard, and Joshi, in that order. In the case of single

oil horizontal well in pseudo-steady state (Figure 10), the deviation increases up to almost 25%, as horizontal length reaches3000 ft. Joshi still gives the most conservative result, and Babu (and Odeh) and Economides, in that order, produce more

optimistic PI results. For single gas well, there is only one correlation for each state. Figures 11 and12shows the results

which are in the same pattern as single oil well cases.Figures 13 and14show the case study results for multilateral wells. Figure 13displays the results for two different

correlations in single-oil steady state as a function of number of laterals: for planar laterals and for stacked laterals. As thenumber of laterals increase, PI increases logarithmically due to the increased interference between laterals. Anotherobservation is that the PIs for stacked laterals are higher than those of planar laterals. Figure 14 shows the results forEconomides correlation in the cases of 4 and 8 spokes in single-oil pseudo-steady state. The overall trend cannot be

observed because only two data points are available in this correlation. However, it is reasonable to observe that the PI valueat 4 spokes in this case is quite similar to that in pseudo-steady state shown in Figure 13.

I nf low Perf ormance Relationship

The comparison results for the IPR case studies are shown in Figures 15 through 31. Please refer to Table 5for thedetailed case descriptions. Because the same correlation for PI is used in calculating IPR, the conservative or optimistic

trends among correlations are exactly the same. For the oil wells (Figures 15 through22), the results confirm a straight-line

IPR because the oil flow rate is directly proportional to the pressure drawdown. On the other hand, the cases of gas wellshave a curve IPR because the pressure drawdown is expressed in the difference of squared pressure (Figures 23 through28).


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8 SPE 116580

The results for oil flow in two phases are newly added in the IPR case studies. The bubble point pressure is set below

the average reservoir pressure to identify oil-phase and two-phase zones that occurred due to pressure changes during fluidflow. Above the bubble point, the fluid is in oil phase which produces a straight IPR, while a curve IPR is generated below

the bubble point because gas becomes more and more evolved as the pressure decreases. All the results for two-phase oil

wells were generated under the assumption of 1000 psi bubble point in the operating pressure range between 0 to 2500 psi.Four correlations - Vogel, Fetkovich, Harrison, and Cheng are capable of calculating the IPR for vertical and horizontal

wells. For slanted wells, only Cheng is available. From the results, Vogel gives the most optimistic IPR, followed by

Fetkovich, Cheng, and Harrison, in that order.Based on the case studies, the selection guideline has been prepared for analytical IPR or PI correlations as shown in

Table 6. Please note that the guideline is confined to the specific input values and it is only for a reference.

ConclusionsThe following conclusions are made:

Analytical solutions for productivity index (PI) or inflow performance relationship (IPR) have been studied throughcomprehensive literature survey, and summarized by the category of well deviation, fluid phase, and time

dependence. Well deviation is divided into vertical, slanted, horizontal and multilateral. Fluid phase includes singleoil, single gas, and oil flow in two phases, and steady state and pseudo-steady state are the variables for time

dependence.

Case studies for PI or IPR have been performed for the properly selected variables to provide the effective range,trends and limitations of the solutions for each category. Since these case studies have been done for the specific

input parameters, the results may be different for other conditions. This study would be a good reference to the researchers who want to study analytical PI / IPR solutions. No solutions for single-phase gas flow, and oil flow (in two phases) in multilateral well have been found in the

literature. Rigorous models should be used for this case.

No solutions for slanted well geometry in single-phase well have been found. Three skin correlations for slantedgeometry: Cinco-Ley, Besson, and Rogers and Economides are used to be combined with any correlation of verticalgeometry to provide the solutions.

AcknowledgementsThe authors would like to thank Chevron management to permit this publication.

NomenclaturesEnglish Symbols

a = correlation constants

A = drainage areaB = formation volume factorC = shape factor

D = non-darcy factorF = Constant

h = pay thickness or interval

J = productivity index

k = PermeabilityL = Length

m = number of elevations or levelsn = number of spokes

p = Pressure

q = flow rate

r = Radiuss = skin factor

T = Temperature

X = correlations between rwandAor refor various drainage areas and well locationsx, y, z = distance in x, y, or z direction

Z = z-factor

Greek Symbols

= non-darcy factor

= Eulers constant (1.78) = angle

= viscosity

= density


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SPE 116580 9

Subscripts

A = area

D = dimensionless

e = external

g = gash = horizontal

p = perforate

o = oilr = average

R = partial penetration of the horizontal well in the areal plane

v = verticalx = x-direction

y = y-direction

w = wellbore

wf = flowing bottom hole = slanted

References1. Aronofsky, J. S., and Jenkins, R.: A Simplified Analysis of unsteady Radial Gas Flow, Trans. AIME, 201, 149 154,

19542. Barrios, L.: Integrated Computational Model for Overall Skin Factor Estimation, Master Project Report, University of

Tulsa, Summer 20043. Besson, J.: Performance of Slanted and Horizontal Wells on an Anisotropic Medium, SPE 20965, October 19864. Borisove. Ju P.: Oil Production Using Horizontal and Multiple Deviation Wells, Nedra, Moscow, 1964. Translated

into English by J. Strauss, edited by S.D. Joshi, Philips Petroleum Company the Red Library Translation, Bartlesville,

OK, 19845. Brown, K. E.: The Technology of Artificial Lift Methods, Vol. 4, PennWell Publishing Company, The University of

Tulsa, p 7-106. Cheng, A. M.: Inflow Performance Relationships for Solution-Gas-Drive, SPE 20720 presented at the 65th Annual

Technical Conference and Exhibition of the Society of Petroleum Engineering held in New Orleans, LA, September 23-

26, 1990

7. Cinco, H., Miller, F. G., and Ramey, H. J.: Unsteady-State Pressure Distribution Created By a Directionally DrilledWell, JPT, p1392~1400, November, 1975

8. Dietz, D. N.: Determination of Average Reservoir Pressure from Build-up Survey, JPT, 955-959, August, 19659. Economides, M. J., Braud, C.W., and Frick, T.P., :Well Configurations in Anisotropic Reservoirs, SPE 27980

presented at the 1994 University of Tulsa Centennial Petroleum Engineering Symposium held in Tulsa, 29-31 August,

1996

10. Economides, M. J., Deimbacher, F. X., Brand, C. W., and Heinemann, Z. E.: Comprehensive Simulation of HorizontalWell Performance, SPE 20717, 1990, and SPEFE, 418-426, December 1991

11. Economides, M. J., Hill, A. D., and Ehlig-Economides, C.: Petroleum Production Systems, Prentice Hall Inc., UpperSaddle River, 1994

12. El-sayed A. H., and Amro, M. M., :Production Performance of Multilateral Wells, SPE 57542 presented at the 1999SPE/IADC Middle East Drilling Technology Conference held in Abu Dhabi, UAE, 8-10 November 1999.

13. Fetkovich, M. J.: The Isochronal Testing of Oil Wells, SPE 4529, 197314. Giger, F. M., Reiss, L. H., and Jourdan, A. P.: The Reservoir Engineering Aspect of Horizontal Drilling, SPE 13024

presented at the SPE 59th Annual Technical Conference and Exhibition, Houston, Texas, Sept. 16-19, 1984

15. Mathew, C. S. and Russell, D. G.: Pressure Buildup and Flow Tests in Wells, Monograph Series, Society of PetroleumEngineers of AIME, Dallas (1967) 1, 110

16. Renard, G. I., and Dupuy, J. M.: Influence of Formation Damage on the Flow Efficiency of Horizontal Wells, SPE19414 presented at the Formation Damage Control Symposium, Lafayette, Louisiana, Feb. 22-23, 1990

17. Goode, P.A., and Kuchuk, F.J. :Inflow Performance of Horizontal Wells, SPE Reservoir Engineering, p 319-323,August 1991

18. Huang, B., and Prada, M., :Multilateral Wells Modeling and Production Prediction, Chevron presentation material,2002

19. Joshi, S. D.: Horizontal Well Technology PennWell Publishing Company, Tulsa, Oklahoma, 199120. Jones, L., Blount, E., and Glaze, O.: Use of Short Term Multiple Rate Flow Tests to Predict Performance of Wells

Having Turbulence, SPE 6133, October 1976

21. Kampkom, R.: Analysis of Two-Phase Inflow Performance in Horizontal Wells, Master Report, University of Texas atAustin.


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10 SPE 116580

22. Odeh, A. S.: Pseudosteady-State Flow Equation and Productivity Index for a Well with Noncircular Drainage Area,Journal of Petroleum Technology, Vol. 30, No 11, pp 1630-1632, 1978

23. Roger, E., and Economides, M.: The Skin due to Slant of deviated Wells in Permeability-Anisotropic Reservoirs, SPE37068, November 1996

24. Van Der Vlis, A. C., Duns, H., and Luque, R. F.: Increasing Well Productivity in Tight Chalk Reservoir, Proc., vol. 3,pp. 71-78, 10th World Petroleum Congress, Bucharest, Romania, 1979

25. Van Everdingen, A.F.: The Skin Effect and its Influence on the Productive Capacity of a Well, Trans., AIME (1953)

198, 171-7626. Vogel, J. V.: Inflow Performance Relationships for Solution-Gas Drive Wells, JPT, Vol 20, No 1, pp 83-92, 1968

Single Oil Well Single Gas Well Two Phase Oil Well

Steady State Pseudo-Steady Steady State Pseudo-Steady Pseudo-Steady

Vertical

(< 15)

Darcy, 1856 Dake, 1978 Dietz, 1965 (shape

factor)

Jones, 1976 (factor)

Darcy, 1856 (Dfactor)

Odeh, 1978 (shapefactor)

Aronofsky andJenkins, 1954

Aronofsky andJenkins, 1954

Jones, 1976 (factor)

Odeh, 1978 (shapefactor)

Vogel, 1968 Fetkovich, 1973 Harrison Cheng, 1990

Slanted

(< 60)

Cinco-Ley, 1975 Besson, 1990 Rogers and Economides, 1996

Cheng, 1990

Horizontal

(< 90)

Joshi, 1988 andEconomides, 1990

Borisove, 1964

Giger, 1983 Renard and Dupuy,1990

Joshi, 1988 andEconomides, 1990

Economides, 1994

Babu and Odeh,1989

Economides, 1994 Economides, 1994 Vogel, 1968 Fetkovich, 1973 Harrison

Cheng, 1990

Multilateral Borisove, 1984 forplanar laterals

Borisove andClonts-Ramey,1984 for stackedlaterals

Economides, 1994

Table 1: Summary Table for PI or IPR Correlations


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SPE 116580 11

No Variable Description Unit Default Values

1 pe External Boundary Pressure psi 3500

2 pr Average reservoir pressure psi 2500

3 k Average Permeability md 100

4 kv Vertical Permeability md 80

5 kh Horizontal Permeability md 125

6 kx Permeability in the x-direction md 168

7 ky Permeability in the y-direction md 93

8 h Pay thickness ft 70

9 hp Perforate interval ft 50

10 re External boundary radius ft 2980

11 xe Reservoir half length in the direction parallel to the wellbore ft 2640

12 ye Reservoir half length in the direction perpendicular to the wellbore ft 2640

13 CA Shape factor - 31.6

14 Bo Formation volume factor bbl/STB 1.1

15 o Viscosity cp 1.7

16 o Oil density lb/ft3 62.4

17 qo Oil flow rate STB/d 500

18 g Oil viscosity cp 0.0244

19 Z Compressibility factor - 0.945

20 T Temperature R 640

21 g Gas specific gravity - 0.7122 qg Gas flow rate Mscf/d 1000

23 rw Wellbore radius ft 0.328

24 L Horizontal well length ft 1750

25 xw Distance from the side wall boundary perpendicular to the wellbore to the midpoint of wellbore ft 1600

26 yw Distance from the side wall boundary parallel to the wellbore to the center of wellbore ft 1089

27 zw Distance from bottom or top boundary to the center of wellbore ft 35

28 Azimuth of well trajectory to x-axis deg 0

29 s Skin - 0

30 Non Darcy factor 1/ft 0

31 Do Oil non Darcy factor D/STB 0

32 Dg Gas non Darcy factor D/Mscf 0

33 n Number of spokes (laterals) - 1

34 m Number of elevations or levels at which laterals are drilled - 1

35 A' Drainage parameter - 0.750

36 Jo Productivity Index - 2.0

37 Pb (PVT) Bubble pressure psi 1000

38 (slanted) Slanted angle 50

Table 2: Default Input parameters for the case studies

Category Variable Range

Vertical External Boundary Radius (re) / Well Bore Radius (rw) 10 to 100000

Slanted Slanted Angle () 0 to 90

Horizontal Horizontal Well Length (L) 50 to 3000 ft

Multilateral Numbers of Spoke (n) 1, 2, 3, and 4

Table 3: Variables for PI case studies


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12 SPE 116580

Category Case Description Variable Figure Locations

Vertical Single Oil Well in Steady State re/rw Figure 1

Vertical Single Oil Well in Pseudo-Steady State re/rw Figure 2

Vertical Single Gas Well In Steady State re/rw Figure 3Vertical Well

Vertical Single Gas Well In Pseudo-Steady State re/rw Figure 4

Slanted Single Oil Well in Steady-State Figure 5

Slanted Single Oil Well in Pseudo-Steady State Figure 6

Slanted Single Gas Well in Steady-State Figure 7Slanted Well

Slanted Single Gas Well in Pseudo-Steady State Figure 8

Horizontal Single Oil Well in Steady-State L Figure 9

Horizontal Single Oil Well in Pseudo-Steady State L Figure 10

Horizontal Single Gas Well in Steady-State L Figure 11Horizontal Well

Horizontal Single Gas Well in Pseudo-Steady State L Figure 12

Multilateral Single Oil in Steady State n Figure 13Multilateral Well

Multilateral Single Oil in Pseudo-Steady State n Figure 14

Table 4: Detailed case descriptions with the corresponding variables and figure locations for the PI case studies

Category Case Description Figure Locations

Vertical Single Oil Well in Steady State Figure 15

Slanted Single Oil Well in Steady State Figure 16

Horizontal Single Oil Well in Steady State Figure 17

Single Oil Wellin Steady State

Multilateral Single Oil in Steady State Figure 18

Vertical Single Oil Well in Pseudo-Steady State Figure 19

Slanted Single Oil Well in Pseudo-Steady State Figure 20

Horizontal Single Oil Well in Pseudo-Steady State Figure 21

Single Oil Wellin Pseudo-Steady State

Multilateral Single Oil in Pseudo-Steady State Figure 22

Vertical Single Gas Well In Steady State Figure 23

Slanted Single Gas Well in Steady State Figure 24Single Gas Wellin Steady State

Horizontal Single Gas Well in Steady State Figure 25

Vertical Single Gas Well In Pseudo-Steady State Figure 26

Slanted Single Gas Well in Pseudo-Steady State Figure 27Single Gas Well

in Pseudo-Steady State

Horizontal Single Gas Well in Pseudo-Steady State Figure 28

Vertical Two Phase Oil Well In Pseudo-Steady State Figure 29

Slanted Two Phase Oil Well in Pseudo-Steady State Figure 30Two Phase Well

in Pseudo-Steady StateHorizontal Two Phase Oil Well in Pseudo-Steady State Figure 31

Table 5: Detailed case descriptions with corresponding figure locations for the IPR case studies


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SPE 116580 13

Category Conservative Optimistic

Vertical Single Oil Well in Steady State Darcy

Vertical Single Oil Well in Pseudo-Steady State Darcy Dake / Dietz / Jones / Odeh

Vertical Single Gas Well In Steady State Aronofsky

Vertical Single Gas Well In Pseudo-Steady State Odeh Aronofsky / Jones

Vertical Two Phase Oil Well In Pseudo-Steady State Harrison Fetkovich Vogel / Cheng

Slanted Single Phase Well Rogers Cinco / Besson

Slanted Single Two Phase Oil Well in Pseudo-Steady State Cheng

Horizontal Single Oil Well in Steady-State Joshi Renard Borisove Giger

Horizontal Single Oil Well in Pseudo-Steady State Joshi Babu Economides

Horizontal Single Gas Well Economides

Horizontal Two Phase Oil Well in Pseudo-Steady State Harrison Cheng Fetkovich Vogel

Multilateral Single Oil in Steady State Borisove for Planar Borisove for Stacked

Multilateral Single Oil in Pseudo-Steady State Economides

Table 6: Selection Guideline for PI/IPR correlations


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14 SPE 116580

0

2

4

6

8

10

12

14

10 100 1000 10000 100000

External Boundary Radius (re) / Well Bore Radius (rw)

Productivity

Index,STB/(day-psi)

Darcy

Figure 1: Productivity Index as a variable of external boundaryradius (re) / well bore radius (rw) for vertical single-oilwell in steady state

0

2

4

6

8

10

12

14

16

18

10 100 1000 10000 100000


ProductivityIndex,

STB/(day-psi)

Dake

Dietz (Shape factor)

Jones (Beta factor)

Darcy (D factor)

Odeh (Shape factor)

Figure 2: Productivity Index as a variable of external boundaryradius (re) / well bore radius (rw) for vertical single-oilwell in pseudo-steady state

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

10 100 1000 10000 100000External Boundary Radius (re) / Well Bore Radius (rw)

Productiv

ityIndex,MSCF/(day-psi)

Aronosfsky

Figure 3: Productivity Index as a variable of external boundaryradius (re) / well bore radius (rw) for vertical single-gaswell in steady state

0

0.05

0.1

0.15

0.2

0.25

10 100 1000 10000 100000


ProductivityIndex,

MSCF/(day-psi) Aronofsky

Jones

Odeh

Figure 4: Productivity Index as a variable of external boundaryradius (re) / well bore radius (rw) for vertical single-gaswell in pseudo-steady state

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80

Slant Angle, degree

ProductivityIndex,STB/(day-psi)

Darcy (IPR) + Cinco (Skin)Darcy (IPR) + Besson (Skin)

Darcy (IPR) + Rogers (Skin)

Figure 5: Productivity Index as a variable of slanted angle () forslanted single-oil well in steady state

0

2

4

6

8

10

12

14

0 20 40 60 80

Slant Angle, degree


Dake (IPR) + Cinco (Skin)

Dake (IPR) + Besson (Skin)

Dake (IPR) + Rogers (Skin)

Figure 6: Productivity Index as a variable of slanted angle () forslanted single-oil well in pseudo-steady state


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SPE 116580 15

0

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80

Slant Angle, degree

ProductivityIndex,

MSCF/(day-psi) Aronofsky (IPR) + Cinco (Skin)

Aronofsky (IPR) + Besson (Skin)

Aronofsky (IPR) + Rogers (Skin)

Figure 7: Productivity Index as a variable of slanted angle () forslanted single-gas well in steady state

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 20 40 60 80

Slant Angle, degree

ProductivityIndex,MSCF/(day-psi) Aronofsky (IPR) + Cinco (Skin)



Figure 8: Productivity Index as a variable of slanted angle () forslanted single-gas well in pseudo-steady state

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500 3000

Horizontal Well Length, ft

ProductivityIndex,ST

B/(day-psi)

Joshi

Borisove

Giger

Renard

Figure 9: Productivity Index as a variable of horizontal well length(L) for horizontal single-oil well in steady state

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500 3000


Productivity

Index,STB/(day-psi)

Joshi

Economides

Babu and Odeh

Figure 10: Productivity Index as a variable of horizontal well length(L) for horizontal single-oil well in pseudo-steady state

0

0.05

0.1

0.15

0.2

0.25

0.3

0 500 1000 1500 2000 2500 3000Horizontal Well Length, ft

ProductivityIndex,MSCF/(day-psi) Economides

Figure 11: Productivity Index as a variable of horizontal well length(L) for horizontal single-gas well in steady state

0

0.05

0.1

0.15

0.2

0.25

0.3

0 500 1000 1500 2000 2500 3000


ProductivityIndex,MS

CF/(day-psi) Economides

Figure 12: Productivity Index as a variable of horizontal well length(L) for horizontal single-gas well in pseudo-steady state


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16 SPE 116580

10

12

14

16

18

20

22

24

26

1 1.5 2 2.5 3 3.5 4

Number of Spokes (Laterals)

Productivity

Index,STB/(day-psi)

Borisove for planar laterals

Borisove for stacked laterals

1 2 3 4

Figure 13: Productivity Index as a variable of number of spokes (n)for multilateral single-oil well in steady state

23.2

23.4

23.6

23.8

24.0

24.2

24.4

24.6

4 4.5 5 5.5 6 6.5 7 7.5 8

Number of Spokes (Laterals)


Economides

4 8

Figure 14: Productivity Index as a variable of number of spokes (n)for multilateral single-oil well in pseudo-steady state

0

500

1000

1500

2000

2500

3000

3500

4000

0 2000 4000 6000 8000 10000 12000

Oil Flow Rate, BBL/DAY

FlowingBottomHolePressure,psia

Darcy

Figure 15: Inflow Performance Relationship for vertical single-oilwell in steady state

0

500

1000

1500

2000

2500

3000

3500

4000

0 2000 4000 6000 8000 10000 12000 14000


FlowingBotto

mHolePressure,psia Darcy (IPR) + Cinco (Skin)

Darcy (IPR) + Besson (Skin)

Darcy (IPR) + Rogers (Skin)

Figure 16: Inflow Performance Relationship for slanted single-oilwell in steady state

0

500

1000

1500

2000

2500

3000

3500

4000

0 10000 20000 30000 40000 50000 60000


FlowingBottomHolePressure,psia

JoshiBorisoveGigerRenard

Figure 17: Inflow Performance Relationship for horizontal single-oilwell in steady state

0

500

1000

1500

2000

2500

3000

3500

4000

0 10000 20000 30000 40000 50000


FlowingBottomH

olePressure,psia Borisove for planar laterals

Borisove for stacked laterals

Figure 18: Inflow Performance Relationship for multilateral single-oil well in steady state


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SPE 116580 17

0

500

1000

1500

2000

2500

3000

0 2000 4000 6000 8000 10000


FlowingBotto

mH

olePressure,psia Dake

Dietz (Shape factor)

Jones (Beta factor)

Darcy (D factor)

Odeh (Shape factor)

Figure 19: Inflow Performance Relationship for vertical single-oilwell in pseudo-steady state

0

500

1000

1500

2000

2500

3000

0 2000 4000 6000 8000 10000 12000


FlowingBottomHolePressure,psia Dake (IPR) + Cinco (Skin)

Dake (IPR) + Besson (Skin)

Dake (IPR) + Rogers (Skin)

Figure 20: Inflow Performance Relationship for slanted single-oilwell in pseudo-steady state

0

500

1000

1500

2000

2500

3000

0 10000 20000 30000 40000 50000


FlowingBotto

mH

olePressure,psia Joshi

Economides

Babu and Odeh

Figure 21:Inflow Performance Relationship for horizontal single-oilwell in pseudo-steady state

0

500

1000

1500

2000

2500

3000

0 10000 20000 30000 40000 50000


FlowingBotto

mHolePressure,psia Economides

Figure 22: Inflow Performance Relationship for multilateral single-oil well in pseudo-steady state

0

500

1000

1500

2000

2500

3000

3500

4000

0 100000 200000 300000 400000 500000

GAS Flow Rate, MSCF/DAY

FlowingBottomHolePressure,psia Aronosfsky

Figure 23: Inflow Performance Relationship for vertical single-gaswell in steady state

0

500

1000

1500

2000

2500

3000

3500

4000

0 100000 200000 300000 400000 500000 600000

GAS Flow Rate, MSCF/DAY

FlowingBottomHolePressure,psia Aronofsky (IPR) + Cinco (Skin)



Figure 24: Inflow Performance Relationship for slanted single-gaswell in steady state


18/19


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SPE 116580 19

0

500

1000

1500

2000

2500

3000

0 5000 10000 15000 20000 25000 30000 35000


FlowingBotto

mHolePressure,psia Vogel

Fetkovich

Harrison

Cheng

Figure 31: Inflow Performance Relationship for horizontal two-phase oil well in pseudo-steady state

a comprehensive comparative study on analytical pi-ipr correlations

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