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Copyright 2000, SPE/PS-CIM International Conference on Horizontal Well Technology

This paper was prepared for presentation at the 2000 SPE/Petroleum Society of CIM Inter-national Conference on Horizontal Well Technology held in Calgary, Alberta, Canada, 6-8November 2000.

This paper was selected for presentation by an SPE/PS-CIM Program Committee followingreview of information contained in an abstract submitted by the author(s). Contents of thepaper, as presented, have not been reviewed by the Society of Petroleum Engineers or thePetroleum Society of CIM and are subject to correction by the author(s). The material, aspresented, does not necessarily reflect any position of the Society of Petroleum Engineers, thePetroleum Society of CIM, their officers, or members. Papers presented at SPE/PS-CIM

meetings are subject to publication review by Editorial Committees of the Society of PetroleumEngineers and Petroleum Society of CIM. Electronic reproduction, distribution, or storage ofany part of this paper for commercial purposes without the written consent of the Society ofPetroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstractof not more than 300 words; illustrations may not be copied. The abstract must containconspicuous acknowledgment of where and by whom the paper was presented. WriteLibrarian, SPE, P.O. Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

Abstract

This paper presents a comprehensive study on the productivity

and flow efficiency of horizontal wells completed with

slotted-liners or perforations. The study is based on a semi-

analytical model that couples the flow equations in the

reservoir and wellbore. The reservoir model takes into account

the 3D convergence of flow around perforations and slots. The

wellbore flow model considers the pressure losses inside the

horizontal well and the effect of axial influx at the perforations

and slots. A new, experimental apparent friction factor

correlation is used for horizontal wellbore flow computations

with perforations and slots. The model is capable of

incorporating the effects of selective completion and non-

uniform skin distribution.

The results of this study indicate that software based on

detailed semi-analytical models can provide a powerful tool to

design, predict, and optimize horizontal well completions. It is

also shown that horizontal wells deserve genuine guidelines to

optimize their completions. For example, horizontal wells are

shown to require significantly lower slot and perforation

densities to accomplish optimum PI compare to vertical wells.

Similarly, in horizontal wells, the effect of slot or perforationphasing becomes more important as the anisotropy of the

formation increases.

Introduction

Horizontal wells are one of the most important strategic tools

in petroleum exploitation.1 As a result of the advances in

drilling and completion technologies in the last two decades,

the efficiency and economy of horizontal wells have

significantly increased. Today, horizontal well technology is

applied more often and in many different types of formations

The state of the art applications of horizontal well technology

require better completion designs to optimize production rates

long-term economics, and ultimate producible reserves.

Horizontal well completions can be categorized as

natural completion, sand-control completion, and stimulation

completion. Natural completion includes open-hole, slotted-linear, and cased and perforated completions. Sand-screens

prepacked screens, and gravel packing are the completions

used for sand-control. Stimulation completion includes

completion with hydraulic fracturing and fracturing with

gravel packing (fracpack or stimpack). All of these completion

methods have been used in practice under different reservoir

conditions.2,3,4

In a horizontal well, depending upon the completion

method, fluid may enter the wellbore at various locations and

at various rates along the well length. Fig. 1 illustrates the

interplay between the pressure and flux distribution along the

wellbore through the completion openings. The complex

interaction between the wellbore hydraulics and reservoir flowperformance depends strongly on the distribution of influx

along the well surface and it determines the overal

productivity of the well. Therefore, the optimization of wel

completion to improve the performance of horizontal wells is

a complex but very practical and important problem. The

complexities of the numerical simulation of horizontal wel

completions make analytical models extremely attractive

However, the inherent difficulties of the analytical solutions

caused by the complex flow geometries, excessive number o

perforations or slots, and non-uniform distribution of flux

along the horizontal well calls for the challenging task of

developing efficient computational algorithms.

In this study we concentrate on the natural completion of

horizontal wells with perforations and slotted liners. We

develop a comprehensive semi-analytical model to investigate

the effect of well completion on horizontal well performance

The model couples the reservoir and wellbore flow equations

For the reservoir part, perforations and slots are modeled as

line sources. The method of sources and sinks and principle o

superposition are used to derive the expression for the

reservoir pressure drop along the well surface. These

expressions take into account the 3D convergence of flow

SPE/Petroleum Society of CIM 65516

Performance of Horizontal Wells Completed with Slotted Liners and PerforationsYula Tang, SPE, U. of Tulsa, Erdal Ozkan, SPE, Colorado School of Mines, Mohan Kelkar, SPE, U. of Tulsa, Cem Sarica,SPE, Pennsylvania State U., Turhan Yildiz, SPE, U. of Tulsa

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2 Y. TANG, E. OZKAN, C. SARICA, M. KELKAR, AND T. YILDIZ SPE/PS-CIM 65516

toward perforations and slots. The wellbore flow model

considers the pressure losses inside the well and the effect of

influx from the reservoir into the horizontal well. A newexperimental apparent friction factor correlation is utilized to

incorporate the effect of ingress from the slots or perforations.

The final coupling of the reservoir and wellbore flowcomponents is accomplished by using the continuity

requirement of the pressure and flux at the reservoir-wellboreinterface. The coupled flow equation is, then, discretized and

the resulting nonlinear matrix problem is solved by an iterativetechnique. Selective completion and non-uniform skin

distribution are included in the model. We have also furnished

the computational code for the model with a graphical user

interface for pre- and post-processing and with windows

programming features. The end product is a user-friendlysoftware, called HORCOM,5,6 that can be used as a horizontalwell design, prediction, and completion optimization tool.

Using the semi-analytical model, this study presents results

to demonstrate the influence of perforations and slots, partial

completion, non-uniform skin distribution, and wellbore

hydraulics on horizontal well performance. Guidelines to

optimize the completion of horizontal wells are also

documented.

BackgroundThe petroleum industry started investigating horizontal

wellbore hydraulics in the late 1980s. New friction factor

correlations for horizontal wellbore were proposed by Refs. 7-

12. The apparent friction factor correlations obtained from the

experimental works of Yuan et al.9,10 and Jang11 that can

account for the influences of perforations and slots are used in

this study for the wellbore hydraulics calculations.The effect of wellbore hydraulics on the performance of

open-hole completed horizontal wells has been investigated in

several studies.13-16 Ozkan et al.14 used the physical couplingconditions (pressure and flux continuity at the well surface) to

model the performance of open-hole completed horizontal

wells. In this study, we extend the work of Ozkan et al toincorporate the effects of slots and perforations.

Ref. 17-19 investigated the influence of selectivecompletion on horizontal well performance. In those studies,

however, the wellbore pressure losses were neglected.

Landman et al.20studied the optimization of perforation

distribution for horizontal wells. In this study, the perforated

well was treated like a pipe manifold with T-junctionsrepresenting the perforations along the wellbore. A simple

approximation was used for the reservoir flow and the effect

of perforation distribution on flow pattern and mechanism wasnot taken into account. Guevera and Camacho21 investigatedthe performance of horizontal wells in the presence of multi-

phase flow and perforations. They simplified the reservoir andwellbore flow solutions to obtain a tractable computational

algorithm. Ozkan et al.22 investigated the transient pressure

behavior of perforated slant and horizontal wells. Their study

indicated that the density and phasing of perforations used to

complete horizontal wells could not be readily deduced from

the experiences with vertical wells. The 3D model presented

in Ref 22 provides the basis for our long-time asymptotic

solution for perforated horizontal wells.

In this study, using the information available in the

literature, filling the gaps, and improving some of the aspectswe have developed a comprehensive wellbore-reservoir mode

that can be used to investigate the performance of slotted-linecompleted or perforated horizontal wells. We have developed

efficient algorithms to numerically evaluate the complexanalytical expressions that result from the rigorous modeling

of wellbore and reservoir flow conditions. Below, we providea brief introduction of the semi-analytical model used in this

study.

Semi-Analytical ModelAs we noted before, in this study we derived a semi-

analytical flow model that couples wellbore and reservoir flow

equations. Here, we introduce the reservoir and wellbore flowmodels and the coupled flow equation. We also briefly discuss

the apparent friction factor correlation used in the wellbore

flow model and the semi-analytical solution technique of the

coupled flow equation. First, we explain the assumptions used

in our model and introduce the dimensionless variables tha

will be used in the discussions for convenience.

Assumptions and Dimensionless Variables. We consider the

flow of a slightly compressible fluid of constant

compressibility, ct, and viscosity, , in a homogenous bu

anisotropic reservoir. 2D anisotropy is considered; rk and zk

represent the permeabilities in the radial and vertica

directions, respectively. The top and bottom boundaries of the

reservoir are assumed to be impermeable. Cylindrica

reservoir geometry is assumed with the boundaries in the

lateral extent of the reservoir at infinity, closed, or at a

constant pressure equal to the initial pressure, pi. The initiapressure is uniform throughout the reservoir and equal to ip .

We define the following dimensionless variables for

convenience. The dimensionless pressure is defined by

( )ppqB

hkp i

rD =

2.141(1)

and the dimensionless time by

tc

k.t

t

rD 2

0002640

A= . (2)

We define the dimensionless horizontal distances by

A

xxD = ,

A

yyD = , DDD yxr += , (3)

and the dimensionless vertical distance and the formation

thickness, respectively, by

z

rD

k

kzz

A= ,

z

rD

k

khh

A= . (4)

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SPE/PS-CIM 65516 PERFORMANCE OF HORIZONTAL WELLS COMPLETED WITH SLOTTED LINERS AND PERFORATIONS 3

In Eq. 1, q is the total flow rate of the well and in Eqs. 2-4, A

is some characteristic length, which we choose as the half

length of the horizontal well,Lh/2. We also define B as the

formation volume factor, h as the formation thickness, as

the porosity, andct as the total system compressibility. Eqs. 1

and 2 are in practical oil-field units.

We let Lpi denote the length of the i-th perforation andiits inclination angle (with respect to vertical). Then thedimensionless perforation length is defined as

ii

z

rPiPDi sincos

k

kLL 22 +=

A. (5)

The dimensionless flux through the i-th perforation is definedby

q

)L(qq

iPDi

iD

A= , (6)

or the dimensionless flux through the m-th slot of length lm is

defined as

q

lqq mmmD = . (7)

In Eqs. 6 and 7, qi and qm are the fluxes along the i-th

perforation and the m-th slot (b/d/ft), respectively.

Horizontal Well Flow Model. Ref. 14 explains the derivation

of the horizontal well flow equation taking into account the

wellbore hydraulics and coupled with the reservoir flow

equation. Here, we do not elaborate on the derivation of thisequation but present it below for convenience.

]2[

16)()(

'"

Re0 0

Re,,

'

DDhDtt

x x

DhD

ttDDDDwD

dxdxqfN

Dx

C

Nftxptp

D D

=

, (8)

wherepwD is the dimensionless pressure at the heel end of the

well andpD is the pressure at some point xD along the surface

of the well. ChD in Eq. 8 is the dimensionless well conductivitydefined by

h

whD

Lkh

rC

41310395.7 = , (9)

andD is a function of the Reynolds number, NRe, and the

friction factor,f, defined by

fNdN

dfND Re

Re

Re 22 += . (10)

In field units, the Reynolds number is given by

w

Rer

q.N

2101576 = . (11)

ft andNRet in Eq. 8 stand for the friction factor and the

Reynolds number at the heel of the well (based on the total

flow rate, q, from the well).In Eq. 8, qhD is the dimensionless flux defined by the

following relation:

q

Lqq hhhD = . (12)

Here, hq , is the flux entering the well per unit length at a

given point andLh is the horizontal well length.Note that, in Eq. 8, by the continuity of pressure and flux

along the well surface,pD is equal to the reservoir pressure a

the sand-face andqhD is equal to the reservoir flux. Thereforewe substitute the appropriate reservoir solution forpD for an

arbitrary flux distribution, qhD, into Eq. 8. For Eq. 8 to accoun

for the specific details of well completion, the reservoir

solution must consider the convergence of flow toward

completions and the friction factor,f, should take into accoun

the fluid influx at completions. Below, we discuss theappropriate reservoir flow equations and friction factor

correlation to be used in Eq. 8. We must also note here thatEq. 8 is nonlinear and requires an iterative solution technique.

Reservoir Flow Equation. In this study, we use the

pseudoskin concept to simplify the analytical and numerica

treatment of the problem. Because the convergence of flow

toward the wellbore openings (perforations and slots) takesplace in the near vicinity of the well, the outer portions of the

reservoir including the boundaries will not be affected by the

existence of these openings. Therefore, the additional pressure

drop due to flow convergence toward wellbore openings can

be treated as a pseudoskin term.

To obtain the pseudoskin due to completions, we first

derive the pseudoradial flow solutions for perforated andslotted-liner completed horizontal wells in an infinite slab

reservoir as a function of an arbitrary flux distribution. Thencomparing these solutions with the corresponding solution for

the open-hole completion case, we obtain the completion

pseudoskin for the arbitrary flux distribution. Once we have

the completion pseudoskin, we add this to the appropriate

pseudosteady state or steady state solution for the open-hole

case. This yields the dimensionless pressure, pD, for theperforated or slotted-liner completed horizontal well to be

used in Eq. 8 under pseudosteady or steady state conditions.Before we present the analytical expressions, we should

note that one can derive the bounded reservoir solutions for

perforated and slotted-liner completed horizontal wells and

directly substitute them into Eq. 8 forpD. These solutionshowever, are extremely complex and are not suitable for the

iterative solution of Eq. 8.

Completion Pseudoskin. As we noted above, we obtain

the completion pseudoskin from the pseudoradial flow

equations. Because of the length of the derivations, here weonly mention the methods used to derive the solutions and the

final form of the equations. We refer the reader to Refs. 5, 6and 22 for further details.

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To derive the pseudoradial flow equation, we first consider

a single perforation or slot and use the method of sources and

sinks. We, then, use the principle of superposition to add theeffect of the other perforations and slots.6,22 In this model, the

perforations and slots are assumed to be line sources. For

simplicity, we assume that the flux along the i-th perforationor slot, qi (b/d/ft), is uniform. To deal, numerically, with a

large number of perforations and slots along a horizontal well,this is a practical approach. (An alternative approach is to

assume infinite conductivity along each perforation/slot as inRef. 22. This, however, increases the computational difficulty

and the CPU time without any significant difference in

pressure values.)

Figs. 2 and 3 show horizontal wells with multiple

perforations and slots, respectively. Using the methodexplained above, the dimensionless pressure at the j-th

perforation/slot generated by NM number of openings is

obtained as follows:

iD

NM

i

DD qijIijIAtjp ++= =

]),(2),([),(

1

21 , (13)

where

( )80907.0ln2

1+= DtA . (14)

In Eq. (13), NM= NP for perforating completion (NP is the

total number of perforations), andNM= MS for slotted-liner

completion (MS is the total number of slots). ( )ijI ,1 and

( )ijI ,2 represent the effect of the i-th opening on the pressureresponse at thej-th opening (the influence functions).I1andI2are functions of wellbore and perforation/slot geometry, as

well as the reservoir parameters. The appropriate expressions

and computational remarks forI1 andI2 are given in theAppendix.

For computational purposes (see the section on thesolution of the coupled flow equation below), we discretize

the wellbore length into 2M segments and assume uniformpressure in each segment. The segment pressure is obtained

from the arithmetic average of the pressures of all the

perforations/slots in the segment. (The comparison from the

computational results indicates that this assumption is accurateenough for large number of segments). Then, also considering

the mechanical skin on theJ-th segment, )(JSHK , the average

pressure in theJ-th segment can be expressed as

)()()(),( ,

2

1

JSJqSIqAtJp HKhDIJ

M

I

hDDD ++= =(15)

where

)( ,,, IJIJIJ FS += , (16)

JsgmIsgm

Im

Imm

Jn

Jnnj

IJMPMPM

mnjI

__

)(

)(

1

)(

)(

,2

),(1

0

1

0

=

==

, (17)

and

JsgmIsgm

Im

Imm

Jn

Jnnj

IJMPMPM

mnjI

F__

)(

)(

2

)(

)(

,2

),(2

1

0

1

0

= == . (18)

In the above expressions, m0(I) andm1(I) are the starting and

the ending sequential numbers of the openings in the I-th

segment respectively, and n0(J) andn1(J) are the staring and

ending sequential numbers of the openings in the J-th

segment, respectively. MPsgm_I is the number of the openingsin the I-th wellbore segment [MPsgm_I = m1(I)-m0(I)], and

MPsgm_J is the number of openings in the J-th wellbore

segment [MPsgm_J= n1(I)-n0(I)]. qhD is the dimensionless flux

given Eq. 12. Here, qh is the apparent flux along the wellboreNote that the actual flux is only through the perforations/slots

For each segmentI, however, the sum of the fluxes from all of

the openings in this segment can be treated as the apparent

flux. Then, the apparent flux, qh(I), can be related to

perforation or slot flux, respectively, by

)2/(

)()(

_

ML

MPlLIqIq

h

IsgmmPDm

h

= , (19)

)2/(

MS)()(

sgm_I

ML

lIqIq

h

mm

h

= , (20)

where, lm is the slot length in theI-th wellbore segment.

The mechanical skin, )(JSHK , in Eq. 15, is defined as the

dimensionless skin pressure drop scaled by ( hLh ). The skin

pressure drop is a sum of the skin pressure drops caused by

various factors, such as formation damage, crushed zonearound perforations, or blocking of the perforations or slots. In

our model, we allow the mechanical skin to vary from

segment to segment.

Before we present the equation for the openhole

completion case and derive the completion pseudoskinexpression, a final remark is in order. It must be noted that the

brute-force application of the principle of superposition leadsto a solution where the wellbore itself is filled with the porous

medium; that is, the condition that the well surface is a noflow boundary is not satisfied. This, however, is not a

shortcoming for the perforated well solution because the effec

of the wellbore radius becomes negligible for the practica

times of interest. (The assumption here is akin to the line

source, horizontal well assumption and is valid for the sameranges of time.) For the slotted-liner completion case, on the

other hand, a correction may be required in Eq. 15 to take into

account the no-flow wellbore surface. Similar to Ref. 23, an

empirical correction to Eq. 15 may be used to ensure that the

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perforation/slot correlations developed by Yuan et al.9,10 to

compute the apparent friction factor.

Solution of the Coupled Flow Equation (Eq. 15). The

solution of the coupled flow equation given in Eq. 15 should

be obtained by an iterative procedure because it is nonlinear.By dividing the wellbore into 2M segments we obtain the

discrete form of Eq. 15 for the pressure response at the J-thsegment

}8

1)

2

12({

168),()(

2

,,,

1

1

Re,

M

qDqD

MM

Ix

CC

xNftJptp

JhDJIhDIDJ

J

I

hDhD

DJttDDDwD

+

=

=

(35)

SubstitutingpD given by Eq. 33 into Eq. 35 and evaluating the

resulting expression at the center of each segment, we obtain

2M equations in 2M+1 unknowns (pwD and qhD,J). An

additional equation is obtained by using the condition that the

sum of the fluxes from the segments should equal theproduction rate, q, of the well. Then, an iterative procedure is

used to solve the resulting non-linear matrix equation. The

details of the solution procedure are given in Refs. 6 and 14.

Model Verification. We verified our semi-analytical model

by comparing its asymptotic results with those for openhole

completed wells. We used high perforation/slot densities in

our model and obtained reasonably good agreements with thepressure and flux distributions of the corresponding openhole

completion cases. Because of space restrictions, we do not

show these results here. Details can be found in Ref. 6.

Discussion of ResultsHere we discuss the results obtained by using our semi-

analytical model. We consider the effects of variouscompletion parameters on the performances of perforated or

slotted-liner completed horizontal wells.

Effects of Perforation Density, Phasing, and Length. To

investigate the effects of perforation density, phasing, and

length on horizontal well productivity, we use the example

data shown in Table 2. Fig. 4 shows the productivity index(PI) as a function of the perforation density. We can determine

from this figure that the perforation density has significant

effect onPIuntil the density reaches 0.5 shot/ft. Increasing the

density beyond 0.5 shot/ft does not provide significantincrease inPI. Beyond the density of 1 shot/ft, the gain in PI

becomes negligible for practical purposes.Fig. 5 shows the effect of perforation length (penetration)

on PI. Considering the fact that the penetration depth of

shaped charges used in perforating is less than 20 inches, it

can be concluded from Fig. 5 that perforation penetration is

one of the key parameters for horizontal well performance.

The effect of perforation phase angle on PI is shown inTables 3a and 3b. The examination of the results shown inthese tables indicates that perforation phase angle has

negligible effect onPIif the formation is isotropic (Table 3a).

When anisotropy is severe, however, the influence of phasing

becomes noticeable, as shown in Table 3b. For example, for a

given perforation density of 0.1 spf, the PIfor90phasing is

225.8 b/d/psi, while it is 252 b/d/psi for 360 phasing. Ingeneral, 360 or 180 phasing is better than 90 phasing

(assuming that zr kk > ). This is because the late time flow inthe reservoir (pss or ss) takes place mainly in the horizontal

direction and zr kk > . Therefore, the perforations with 360o180phasing are normal to the main flow direction and more

productive compared to the perforations with 90 phasing

(For 90 phasing, only half of the perforations are in the

vertical plane and normal to the main flow direction. The othe

half of the perforations lie in the horizontal plane and receive

less flux.)The above results indicate that the perforation density and

penetration have more influence on PI of the well than that of

the perforation phasing.

Effect of Slot Parameters. To investigate the effect of slo

parameters on the productivity of a slotted-liner completed

horizontal well, we use the example data set in Table 2. For aslotted-liner, we define the slot penetration ratio (SPR) asfollows:

100=lengthwellbore

slotsoflengthlcumulativeSPR (36)

The slot phase angle (phasing) is defined as the angle betweentwo adjacent slot arrays (slot clusters) on the projection plane

that is perpendicular to wellbore axis, measured in degrees.

The Effect of Slot Phasing Angle. Fig. 6 shows tha

phasing angle has significant influence onPI, especially when

the formation is anisotropic. The smaller the phase angle, the

bigger the PI. Note that the effect of the phasing angle on thePIof the slotted-liner completion is more important compared

to the perforated horizontal well case examined above. This isbecause perforations are, normally, in spiral configuration

while slots are in staggered alignment. For perforating case

changing the phase angle does not change the total number o

perforations. For a slotted-liner, however, decreasing the

phase angle increases the number of slots in each slot ring (o

slot plane). This yields more uniform distribution of flux along

the well and thus less pressure loss.

The effect of phasing is also dependent on the slopenetration ratio (SPR). We have observed that the effect oslot phasing becomes more significant when the slo

penetration ratio is small, as shown in Fig. 7. Focompleteness, we have also checked the influence of slot

clustering on well PI. (We define the slot clustering by the

number slots concentrated at one location. Fig. 3, for exampleshows slot clusters with three slots at each location.) Our

computations indicated that slot clustering has little effect on

productivity.

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Effect of Slot Size. Fig. 8 shows the effect of slot size on

PIfor a fixed slot penetration ratio (SPR) of 50%. We can see

from this figure that decreasing the slot size increases the PI.Note that because we fix SPR in this example, when we have

very long slots, we have long intervals between them. This

causes severely nonuniform distribution of flux along the welllength (this is akin to selective completion effect). On the

other hand, when we use shorter slots, they are distanced byshorter intervals that yield more uniform distribution of flux

along the well length with shorter radius of flow convergencezone around the slots (the smaller the radius of the flow

convergence zone, the smaller the completion pseudoskin).

Effect of Slot Length, Interval, and Penetration Ratio.

Fig. 9 shows that increasing the slot length increases thePIfora fixed slot interval of 6.25. The gain in the PI, however,

becomes marginal when the slot length exceeds 6.25 (SPR =

50%). On the other hand, Fig. 10 shows that thePIdecreases

when the slot interval is increased for a fixed slot length of

6.25.

We can combine the results shown in Figs. 9 and 10 by

defining a PI recovery (PIR) term as follows

100max

=PI

PIPIR (37)

Fig. 11 shows that PIR increases with slot penetration ratio

(SPR). Beyond 50% SPR, the gain in PIR becomes

insignificant. When SPR is reduced below 40%, the PIR

decreases sharply.

Effect of Partial Completion and Perforation Strategy. For

slotted-liner completed or perforated horizontal wells, we mayhave some blind-pipe segments that do not permit flow from

the reservoir. This is akin to selective completion of the well.

Below we discuss the effect of selective completion on

horizontal well performance.

We use the data from the Troll Field given in ref. 14, and

assume pseudo-radial flow with Q = 30000 b/dat t = 300 hrs.

We consider a perforating completion with the perforation

length, Lp = 12. Figure 12 shows the results for variouscompletion scenarios. In Case A, we space perforationsuniformly along the full length of the wellbore with a density

of 0.25 spf. The corresponding PI is obtained to be 503

b/d/psi. In Case B, we divide the well length into five equal

length intervals. Sections 1, 3, and 5 are open with uniform

perforation density ofSPF= 0.25 in each interval. Sections 2and 4 are closed. Compared to Case A, we obtain a lowerPI

of473 b/d/psi.It is also possible to improve PI by increasing perforation

density on one of the open intervals, as shown in Cases C, D,

and E of Fig. 12. As expected, increasing the perforation

density close to the heel end of the well yields good results

(the PI approaches that of the fully open well case). On theother hand, using high perforation density in the middle or the

toe end of the well does not significantly improve the PI.

Figure 13 shows the flux distribution characteristics for

each of the situations examined in Fig. 12. The flux

distribution inside each segment is a skewed U-shaped curve

The segments closer to the heel end of the well receive moreflux because of wellbore hydraulics. Thus, increasing the

number of perforations near the heel end of the well yields thehigher gain in PI. From the examination of the results shown

in Fig. 13, we reach the conclusion that both the open segmenlocations and perforating strategy have significant effect on

the productivity of the horizontal well.

ConclusionsIn this study, we have developed detailed reservoir flow

models for perforated and slotted-liner completed horizonta

wells. These models allow us to rigorously couple reservoirand wellbore flow performances and capture the effect of the

complex interaction between the reservoir and wellborethrough limited openings on the well surface. By using the

new model, we developed useful guidelines for horizontal wel

completions. These guidelines are documented below:

(1) Both perforation penetration (length) and density havesignificant impact on horizontal-well productivity

Beyond the density of 0.5 shot/ft, the gain in PI becomesmarginal. This is significantly lower from the perforation

densities required for vertical wells.

(2) For perforating in isotropic formations, we recommend to

use 90 phasing because it has the smaller pressure dropdue to flow convergence around perforations and less

frictional pressure loss in the wellbore. For severely

anisotropic reservoirs, perforating with 360 or 180phasing along vertical direction yields the larges

productivity. This is also different from the perforation

practices for vertical wells

(3) For slotted-liner completion, the use of small phasing

angles, such as 30, 60, or 90 is recommended. As theanisotropy becomes more severe, smaller phasing anglesshould be used.

(4) The slot penetration ratio (slotted section length over thetotal section length) has significant effect on the

productivity of the horizontal well. The slot-penetration

ratio should approach 50% to guarantee sufficient

productivity.

(5) Horizontal well productivity is also significantly affectedby the slot size. Under fixed slot penetration ratio, smalle

slot size should be preferred.(6) Using higher perforation density and deeper penetration

near the heel end of the well helps to obtain the highest

gain of productivity.The above guidelines are different from those for vertica

wells and also serve as a justification for this research. We

finally note that the completion optimization software

developed in this study proves to be a practical and valuable

tool for well completion engineers.

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References1. Biglarbigi, K., Mohan, H., Ray, R., and Meehan, D.: Potential

for the Horizontal Well Technology in the U.S., JPT (May

2000), 23-31.2. Navarro, J. B.: Slotted-Liner Completions used in the First

Horizontal Wells in Mexico, paper SPE 37110 presented at the1996 SPE International Conference on Horizontal Well

Technology held in Calgary, Canada, Nov. 18-20.

3. Kleppa, E., Svane, E., and Varhaug, H.: Innovative Live WellPerforating System Used in the Statfjord Field, paper SPE50588 presented at the 1998 SPE European Petroleum

Conference held in The Hague, The Netherlands, Oct. 20-22.4. Foster, J., Grigsby, T., LaFontaine, J.: The Evaluation of

Horizontal Completion Techneques for the Gulf of Mexico.Where have We and Where Are We Going, paper SPE 53926

presented at the 1999 SPE Latin American and CaribbeanPetroleum Engineering Conference held in Caracas, Venezuela,

April 21-23.5. Ozkan, E., Sarica, C., Kelkar, M., and Tang, Y: Optimization of

Horizontal Well Completion, Joint Industry Project Report, TheUniversity of Tulsa, Tulsa, OK, Feb. 2000.

6. Tang, Y.: Optimization of Horizontal Well Completion, Ph.D.dissertation, The University of Tulsa, Tulsa, OK (to be

completed).7. Asheim, H., Kolnes, J., and Oudeman, P.: "A Flow Resistance

Correlation for Completed Wellbore," Journal of PetroleumScience and Engineering, 8 (1992), 97-104.

8. Kloster, J.: Experimental Research on Flow Resistance inPerforated Pipe, M.S. Thesis, Norwegian Institute of

Techjnology, 19909. Yuan, H., Sarica, C., and Brill, J.: Effect of Perforation Density

on Single-Phase Liquid Flow Behavior in Horizontal Wells,SPEPF (Aug. 1999), 203-209.

10. Yuan, H., Sarica, C., and Brill, J.: Effect of CompletionGeometry and Phasing on Single-Phase Liquid Flow Behavior inHorizontal Wells, paper SPE 48937 presented at the 1998 SPE

Technical Conference and Exhibition held in New Orleans, LA,Sept. 27-30.

11. Jang, W.: An Investigation of the Effects of CompletionsGeometry on Single-Phase Liquid Behavoirs in Horizontal

Wells, M.S Thesis, The University of Tulsa, Tulsa, OK, 199912. Ouyang, L. B. and Aziz, K.: A Mechanistic Model for Gas-

Liquid Flow in Pipes with Radial Influx or Outflux, paper SPE56525 presented at the 1999 SPE Annual Technical Conference

and Exhibition, Houston, TX, Oct. 3-6, 1999.13. Dikken, B.J.: Pressure Drop in Horizontal Wells and Its Effect

on Production Performance,JPT(Nov. 1990) 1426-1433.14. Ozkan, E., Sarica, C., Haci, M.: Influence of Pressure Drop

along the Wellbore on Horizontal- Well Productivity, SPEJ(Sep. 1999), 288-301.

15. Suzuki, K.: Influence of Wellbore Hydraulics on HorizontalWell Pressure Transient Behavior, SPEFE (Sept. 97) 175.

16. Penmatcha, V., Aziz, K.: Comprehensive Reservoir/WellboreModel for Horizontal Wells, SPEJ (Sep. 1999), 224-23417. Kamal, M. M., Buhidma, I., M., Smith, S. A., and Jones, W. R.:

Pressure Transient Analysis for a Well with Multiple

Horizontal Sections, paper SPE 26444 presented at the SPE1993 Annual Technical Conference and Exhibition, Houston,

TX, Oct. 3-6, 1993.18.Yildiz, T., Ozkan, E.: Transient Pressure Behavior of Selectively

Completed Horizontal Wells, paper SPE 28388 presented at theSPE 1994 Annual Technical Conference and Exhibition held in

New Orleans, LA, Sept. 25-28.

19. Retnanto, A., Economides, M. J., Ehlig-Economides, C. A, andFrick, T. P.: Optimization of the Performance of Partially

Completed Horizontal Wells, paper SPE 37492 presented at theSPE Production Operations Symposium, Oklahoma City, OK

March. 9-11, 1997.20. Landman, M.J., Goldthorpe, W.H.: Optimization of Perforation

Distribution for Horizontal Wells, paper SPE 23005 presentedat the SPE Asia-Pacific Conference held in Perth, Western

Australia, Nov. 4-7, 1991.21. Gonzles-Guevara, J. A. and Camacho-Velzquez, R.: A

Horizontal Well Model Considering Multiphase Flow and thePresence of Perforations, SPE 36073 presented at the 4th Latin

American and Caribbean Petroleum Engineering ConferencePort of Spain, Trinidad and Tobago, April 23-26, 1996.

22. Ozkan, E., Yildiz, T., and Raghavan, R.: Pressure-TransienAnalysis of Perforated Slant and Horizontal Wells, paper SPE

56421 presented at the 1999 SPE Annual Technical Conferenceand Exhibition held in Houston, TX, Oct. 3-6.

23. Spivak, D., and Horne, R.N.: Unsteady-State Pressure Responsedue to Production with a Slotted Liner Completion, paper SPE

10785 presented at the 1982 SPE California Regional MeetingSan Francisco, CA, March. 24-26.

AcknowledgmentsThe support provided by the Department of Energy

Minerals Management Service, and the member companies of

the Joint Industry Project titled Optimization of HorizontaWell Completion is gratefully acknowledged. Parts of this

paper will appear in the Ph.D. dissertation of Yula Tang.

Nomenclaturea, b = defined by Eqs. (A-17) and (A-18)a, b, and Cn = coefficients in friction factor correlation defined

in Eq. (34)

A = defined by Eq. (14) for pseudo-radial flow

A0= defined by Eq. (29) for pss and ss flow

B =formation volume factor, bbl/stbChD = dimensionless well conductivity defined by Eq. 9Cm1 and Cm2 = defined by Eqs. (A-15) and (A-16) or by Eqs

(A-33) and (A-34)

tc =total compressibility, psi-1

d= pipe diameter, ft

D = defined by Eq. 10

E1 = defined by Eq. (A-6)

f = fanning friction factor

cF1 = defined by Eq. (A-40)

IJF , = defined by Eq. 18

oIJF , = defined by Eq. 25 for openhole, pseudo-radial flow

h = formation thickness, ft

i = the i-th perforation/slot sourth

I = the I-th wellbore section

1I, 2I =influence functions for openings defined by Eqs. A-1,

A-2,

100I , 101I , and 102I =defined by Eqs. A-12 ~ 14 for

computingI1

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11I , 12I = defined by Eqs. A-29, A-30,

20I =defined by Eq. A-32,

j = thej-th perforation/slot observation point

J = theJ-th wellbore section

Ki1(x) = function for infinite integration ofK0, defined by Eq.

(A-31)

rk =permeability in radial direction, md

zk =permeability in z-direction, md

0K = modified Bessel function of the first kind of order 0

A = reference length of the system, ft

mA = the m-th slot length, ft

hL = horizontal well length, ft

piL = the i-th perforation length, ft

M= half number of wellbore sections

m0(I), m1(I) = the starting and ending sequential numbers of

the openings in the I-th segment

MPsgm_I = the number of openings in the I-th wellbore segment

MS= total slot numbern0(J), n1(J) = the staring and ending sequential numbers of the

openings in the J-th segment,

NP = total perforation number

NRe = Reynolds number

p =pressure, psi

PI = productivity index, b/d/psi

PIR = PI recovery ratio defined by Eq. (37)

wp =bottomhole pressure at heel end, psi

q = well total production rate, stb/dqi or qm = the i-th perforation or m-th slot flux, b/d/ft

qh = the wellbore flux, b/d/ft

rw or Rw = wellbore radius, ft

r or R = radial distance in horizontal plane, ft

RwDi = the transformed dimensionless wellbore radius at the i-

th perforation (defined by Eq. A-7 )Rw0D = the transformed dimensionless wellbore radius

calculated from Eq. A-7 with i = 0

re = reservoir outer boundary radius, ft

HKS = Hawkins mechanical skin

IJS , = defined by Eq. 16 for pseudo-radial flow

oIJS , = defined by Eq. 30 for pss or ss flow

SPR = slot penetration ratio defined in Eq. (36)

tS = total skin including the mechanical skin and the

pseudoskino

IJT , = defined by Eq. 16 for openhole, pseudo-radial flow

t = time, hrx, y, z = coordinates in the system, ft

wz = location of well center in vertical direction, ft

Greek symbols

= porosity= the slot/perforation density in Eq. (34), shots/ft

= viscosity, cp

= density, lbm/ft3

ji =angle defined by Eq. (A-8) and Fig. 2A , rad

oIJ, = defined by Eq. 24

IJ, = defined by Eq. 17

= defined by Eq. A-15o

IJ, = defined by Eq. 31 for pss flow

= defined by Eq. (A-19)i = inclination angle of the i-th perforation, rad

'i =transformed inclination angle (Eq. A-3), rad

Subscripts and superscriptsD=dimensionless

i= initial

w=wellbore

p=perforation

t = total

AppendixInfluence Functions for Perforations and Slots

As noted in the text, the influence functions, ( )ijI ,1 and

( )ijI ,2 , represent the effect of the i-th opening on thepressure response at thej-th opening given by Eq. 13.I1andIare functions of wellbore and perforation/slot geometry, as

well as the reservoir parameters. Below, we provide the

expressions for the influence functions for perforated and

slotted-liner completed wells.

Influence Functions for Perforations. For computationa

convenience, we distinguish between the cases where the

influencing perforation is inclined and vertical.

Inclined Perforation. When the influencing perforation isinclined, I1 and I2 are given by the following relations

respectively:

( ) ( ) DL

L

DjiipDi

rdRL

ijI

ipDi

ipDi

=

'

'

sin5.

sin5.

1 lnsin

1, (A-1)

and

( )

=

=

D

jDD

D

D

n

L

L

DjiDipDi

h

znrd

h

zn

Rh

nK

LijI

ipDi

ipDi

coscos

sin

1,

1

sin5.

sin5.

02

'

'

(A-2)

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The terms involved in Eqs. A-1 and A-2 are defined by the

following relations:

= i

z

ri

k

k tantan 1' , (A-3)

where i is the inclination angle of the i-th perforation with

respect to vertical, hD and pDiL are defined by Eqs. 4 and 5,respectively, in the text

jiijDDijDDijD rrrrR cos2'22' ++= , (A-4)

2

1

2)( Exxr DiDjijD += , (A-5)

''

1 sin)2

(sin)2

( jPDj

jwDi

iPD

iwD

LR

LRE ++= , (A-6)

ii

z

rwwDi

k

kRR 22 sincos +=

A

, (A-7)

where Rw is the radius of the wellbore and A is the

characteristic length,

[ ]

=

=

02

0/)(tan

1

11

1

Eif

EifExx DiDjji

, (A-8)

'cos)5.0( jpDjjWDDwDj LRzz ++= , (A-9)

and

)2

(coscot'''' iPD

iDwiDiDwD

LRrzz +++= . (A-10)

In Eqs. A-9 and A-10, zwD is defined by Eq. 4 in the text with

zw denoting the elevation of the horizontal well axis from the

bottom boundary of the formation.

The numerical evaluation of functionsI1 andI2 may not bestraightforward.I2 given in Eq. A-2 includes the integration of

the product of a modified Bessel function and a cosine

function. No closed form analytical expression can be derived

for this integration. We have used Chebyshev polynomials to

numerically evaluate this integral. For functionI1 given in Eq.A-1, however, a closed form expression can be derived as

follows:

[ ] 11

101

2

102101

'1

)25.0(

sin2

1

),(

m

m

Cx

Cx

mpDm

IabII

LmnjI

=

=

= , (A-11)

where

)ln(2

101 xbxaxI ++= , (A-12)

)ln(5.022

102 xbxabxI ++= , (A-13)

=+

+

=

02

2

02

tan2 1

100

ifxb

ifxb

I , (A-14)

'sin5.0 mmPDml LC = , (A-15)

'

2 sin5.0 mmPDm LC = , (A-16)

2

,njDmra = , (A-17)

njmnjDmrb ,, cos2 = , (A-18)

and

24 ba = . (A-19)

Vertical Perforation. For vertical perforations ( i and

thus 0=i ), the influence functions are obtained byevaluating the limiting forms of Eqs. A-1 and A-2 as 0iThe resulting expressions are as follows:

( ) )DjiRijI ln,1 = , (A-20

and

( )

+

=

=

D

pDi

D

pDiDwwD

D

jD

n

Dji

DipDi

D

h

Ln

h

LRzn

h

znR

h

nK

nL

hijI

2sin

5.0cos

cos12

,

0

1

02

, (A-21)

where,Rw0D stands for the transformed dimensionless wellbore

radius calculated from Eq. A-7 with i = 0.

Variables involved in Eqs. A-20 and A-21 are defined by

Eqs. A-4 ~ A-10 by using 0=i . For example, RDji can be

calculated directly by A-5 (rDji), and'

Dz by A-10 (without the

second term on the right hand side of Eq. A-10). There is a

sign in Eq. A-21. We choose + for 'i = 0 (upward

vertical perforation) and for 'i = (downward vertica

perforation).

Influence Functions for Slots. We define the influence

functions, I1 andI2, for slots by the following relationsrespectively:

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( )

( ) ( ) Dlx

lx

mDjDDjD

mD

xdyyxx

lmjI

mDmD

mDmD

+

=

+

5.0

5.0

22

1

ln

1,

, (A-22)

and

( )

( ) ( )

+

=

=

+

D

mD

D

jD

n

D

lx

lx

mDjDDjDD

mD

h

zn

h

zn

xdyyxxh

nK

lmjI

mDmD

mDmD

coscos

1,

1

5.0

5.0

220

2

.

(A-23)

In Eqs. A-22 and A-23, mDl is the dimensionless length of the

m-th slot defined by

A

mmD

ll = , (A-24)

where lm is the length of the m-th slot, andzjD andzmD are

given, respectively, by

z

rjjD

k

kzz

A= (A-25)

and

z

rm

mD k

kzz

A=(A-26)

Numerical evaluation of I1 and I2 is not straightforward.

Below, we provide some details regarding the computation of

these functions.

Evaluation of I1(j, m). The following expressions can beused to computeI1.

If 0y-y mDjD = :

[ ] 1)ln(2

1),(

)2

(

)2

(

2

1 ==

+=

mDmDjD

mDmDjD

lxxu

lxxu

mD

uul

mjI . (A-27)

If 0y-y mDjD :

)2/()(),( 12111 mDlIImjI = , (A-28)

where

)2

(

)2

(

22

11 ])(ln[mD

mDjD

mDmDjD

lxxu

lxxu

mDjD y-yuuI=

+=+= , (A-29)

and

)2

(

)2

()(

1

12

tan)(2

2

mDmDjD

mDmDjD

mDjD

lxxu

lxxu

y-yu

mDjD

mD

y-y

lI

=

+=

=

(A-30)

Evaluation of I2(j, m). The computation ofI2 can be

accomplished by using the following formulas:0y-y mDjD = :

We can use the Ki1(x) function for the computation of the

integral of Bessel function in Eq. A-23. The Ki1(x) function is

given by

=x

i duuKxK )()( 01 . (A-31)

Let us defineI20 as follows:

'22'020

2

1

])()([ D

C

C

mDjDDjDD

dxyyxxhnKI

m

m

+= , (A-32)

where

,2

1mD

mDm

lxC = (A-33)

and

22

mDmDm

lxC += . (A-34)

If 21 mjDm CxC

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)cos()cos(1

1

1

D

mD

D

jD

n

D

mD

Ch

zn

h

zn

n

h

lF

=

=

. (A-40)

To solve this problem, we use the formula

)cos1(2

1ln

2

1)cos(

1 xk

kx

n =

=

, (A-41)

and recastF1Cinto the following form:

)]}cos(1ln[

)]cos(1ln[4{ln4

1

D

mDjD

D

mDjD

mD

DC

h

zz

h

zz

l

hF

++

+=

. (A-42)

This expression significantly accelerates computations.

0y-y mDjD .

For this condition, the computation of the integrals by the

Chebyshev polynomials is too slow. We have used the mean-

value theorem for the evaluation ofI20, and obtained thefollowing expression:

2]})()([

])()([{

22'

0

22'

020

mDmDjDmDmDjD

D

mDjDmDmDjD

D

lyylxx

h

nK

yylxxh

nKI

+

+++=

, (A-43)

where has been determined by comparing the results of Eq.A-43 with the results of Chebyshev integration. We have

found that in Eq. A-43 should be chosen as 0.08. Ourcomputations indicate that the use of Eq. A-43 instead of

Chebyshev integration causes less than 2% error for all

practical cases but the computation speed is improved 17~48

times.

TABLE 1 COEFFICIENTS FOR APPARENT FRICTION FACTOR

A. SLOTTED-LINER

B. PERFORATED PIPE

*given by Yuan

TABLE 2 INPUT DATA FOR THE EXAMPLE CASE

TABLE 3 PERFORATION PHASE ANGLE EFFECT

A. ISOTROPIC FORMATION

B. ANISOTROPIC FORMATION

Fig. 1 Effect of horizontal well completion on reservoir influxand wellbore hydraulics

No. PHA SPF a b Cn

1 90 3 0.393 -0.266 2.000

2 180 3 0.171 -0.189 2.411

3 90 4.5 0.130 -0.175 1.781

4* 180 4.5 0.317 -0.258 2.000

5* 360 4.5 0.318 -0.251 2.200

6* 90 9 0.501 -0.300 2.300

7 180 9 0.172 -0.156 1.768

No. SPF PHA a b Cn

1 5 90 0.873 -0.341 2.344

2 5 180 0.622 -0.287 2.028

3* 5 360 0.641 -0.312 2.169

4 10 90 0.159 -0.155 1.489

5* 10 180 0.363 -0.266 2.26 10 360 0.755 -0.308 3.901

7* 20 90 1.297 -0.421 2.2

8 20 180 4.532 -0.505 2.332

9 20 360 1.078 -0.346 2.694

ps eu do -r ad ial

flow

t = 300 hrs.

Q t = 3000 b/d

Lh =

1000 ft

h = 75 ft,

rw = 3 in. zw = 25 ft = 0.25 kr= 6 d,kz = 1.5 d

= 1 .43 cp o = 55lbm/cuftB

o= 1.16 C = 6.9 10

-6

ps i-1

Lp = 12 in. Pha = 360 SPF =0.1

dp = 0.75 in

36 0 90 36 0 90 Density (spf) 0.04 0.04 0.1 0.1

PI (b/d/psi) 49.22 49.44 76.97 77.18

3 6 0 9 0 3 6 0 9 0 D en sity (sp f) 0 .0 4 0 .0 4 0 .1 0 .1

P I (b/ d / ps i) 1 6 2 1 3 6 2 5 2 2 2 5 .8

Flow rate

Flux ingress

Wellbore hydraulics

x

z

Pressure

Heel

T

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2A . Inclined perforation source

2B . Vertical perforation source

Fig. 2 Geometry of perforations along a horizontal well.

Fig. 3 slotted-liner completed horizontal well

Fig. 4 Effect of perforation density on PI

Fig. 5 Effect of perforation penetration on PI

Fig. 6 Slot phasing Effect under different anisotropy

(Lp = 12", Vertical Perf., Lh=1000',Qt=3000b/d)

0

50

100

150

200

250

300

350

400

0 0.2 0.4 0.6 0.8 1 1.2

Perf. Density (shots/ft)

PI(b/d

/ps

i)

(SPF = 0.1, Ve rtical Perf., Lh=1000', Qt=3000b/d)

0

50

10 0

15 0

20 0

25 0

30 0

35 0

40 0

0 2 0 4 0 6 0 8 0 1 0 0 1 2 0

Lp (in.)

PI(b/d/psi)

(ls = 9.375", ee = 18.74")

0

50

100

150

200

250

300

350

400

060120180240300360

Slot Phasing (deg.)

Pro

duc

tiv

ity

Index

(b/d/ps

i)

PI (Kr = 6000, Kz = 1500)

PI (Kr =Kz = 1500)

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Fig. 7 Slot phasing effect under different SPR

Fig. 8 Slot size effect under constant SPR

Fig. 9 Slot length effect on PI

Fig. 10 Slot interval effect on PI

Fig. 11 Slot penetration ratio effect on PI

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Slot Penetration Ratio (SPR)

PIc

hange

by

Phas

ing

(%)

Kz/Kr=4

Kz/Kr=1

SPR = 50%

250

270

290

310

330

350

0 50 100 150 200 250 300

Slot Size (in.)

Pro

duc

tiv

ity

(b/d/ps

i)

Slot Interval = 6.25"

0

50

10 0

15 0

20 0

25 0

30 0

35 0

40 0

0 10 20 30 40 50

Slot Length (in.)

P

roductivity

(b/d/psi)

0

100

200

300

400

0 50 100 150 200

Slot Interval (in.)

PI(b/d/ps

i)

Slot Length = 6.25"

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0.00 20.00 40.00 60.00 80.00 100.00

Slot Penetration Ratio (SPR)

PIRecovery

(%)

ls = 6.25"

ee = 6.25"

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Fig. 12 Partial completion and perforation Cases

Fig. 13 Flux distribution for partial completion and different

perforating cases

0

0.5

1

1.5

2

2.5

3

3.5

0 500 1000 1500 2000 2500 3000

x (Distanse from the heel, ft)

Norma

lize

dFlux

fully open, uniform perf.

uniform perf

high SPF at heel

high SPF at center

high SPF at toe

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