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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.
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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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4 Y. TANG, E. OZKAN, C. SARICA, M. KELKAR, AND T. YILDIZ SPE/PS-CIM 65516
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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6 Y. TANG, E. OZKAN, C. SARICA, M. KELKAR, AND T. YILDIZ SPE/PS-CIM 65516
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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SPE/PS-CIM 65516 PERFORMANCE OF HORIZONTAL WELLS COMPLETED WITH SLOTTED LINERS AND PERFORATIONS 7
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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8 Y. TANG, E. OZKAN, C. SARICA, M. KELKAR, AND T. YILDIZ SPE/PS-CIM 65516
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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SPE/PS-CIM 65516 PERFORMANCE OF HORIZONTAL WELLS COMPLETED WITH SLOTTED LINERS AND PERFORATIONS 9
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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10 Y. TANG, E. OZKAN, C. SARICA, M. KELKAR, AND T. YILDIZ SPE/PS-CIM 65516
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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SPE/PS-CIM 65516 PERFORMANCE OF HORIZONTAL WELLS COMPLETED WITH SLOTTED LINERS AND PERFORATIONS 1
( )
( ) ( ) 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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12 Y. TANG, E. OZKAN, C. SARICA, M. KELKAR, AND T. YILDIZ SPE/PS-CIM 65516
)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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SPE/PS-CIM 65516 PERFORMANCE OF HORIZONTAL WELLS COMPLETED WITH SLOTTED LINERS AND PERFORATIONS 1
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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14 Y. TANG, E. OZKAN, C. SARICA, M. KELKAR, AND T. YILDIZ SPE/PS-CIM 65516
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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SPE/PS-CIM 65516 PERFORMANCE OF HORIZONTAL WELLS COMPLETED WITH SLOTTED LINERS AND PERFORATIONS 1
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