(SPE 89414 MS) A Simple Approximate Method to Predict Inflow Performance of Selectively Perforated Vertical Wells

Abstract This paper presents an approximate model to forecast the productivity of selective perforated wells. The model includes algebraic equations and it could be easily computed using a programmable calculator or spreadsheet program. The model has been compared against the rigorous 3D semianalytical model and software existing in the literature. The approximate model compares well against the 3D model and the software. The model is useful for designing perforation parameters.

SPE 89414

A Simple Approximate Method to Predict Inflow Performance of Selectively Perforated Vertical Wells

E Guerra, SPE, and T Yildiz, SPE, Colorado School of Mines

Copyright 2004, Society of Petroleum Engineers Inc

This paper was prepared for presentation at the 2004 SPE/DOE Fourteenth Symposium on

Improved Oil Recovery held in Tulsa, Oklahoma, U.S.A., 17–21 April 2004

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Abstract

This paper presents an approximate model to forecast the

productivity of selective perforated wells The model includes

algebraic equations and it could be easily computed using a

programmable calculator or spreadsheet program The model

has been compared against the rigorous 3D semi-analytical

model and software existing in the literature The approximate

model compares well against the 3D model and the software

The model is useful for designing perforation parameters

Introduction

Oil and gas wells are generally perforated at multiple intervals

along the well trajectory The principal objective of

perforating is to open flow channels across the casing for

formation fluid entry The well completed with perforations at

multiple segments along the wellbore are referred to as the

selectively perforated well (SPW)

The productivity of the wells is controlled by the well

completion type and formation damage Formation damage is

the result of the permeability impairment in the near wellbore

region Formation damage decreases the well productivity

The influence of formation damage is localized in the near

wellbore region The formation damage effect is quantified in

terms of the mechanical skin factor, s d

In openhole completed vertical wells, the complete

formation produces uniformly and the specific productivity

index is constant along the wellbore On the other hand, the

selective perforating results in multiple flow convergence

regions along the wellbore This disturbance in the flow

pattern makes flow modeling considerably more difficult The

impact of selective perforating may be quantified in term of

completion pseudoskin Similar to the formation damage

effect, the influence of the well completion is intensified in the

near wellbore region Therefore, the well flow models have to

account for not only the individual impact of the formation damage and well completion but also the dynamic interaction between them in the near wellbore region

The compounded effects of the well perforating and formation damage are expressed in terms of total skin factor, Given the total skin factor, the steady state pressure drop

in the perforated wells could be expressed as

t s

] ) / ( ln [ 2

141

t w e o sct wf

h k

B q p

(1)

If the well produces from a non-radial reservoir under pseudo steady state flow conditions then the pressure drop equation is given as

] ) 2458 2 ( ln 2

1 [ 2

141

~

w A

o sct

r C

A h

k

B q p

(2)

The flow equations expressed in Eqs 1 and 2 are simple and straightforward provided that the total skin factor is accurately related to the perforating parameters and the formation damage based skin factor

Background

In this section, we would like to review the most relevant studies in the literature and set the ground for the model development

penetrated wells (PCW), only a single segment along the wellbore is open to flow The completed segment is considered to be barefoot Partial penetration forms a two-dimensional (2D) flow field in the formation around the wellbore

The effect of partial penetration on well productivity has been investigated in details.1-8 Brons and Marting1 observed

that the partial penetration generates a pseudo damage

reducing the well productivity The others have confirmed that

an additional pressure drop is created by the partial penetration effect.2-8 The additional pressure drop is measured in terms of partial penetration pseudoskin, s pp Graphical results1, analytical solutions2,3, numerical solutions4,5, and empirical equations6-8 have been proposed to compute the pseudoskin resulting from partial completion Among these different

methods, the empirical equations proposed by Odeh6,

Papatzacos7, and Vrbik8 are the most popular due to their

simplicity The analytical solutions are not usually preferred

because of their infectivity with infinite series and special

functions We have compared the empirical methods against

the analytical solutions It has been observed that the results

from the Vrbik method and analytical solutions match very

well.9 The Vrbik method was chosen to build the approximate

model described in this paper The Vrbik model for partial

penetration pseudoskin is recaped in Appendix A

It should be noted that partially penetrating well models

assume that the open interval allows fluid entry at every point

on wellbore surface The partial penetration pseudoskin does

not account for formation damage or additional flow

convergence due to perforations and slots

The simultaneous effects of formation damage and partial

penetration have been modeled using analytical and numerical

techniques.3-5 It has been observed that formation damage

results in a greater productivity loss in partially penetrating

wells The total skin factor combining the individual

contribution of formation damage and partial penetration is

given as below

pp d

p

h

h

s = + (3)

where is the formation damage skin factor formulated by

Hawkins

d

s

10

) / ( ln ) 1 /

s = − (4)

In the derivation of Eq 3, it is assumed that the flow

convergence owing to partial penetration is completed outside

the damaged zone If the flow convergence towards open

segment happens partly outside and partly inside the damaged

zone then the total skin factor is given as

pp d p

h

h

γ

1

(5)

where γ is greater than 1 Odeh3

and Jones and Watts4 proposed simple equations to compute the parameter γ

Selectively Completed Wells In field applications, the wells

are usually completed across several intervals along the well

trajectory The well completed this way is referred as to the

selectively completed well (SCW) Each completed segment

on the SCW is barefoot Therefore, the flow pattern towards

the SCW is 2D

Brons and Marting1 indicated that if the well is completed

symmetrically along the wellbore then the PCW models could

be applied to each symmetrical unit and the productivity of the

SCW could be estimated by multiplying the productivity of

the symmetrical unit by the number of the symmetrical

elements Since then, several 2D analytical models have been

developed to simulate the fluid flow into the SCWs.11-13 These

models are more complex than the analytical models for

PCWs In addition to being contaminated with the special functions and infinite series, the analytical models for SCWs require matrix construction and inversion since the models compute not only the pressure drop but also the rate at each open segment

In this study, we present an approximate model to replace the 2D analytical solutions and to avoid the matrix solution and complicated mathematical functions

Fully Perforated Wells The well perforated completely all

along the well trajectory is referred as to fully perforated well (FPW) The flow efficiency of the FPWs has been the subject

of many investigations.13-25 The effect of ideal perforating has been quantified in terms of perforation pseudoskin, Numerical

p s

14-17 , semi-analytical18-21, and analytical methods22 have been proposed to compute the perforation pseudoskin In the absence of formation damage and rock compaction around the perforation tunnels, the perforation pseudoskin is a function of perforation length, perforation radius, phasing angle, shot density, wellbore radius, and permeability anistropy

) / , , , , ,

p

It has been observed that the productivity of the FPW is not only controlled by the magnitude of perforation pseudoskin but also the skin factors due to formation damage and the rock compaction around the perforation tunnels The skin factor resulting from formation damage, , is characterized by the degree of permeability impairment ( ) and the extent of the damaged zone ( ) as shown in

Eq 4 The skin factor due to rock compaction around the perforation tunnels is expressed as

d s k

) / ln(

)

d cz p

p

k

k k

k L

z

s =∆ − (7)

To predict the productivity of the FPW accurately, the interaction between the three skin terms ( , , and ) has to be formulated properly The combined effect of perforation pseudoskin, formation damage, and rock compaction around perforations is referred as to perforation total skin,

p

s s d s cz

pdc s

Among the many methods for estimating the perforation total skin, the methods proposed by McLeod18 and Karakas and Tariq19, 20 have been very popular due to their simplicity

For the sake of completeness, the Karakas-Tariq method is summarized in Appendix B Additionally, Jones and Slusser21 proposed a simple but accurate method combining perforation pseudoskin and formation damage skin

A three-dimensional (3-D) analytical model to determine the perforation total skin has been proposed in Ref 22 The main advantage of the 3D solution is that it could handle arbitrary perforation distribution and non-uniform perforation parameters

A software package called SPAN is also available for

computing the productivity of the perforated wells.23 SPAN

uses a modified version of the Karakas-Tariq algorithm

described in Refs 19 and 20.24

Recently, the McLeod method, the Karakas-Tariq

algorithm, Jones-Slusser method, SPAN software, and the 3D

analytical solution were compared against the experimental

data.25 It has been shown that 1) the McLeod method

underestimates the perforation total skin, 2) the Karakas-Tariq

method for perforation pseudoskin works fine, however, the

Karakas-Tariq algorithm overpredicts perforation total skin in

the presence of formation damage and crushed zone, 3) the 3D

analytical solution and SPAN software replicate the

experimental data very well, and 4) the Jones-Slusser model,

which does not consider the effect of compacted zone, agrees

well with the 3D analytical solution and SPAN software when

the crushed zone is also ignored in the 3D solution and SPAN

In Ref 25, a modified version of the Jones-Slusser

method, accounting for the skin factor due to rock compaction

around the perforation tunnels, has been proposed The

modified Jones-Slusser method could be analytically derived

if it is assumed that the perforations are terminated inside the

damaged zone and a radial flow geometry exists beyond the

damaged zone The modified Jones-Slusser model compared

well against the experimental results for the special case of

short perforations terminated inside the damaged zone In the

modified Jones-Slusser method, the perforation total skin is

expressed as below

cz p d d

k

k s

s = + + (8)

Partially Perforated Wells If only a segment along a cased

well is completed with perforations then this type of well is

referred as to the partially perforated well (PPW) The flow

convergence and the shape of the streamlines around the PPW

are controlled by the combined effect of partial penetration

and perforations If it is assumed that the flow convergence

due to partial penetration is completed before the fluid feels

the impact of the perforations and the damaged zone then an

analytical expression could be derived to compute the total

skin factor including the simultaneous effects of partial

penetration, perforations, formation damage, and the crushed

zone.13, 20-23, 25 The equation for the total skin factor is

pp pdc p

h

h

s = +

γ

1

(9)

Eq 9 has been verified against the 3D analytical solution.22

Selectively Perforated Wells To the authors’ knowledge,

there exist two studies on the performance of the selectively

perforated vertical wells (SPW) Ref 22 described a general

3D analytical solution considering arbitrary distribution of

perforation and variable perforation properties The solution

involves matrix construction, Bessel functions, numerical and

analytical integration of Besses functions, and infinite series

The size of the matrix is where is the

number of the perforations Therefore, if a large number of perforations are involved, the computation of the 3D analytical solution may demand long CPU time Ref 22 and 23 also presented a pseudo 3D model based on the SCW model of Ref 12 The pseudo 3D model is very efficient even for the wells with tens of thousands of perforations However, the pseudo 3D models of Refs 22 and 23 are still composed of matrix construction, Bessel functions, and infinite series

) 1 ( ) 1 (n p + × n p + n p

The objective of the current paper is to develop a fast and easy-to-use approximate SPW model free of matrix setup, infinite series, and special functions

Approximate Model

The approximate model for selectively perforated wells is broken into two submodels; a perforation total skin model considering unit formation thickness of 1 ft and a SCW with a variable total skin across the perforated intervals We will describe both models

Perforation Total Skin Model A hybrid method is used to

compute the total skin factor combining the individual contributions of perforation pseudoskin, formation damage, and compacted zone around the perforations First, we only determine the perforation pseudoskin, , by using the steps 1-4 of the Karakas-Tariq algorithm The Karakas-Tariq algorithm is appended At this stage, the effects of formation damage and rock compaction are not accounted for yet

p s

For the short perforations ending inside the damaged zone,

we use the modified Jones-Slusser method to estimate the combined effects of perforation pseudoskin, formation damage, and rock compaction The modified Jones-Slusser methos is basically the expression in Eq 8

For the long perforations reaching beyond the damaged zone, we use a modified version of the Karakas-Tariq method provided by Hegeman.24 The modified method is also used in SPAN software, version 6.0.23 There are basically two changes applied to the original Karakas-Tariq method The first modification is that true wellbore radius ( ) instead of the effective wellbore radius ( ) is used in computing term The second modification is in the calculation of term True perforation length ( ) instead of the effective perforation length ( ) is used in computing term

w r w

cz s p

L p L′ s cz

Approximate Model for Selectively Completed Wells For

the simplicity, we will develop the approximate model considering steady state flow However, the methodology could be also applied to the well producing under pseudo steady state flow

Consider a damaged vertical openhole producing under steady state flow conditions The specific productivity index, , for such a well is obtained by rearranging Eq 1

o

J~

d w

e o wf

e

sct o

s r

r B

k p

p h

q J

+

=

−

=

ln

1 2

141 ) (

~

In a vertical openhole, every unit-thickness of the formation

produces the same amount of the fluid Therefore, the specific

productivity index is constant and uniform at the wellbore as

well as inside the formation across its thickness It should be

reminded that the specific productivity index is different from

the flux at the wellbore and these two concepts should not be

interchanged The flux at the wellbore is the rate per unit

length along the wellbore

Now consider a damaged partially penetrating well As

shown on Fig 1, partial completion makes the streamlines

converge around the open segment and creates a 2D flow

field However, the effect of partial completion on the fluid

streamline pattern is concentrated in the near wellbore region

At the locations away from the wellbore and deep inside the

formation, the fluid flow is 1D radial and the streamlines are

parallel to each other and the upper and lower reservoir

boundaries Let’s refer the distance at which the streamlines

start to converge towards the open segment as the radius of

flow convergence ( ) The flow towards a partially

completed well beyond the radius of flow convergence is

almost the same as that towards an openhole

c r

The specific productivity index for a PCW, J~pc, is

t w

e o wf

e

sct pc

s r

r B

k p

p

h

q

J

+

=

−

=

ln

1 2

141 ) (

~

where is the total skin factor as expressed in Eq 3 or 5

Notice that, even for a PCW, the specific productivity index is

defined with respect to formation thickness not the length of

the completed segment Typically, in a PCW, the flux along

the wellbore is discontinous; it is zero at the uncompleted

segments and it varies somewhat along the open segment On

the other hand, if we examine the flux along the formation

thickness at a location beyond the radius of flow convergence

then it can be stated that the flux beyond is constant and

uniform across the formation thickness Similarly, if we

consider the specific productivity index as a measure of

formation capacity and evaluate it at a location beyond not

at the wellbore then the specific productivity index is expected

be constant and uniform across the formation thickness as

well

t

s

c r

c r

At this stage, let’s examine the fluid flow into a selectively

completed well Consider a SCW with number of open

intervals and variable formation damage skin factor across

each open segment as shown on Fig 2 The selective

completion yields multiple flow convergence regions in the

near wellbore region However, the impact of the selective

completion on the flow streamlines is localized Beyond the

radius of flow convergence, the streamlines are parallel to

each other and reservoir bedding plane In SCWs, although the

flux distribution at the wellbore is discontinous and

non-uniform, the flux and the specific productivity index evaluated

at the locations beyond are constant and uniform across the

formation thickness The specific productivity index for a

SCW, , could be written as

s n

c r

sc

J~

tsc w

e o wf

e

sct sc

s r

r B

k p

p h

q J

+

=

−

=

ln

1 2

141 ) (

~

where is the total skin factor representing the effects of selective completion and variable formation damage

Analytical expressions to compute could be found in Refs 12 and 22 In general,

tsc s

tsc s

) , / , , / , , ,

tsc

Here, we would like to offer an alternative method to predict J~sc

Consider a SCW with number of open intervals distributed symmetrically along the wellbore Also assume that the open intervals are subject to the same degree of formation damage In such a case, all the completed intervals produce at the same rate of and flow induced no-flow boundaries parallel to the bedding plane are formed at the center of each uncompleted segment Due to flow and completion symmetry, we can decompose the SCW into number of fictitious partially penetrating wells producing from the same number of independent fictitious reservoirs/layers

Let be the fictitious thickness of the i

s n

sci q

s n

i

Additionally, let , and represent the rescaled location

and the actual length of the i

bi

th

partially completed well

producing only from the i th fictitious reservoir, respectively

The special productivity index for each fictitious PCW could

be written as below

ti w

e o wf

e i

sci pci

s r

r B

k p

p h

q J

′ +

=

−

′

=

′

ln

1 2

141 ) (

~

where is the total skin factor, representing the influences

of partial completion, perforations, formation damage, and

rock compaction, for the i

ti s′

th

fictitious PCW could be computed using Eqs 3, 5, or 9, depending on the completion design Additionally, due to geometry,

ti s′

h h h

h h

h h

s

s n i i n

i′+ + ′ = ′= +

+

′ +

′ +

=1 3

2

number of fictitous PCWs are part of the original whole SCW Now if we evaluate the specific productivity indecies for the SCW and the fictitous PCW beyond the radius

of flow convergence then all the specific productivity indicies should be equal

s n

sc pcn pci

pc pc

J

s

~

~

~

~

~

~

3 2

A comparison of Eqs 12 through 16 reveals that the total

skin factors for the SCW and the number of fictitous

PCWs should be the same

s n

sc tn ti

t t

s

s =

′

=

=

′

=

′

=

′

=

′1 2 3 (17)

Additionally,

i pci n

i

h

J

s

′

′

=

~ 1

~

1

(18)

If the open intervals are symmetrically distributed then all

the fictitious layers has the same thickness

s n

h h

h

h

3

2

1′= ′ = ′ = = ′= = ′ = (19)

Now let’s go back and re-consider a SCW with arbitrary

distribution of open segments and different degree of

formation damage across them as displayed in Fig 3 In such a

case, the actual reservoir and SCW cannot be divided into

number of equivalent layers and equivalent PCWs,

respectively However, even in the case of asymmetric

segment and contrasting formation damage distributions, it is

expected that each completed segment will establish its own

drainage volume; therefore, flow induced no-flow boundaries

will emerge somewhere along the uncompleted segments

between the completed ones not at the center of uncompleted

segments as in case of symmetric completion In the

asymmetric completions, the flow induced no-flow boundaries

may not be completely parallel to reservoir bedding and as

well defined as those in the symmetric completions

Regardless, in case of asymmetric segment and unequal

damage distribution, we could still decompose the actual SCW

in the real reservoir into number of fictitious PCWs in

layers However, each fictitous layer will have a different

fictitious thickhness of assigned to it

s n

s

i h′

In asymmetric completions, the fictitious PPWs are still

part of the real SCW; therefore, when we evalute the specific

productivity indecies beyond the radius of flow convergence,

the actual SCW and the fictitious PPWs all should possess the

same specific productivity index value In other words, Eqs

12 through 18 are also valid for the asymmetric completions

Now, the remaining unresolved issue is how to assign the

fictitious thickness to each fictitious layer Assignment of

individual layer thickness requires an iterative procedure We

suggest allocating the fictitious thickness based on the ratio of

the segment height to the total penetration ratio initially

pt

pi

h′= / (20)

∑

=

= n s

i

pi

h

1

(21)

It is very likely that the initial fictitious thickness distribution based on Eqs 20 and 21 will not satisy the conditions expressed in Eqs 16 – 18 In such a case, in the following iteration, we reallocated the fictitious thickness based on the ratio of specific productivity indicies for the individual PCW and SCW

k i k sc

k pci k

J

J

~ ) 1 (

(22)

where J~sc k is estimated from Eq 18

We presented the iterative approximate model (Eqs 12-18 and 20-22) for a selectively completed well and steady state flow conditions However, the same alghorithm also applies to selectively perforated wells and pseudo steady state flow For selectively perforated wells, we use Eq 9 to estimate the total skin factor instead of Eq 3 or 5 which is for selectively completed wells To invoke the pseudo steady state flow condition, we just need to use Eq 2 in the specific productivity index computation

In Appendix C, a stepwise procedure is given for the iterative solution of the approximate model

Verification of the Approximate Model As mentioned

previously in the text, in the literature, there are 2D and 3D analytical solutions for selectively completed/perforated wells Also, the software SPAN could be used to predict the productivity of the partially perforated wells To verify the approximate model proposed in the current study, we compared it against the analytical solutions of Refs 12, 13, and 22 and the software SPAN

Table 1 shows the comparison of the 2D analytical and approximate models for SCWs only Three completed intervals and two different cases of formation damage were considered in the comparison As can be seen on the table, the results from the simple approximate model compare very well against those from the 2D analytical solution Besides the results shown in Table 1, we also conducted additional extensive comparison of the models For the majority of the cases, the approximate model replicated the results from the 2D analytical model However, for some negative skin values less than -2.3, the approximate model did not work well when the interval with negative skin was very short

The approximate model was also tested extensively against SPAN software by considering PPWs with different completion/perforation schemes An example comparison is shown in Fig 4 Table 2 presents the data used in the comparison depicted in Fig 4 As can be seen on the figure, the results from the approximate model and the software agree very well In some other comparisons, we observed small deviations between the compared models The average difference between results from the approximate model and SPAN was 6%

Although the results are not shown, we also compared the approximate solution against the 3D solution presented in Ref

22 for selectively perforated wells and observed good agreement

Discussion

In this section, we present the application of the approximate

model to selectively perforated wells and investigate the

effects of different perforation designs on the well

performance

Table 3 lists the completion and perforation data

considerd for the SPW The rest of the basic data set is the

same as that tabulated in Table 2 The well is perforated across

three intervals The length and location of each interval are

printed in Table 3

First, we kept perforation length constant and assigned

different values of the shot density, and

However, in all three cases, all the perforated

intervals had the same shot density The results, in terms of

productivity index, total skin factor, and fractional segment

rates, are summarized in Table 4 For comparison purposes,

the results for SCW and damaged and undamaged openhole

completions are also listed in Table 4 The results show that as

the perforation length increases, well productivity is improved

The well with has 1.8 times higher productivity

than that with It should be also noticed that the

SPW with performs slightly better than SCW

"

12

=

p L

12

,

8

,

4

=

spf

n

12

=

spf n

4

=

spf n

12

=

spf

n

We also investigated the impact of perforation length on

the well performance The results for this investigation are

reported in Table 5 For , the total skin factors are

substantially higher The skin factors for are about

four times larger than those for As a result of high

total skin factors due to short perforations, the productivity

index values for are approximately three times lower

than those for It should be noticed that, since the

total skin factors are high, the changes in the skin factors do

not affect the fractional rate distribution The results in Table 5

verifies that the deep penetrating perforations extending

beyond the formation damage zone may improve the well

productivity significantly

"

3

=

p L

"

3

=

p L

"

12

=

p L

"

3

=

p L

"

12

=

p L

Summary and Conclusions

1 A simple approximate model to predict the inflow

performance of selectively completed and selectively

perforated wells has been developed The model is

based on an iterative procedure and uses simple

algebraic equations

2 The model has been compared against the 2D

analytical solution for selectively completed wells,

SPAN software for partially perforated wells, and 3D

analytical solution for selectively perforated wells In

general, the approximate model agrees very well with

the more complicated solutions and the software The

accuracy of the approximate model has been verified

by conducting an extensive comparison study

3 Several novel applications of the approximate model

have been presented The brief sensitivity study

presented verifies that well productivity may be

markedly improved if the perforations pierce through

the formation damage zone and communicate with the undamaged formation beyond the damaged zone

Nomenclature

A = drainage area, ft2

B o = formation volume factor, dimensionless, rbbl/stb

C A = reservoir shape factor

h = formation thickness, L, ft

h b = the distance between the bottom of the completed

interval and reservoir, L, ft

h p = the length of the completed interval, L, ft

h pt = the length of the total completed interval, L, ft

h = formation thickness of the fictitious layer, L, ft ′

J c = productivity of completed well, stb/day/psi

J o = productivity index of open hole, stb/day/psi = specific productivity index of open hole,

stb/day/psi/ft

o

J~

= specific productivity index of partially completed

wells, stb/day/psi/ft

pc

J~

= specific productivity index of the fictitious partially

completed wells, stb/day/psi/ft

pc

J~′

= specific productivity index of selectivley completed

wells, stb/day/psi/ft

sc

J~

k = permeability, L2, md

k cz = permeability of crushed zone, L2, md

k r = permeability in radial direction, L2, md, k x k y

k x = permeability in x-direction, L2, md

k y = permeability in y-direction, L2, md

k z = permeability in z-direction, L2, md

L p = perforation length, L, ft

n s = number of completed segments

n spf = number of shots per foot

p = pressure, m/Lt2, psi

PR = productivity ratio, dimensionless, fraction

q sct = total well flow rate at surface, L3/t, stb/day

q sci = flow rate across the ith segment, L3/t, stb/day

r cz = radius of crushed zone around perforation, L, ft

r e = reservoir radius, L, ft

r p = perforation radius, L, ft

r w = wellbore radius, L, ft

s cz = skin due to rock compaction around perforations in

the presence of formation damage

= skin due to rock compaction around perforations in

the absence of formation damage

cz s′

s d = skin due to formation damage/stimulation

s p = pseudoskin due to perforating

s pc = total skin combining flow convergence towards

perforations and crushed zone skin

s pd = total skin combining flow convergence towards

perforations and formation damage

s pdc = total skin including perforation pseudoskin,

formation damage, and rock compaction around perforation tunnels

s pp = pseudoskin due to partial penetration

s t = total skin factor

s tsc = total skin factor for selectively

completed/perforated well

µ = viscosity, m/Lt, cp

p e = reservoir boundary pressure, m/Lt2, psi

p wf = flowing wellbore pressure, m/Lt2, psi

p = average reservoir pressure, m/Lt

, psi

∆r cz = thickness of the crushed zone, L, ft

∆r d = damaged zone thickness around wellbore, L, ft

∆z p = the vertical distance between perforations, L, ft

θp = perforation phasing angle

Subscripts

cz = crushed zone

d = wellbore damage

p = perforation

t = total

w = wellbore

Acknowledgment

The authors would like to thank Pete Hegeman for providing

the information about the modified Karakas and Tariq method

and a copy of SPAN software

References

1 Brons, F and Marting, V.E.: “The Effect of Restricted Fluid Entry

on Well Productivity”, JPT (February 1961) 172

2 Odeh, A.S.:“Steady-State Flow Capacity of Wells with Limited

Entry to Flow,” SPEJ (March 1968) 43; Trans., AIME, 243

3 Odeh, A.S.:”Pseudosteady-state Flow Capacity of Oil Wells with

Limited Entry and an Altered Zone around the Wellbore,” SPEJ

(August 1977) 271

4 Jones, L.G and Watts, J.W.:”Estimating Skin Effect in a Partially

Completed Damaged Well,” JPT (February 1971) 249

5 Saidowski, R.M.:“Numerical Simulations of the Combined Effect

of Wellbore Damage and Partial Penetration,” paper SPE 8204

presented at the 1979 SPE Annual Technical Conference and

Exhibition, Las Vegas, Nevada, September 23-26

6 Odeh, A.S.:“An Equation for Calculating Skin Factor Due to

Restricted Entry,” JPT (June 1980) 964

7 Papatzacos, P.:”Approximate Partial-Penetration Pseudoskin for

Infinite-Conductivity Wells,” SPERE (May 1987) 227

8 Vrbik, J.:“A Simple Approximation to the Pseudoskin Factor

Resulting from Restricted-Entry,” SPEFE (December 1991) 444

9 Guerra, E., Inflow Performance of Selectively Perforated Vertical

Wells, MS Thesis, Colorado School of Mines, Golden, Colorado

(May 2004)

10 Hawkins, M.F.:”A Note on the Skin Effect,” Trans AIME, (1956)

207

11 Larsen, L.:“The Pressure-Transient Behavior of Vertical Wells

with Multiple Flow Entries,” paper SPE 26480 presented at the

1993 SPE Annual Technical Conference and Exhibition in

Houston, October 3-6

12 Yildiz, T and Cinar, Y.:”Inflow Performance and Transient

Pressure Behavior of Selectively Completed Vertical Wells,”

SPE Reservoir Eng (October 1998) 467

13 Yildiz, T.:”Impact of Perforating on Well Performance and

Cumulative Production” Journal of Energy Resources

Technology, (September, 2002) 163

14 Hong, K.C.:”Productivity of Perforated Completions in

Formations With or Without Damage,” JPT (August 1975)

1027, Trans AIME, 259

15 Locke, S.: “An Advanced Method for predicting the Productivity

Ratio of a Perforated Well,” JPT (December 1981) 2481

16 Tariq, S.M.:”Evaluation of Flow Characteristics of Perforations Including Nonlinear Effects With the Finite Element Method,”

SPEPE (May 1987) 104

17 Dogulu, Y.S.”Modeling of Well productivity in perforated Completions,” paper SPE 51048 presented at the 1998 SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania, November 9-11

18 McLeod, H.:”The Effect of Perforating Conditions on Well

Performance,” JPT (January 1983) 31

19 Karakas, M and Tariq, S.M.:”Semianalytical Productivity Models

for Perforated Completions,” SPEPE (February 1991) 73

20 Bell, W.T., Sukup, R.A., and Tariq, S.M., Perforating, SPE

Monograph Volume 16, Richardson, TX, 1995

21 Jones, L.G and Slusser, M.L.:”The Estimation of Productivity Loss Caused by Perforations – Including Partial Completion and Formation Damage,” paper SPE 4798 presented at the 1974 SPE Second Midwest Oil and Gas Symposium, Indianapolis, Indiana, March 28-29

22 Yildiz, T.:“Productivity of Selectively Perforated Vertical Wells,”

SPEJ (June 2002) 158

23 SPAN user guide, Version 6.0, Schlumberger Perforating and Testing, 1999

24 Hegeman, P., Personal Communication, Schlumberger Product Center, Sugarland, Texas

25 Yildiz, T.: “Assessment of Total Skin Factor in Perforated Wells,” paper SPE 82249 presented at the 2003 SPE European Formation Damage Conference, The Hague, The Netherlands, May 13-14

Appendix A – Vrbik Model for Partial Penetration Pseudoskin

The approximate model for the selectively perforated well is partially based on the partial penetration pseudoskin model proposed by Vrbik.? The Vrbik model is summarized below The details on the Vrbik model can be found in the original publication by Vrbik.?

2 / ) ln 2704 1 ( 1 / 1

)]

( ) ( [ 5 0 ) ( ) ( ) 0

f

) ( ) 2 ln(

) 2 ( ln )

f = + − − + (A-3)

π

π /2) 0.1053 ]/ (

sin [ ln )

D

z1=1−2 (A-5)

pD h D

z2=1−2 + (A-6)

pD h D

z3=1−2 − (A-7)

h h

h pD = p/ (A-8)

h h

h bD= b/ (A-9)

bD

h

D= (1− )/2− (A-10)

h k k r

r wD = w z/ r / (A-11)

Appendix B – Karakas-Tariq Model for Perforation

Pseudoskin

Karakas and Tariq23 proposed the stepwise procedure below to

estimate the pseudoskin due to perforating

1 Compute the pseudoskin due to flow convergence in the

horizontal plane

) /

ln(w we

s = (B-1)

) (

( )

r θ =αθ + (B-2)

Eq B-2 is valid for all the phasing angles except

zero is tabulated as a function of the phasing angle

)

(θp

α

0

=

)

(θp

α r we (θp =0)=L p /4

2 Estimate the pseudoskin due to cylindrical wellbore

) /( w p

w

r = + (B-3)

] ) ( [ exp ) ( )

s θ = θ θ (B-4)

1

c and c2 are tabulated as functions of the phasing angle

3 Compute the pseudoskin due to flow convergence in the

vertical plane

spf

z =1/

∆ (B-5)

p z r p

z =∆ / /

∆ (B-6)

) / 1 ( 2

/

r = ∆ + (B-7)

) ( ) log(

)

a

a= θ + θ (B-8)

) ( )

1 p r pD b p

b

b= θ + θ (B-9)

1

a , , , and are all tabulated as functions of the

phasing angle

2

b pD b pD a

s =10 ∆ −1 (B-10)

4 Determine the perforation pseudoskin

wb v

H

s = + + (B-11)

5 Add crushed zone effect

First, estimate the skin factor due to crushed zone around

the perforation tunnels

) / ln(

) 1 ( '

p cz cz

p

p

k

k L

z

s = ∆ − (B-12)

The simultaneous effects of flow convergence toward perforations and the permeability impairment around the perforations are formulated as below

cz p

pc s s

s = + ′ (B-13)

6 Add formation damage effect

If the perforations are short and terminated inside the damaged zone then the total skin factor is given as below

)

d d

k

k s

s = + + + ′ (B-14)

where s x is negligible for most cases

If the perforations are long and extend beyond the damaged zone, the perforation length and wellbore radius in steps 1 through 5 are replaced with the effective perforation length and effective wellbore radius defined below

d d p

L′ = −(1− / )∆ (B-15)

d d w

r′ = +(1− / )∆ (B-16)

Appendix C – Alghorithm for the Approximate Model

Assume that the reservoir and fluid properties, pressure drop, the number of open segments, the location and length of the completed segments, and the degree of formation damage and perforation variables for all the completed segments are available Given , , , , , , , , , ,

p

∆ µ B o k r k z r e r w h n s h bi pi

h (k d /k)i ∆r di n spfi L pi r pi θpi (k cz/k)i,

czi r

∆

1 Divide the SCW or SPW into number of PCWs

or PPWs For the first iteration, estimate the initial values of for using

s n i

pt pi

h′= / .(C-1)

∑

=

= n s

i pi

h

1 .(C-2)

2 Recalculate the location of the open segments in the fictitious layers

∑−

=

′

−

=

1

i j i bi

h , h b′1 =h b1 (C-3)

If (h bi′ +h pi)>h i′ then

pi i

bi h h

h′ = ′− (C-4)

If then h bi′ =0

pi

i h

h =′ (C-5)

3 If the well is selectively completed then, for the

and , compute and for

using the Vrbik method and Hawkins equation,

respectively Then, estimate the total skin factor,

, for all the fictitious PCWs

r

z k

k / , r w h′ i h′ bi h pi (k d /k)i,

di r

ti

s′

ppi di pi

i

h

h

γ

1

(C-6)

4 If the intervals are perforated then estimate the

perforation total skin, , for 1 by

using the hybrid method described in the body of the

text Combine the influences of partial penetration

and perforation total skin as displayed below

pdci

s ≤ i ≤ n s

ppi pdci p

h

h

s′ = + ′

γ

1

(C-7)

5 Compute the specific productivity indecies for all

the fictitious PCWs/PPWs and the SCW

ti w

e o pci

s r

r B

k J

′ +

=

′

ln

1 2

141

~

µ (C-8)

i pci n i

h J

s

′

′

=

~ 1

~

1

(C-9)

6 Check if

sc pcn pci

pc

J

s

~

~

~

~

~

2

s tn ti

t

s′1 ≈ ′2 ≈ ′ ≈ ≈ ′ (C-11)

7 If the convergence criteria are satisfied then compute the productivity indicies or the rates for individual segments and the SCW

i sc i pci

J′ = ~′ ′=~ ′ (C-12)

h J

J sc =~sc (C-13)

)

pci

q = ′ − (C-14) and

)

sc

q = − (C-15)

8 If the convergence criteria are not satisfied then recalculate the thickness values for each fictitious layers for the next iteration

k i k sc

k pci k

J

J

~ ) 1 (

(C-16)

9 Go to step 2 Iterate on the steps 2-8 until the convergence criteria given in step 6 are satisfied

SI Metric Conversion Factors

bbl × 1.589 873 E-01 = m3

cp × 1.0* E-03 = Pa× s

ft × 3.048* E-01 = m

ft3 × 2.831 685 E-02 = m3

lbf × 4.448 222 E+00 = N lbm × 4.535 924 E-01 = kg

mD × 9.869 233 E-04 = µm2

* Conversion factor is exact

TABLE 1 – COMPARISON OF APPROXIMATE AND

2D ANALYTICAL MODELS FOR SCWs

md 100

=

= z

r k

rbbl/stb 3 1

=

o

'

000

,

1

=

e

r r w=0.25' h=100' h p1 =h b1 =10'

'

30

2 =

p

h h b2=50' h p3 =5' h b3=95' n s=3

0

=

di

s s d1,2,3=6,4,0 2D This study 2D This study

sc

sct

sc q

sct

sc q

sct

sc q

t

1

t

2

t

3

t

TABLE 2 – DATA USED TO COMPARE THE

APPROXIMATE MODEL AND SPAN FOR PPWs

rbbl/stb

,

o

cp

,

ft

,

ft

,

p

ft

,

b

ft

,

e

ft

,

w

md

,

r

r

z k

ft

,

d

r

md

,

d

spf

inches

,

p

inches

,

p

degrees

,

p

inches

,

cz

r

md

,

cz

TABLE 3 – DATA USED FOR THE SPW EXAMPLE WITH THREE PERFORATED SEGMENTS

s

ft ,

ft , 1

ft , 1

ft , 2

ft , 2

ft , 3

ft , 3

h

h pt/ 0.2

spf

inches ,

p

TABLE 4 – THE IMPACT OF SHOT DENSITY ON

SPW PERFORMANCE, L p =12"

Fractional segment rate

spf

n J sc s t

1st 2nd 3rd

OH* 1.100 4.4 OH** 1.703

* Formation damage, ** No formation damage

TABLE 5 – THE IMPACT OF PERFORATION LENGTH ON SPW PERFORMANCE

Fractional segment rate

spf

n L p J sc s t

1st 2nd 3rd