Transport in Porous Media Volume 80 Issue 2 2009 [Doi 10.1007%2Fs11242-009-9357-6] Colin Atkinson;...

8/10/2019 Transport in Porous Media Volume 80 Issue 2 2009 [Doi 10.1007%2Fs11242-009-9357-6] Colin Atkinson; Franck Monmont; Alexander Zazovsky -- Flow Perf

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Transp Porous Med (2009) 80:305328

DOI 10.1007/s11242-009-9357-6

Flow Performance of Perforated Completions

Colin Atkinson Franck Monmont

Alexander Zazovsky

Received: 23 November 2005 / Accepted: 10 February 2009 / Published online: 4 March 2009 Springer Science+Business Media B.V. 2009

Abstract A powerful approximate method for modeling the flow performance of perfo-

rated completions under steady-state conditions has been developed. The method is based on

the representation of the perforation tunnels surrounding a wellbore by the equivalent elon-

gated ellipsoids. This makes possible an analytical treatment of a 3D problem of steady-state

flow in a porousmedium with complex multiple production surfaces. Thesolution is obtained

for a vertical wellbore fully penetrating through a horizontal formation in the presence of

permeability anisotropy. The perforations are oriented horizontally, arranged in almost arbi-trary patterns, repeating along the wellbore, and may have different lengths and shapes. The

hydraulic resistances of perforations flowing inside them as well as the crushed zones around

them with impaired permeability are neglected. The approximate solution found was verified

by comparing the previous analytical/numerical solutions for a small number of perforations.

This approach allows one to determine the local skin or the effective wellbore radius for any

perforated interval, which can then be integrated into the conventional calculations of well

productivity and used for the perforating gun selection during perforation job design.

Keywords Well productivity

Steady-state flow

Skin

Effective wellbore radius

Productivity index Perforation Perforated completion Modeling

List of Symbols

ai , bi , ci Semi-axes of ellipsoids representing cylindrical perforation tunnels

dp Diameter of the perforation tunnel

dw Wellbore diameter

dwp Equivalent openhole wellbore diameter at infinite shot density

C. Atkinson

Department of Mathematics,

Imperial College London, London SW 721, UK

C. Atkinson F. Monmont (B) A. ZazovskySchlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, England

e-mail: [email protected]

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306 C. Atkinson et al.

D Simulation domain

G Dimensionless parameter used in Brooks correlation

h Thickness of perforated interval with repeating perforation pattern

H Perforated formation thickness

k Formation permeabilitykH Horizontal permeability

kV Vertical permeability

l Depth of penetration

lp Length of perforation tunnel

N Shot density equal to number of tunnels per foot of perforated interval

p Pressure

p0 Far-field formation pressure

pi Superposed solution components

pw Wellbore pressure

p Drawdown pressureQ Steady-state production rate

rp Radius of perforation tunnel

rw Wellbore radius

re Radius of far-field formation boundary

rw Effective wellbore radiusS Skin of perforated completion

v Flow velocity

wi j Distance between centers of equivalent ellipsoids representing perforation

tunnels

(xi , yi ,zi ) Coordinates of centers of equivalent ellipsoids

X, Y,Z Cartesian coordinates

k Permeability anisotropy ratio

Porosity

i Strengths of point sources representing far-field flows to ellipsoids

Fundamental solution for steady-state flow into ellipsoidal cavity

Boundary of simulation domain

Productivity index of perforated completion

0 Productivity index of openhole completion

Productivity index of perforated completion at infinite shot density Viscosity

Productivity ratio

Productivity efficiency

1 Background

Drilling wells for the production of hydrocarbons from the oil and gas reservoirs is often

accompanied by the reinforcement of their walls in order to avoid the borehole collapse or

fracturing during drilling. This is usually achieved by setting a casing, i.e., installing a metal

pipe inside a well and cementing it to the borehole wall. After construction of the well, the

hydraulic communication between the wellbore and the reservoir is established by perforat-

ing. A tool string with perforating guns is thus lowered downhole on a wireline cable to create

an array of tunnels that penetrate through the casing, the cement, and through the formation.

The perforating guns used in the oil and gas industry are equipped with shaped charges

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Flow Performance of Perforated Completions 307

Fig. 1 Schematic of perforated wellbore

filled by high explosives compressed behind conical liners, which can produce after initia-

tion cumulative jets of metal particles propagating with velocity up to 7 km/s and penetrating

into the reservoir to distances from a few inches to a few feet (see Fig. 1).

There are currently many ways to control the number, the length, and the shape of tunnels

created during perforating by selecting the appropriate type and size of the gun, as well asthe shaped charges. Generally speaking, the pattern of created perforation tunnels is usually

known. The typical perforating gun consists of repeating gun sections with the same posi-

tioning of shaped charges within each section. The distribution of shaped charges inside the

gun section is usually shown on the unwrapped surface of the gun section as a pattern of

holes created after perforating (see Fig. 2). Although all shaped charges within the same gun

section are identical, the created tunnels can differ in length and shape. This phenomenon is

explained by the fact that the gun diameter is smaller than the inner diameter of casing. For

this reason, the gun position inside the wellbore may not be centralized. For example, if the

borehole deviates from the vertical direction, the gun conveyed downhole on wireline or drillpipe is resting on its lower wall. This kind of gun positioning leads to different clearances

between the casing and the shaped charges distributed circumferentially resulting in different

penetration depths and tunnel shapes after perforating (see Fig. 3).

After perforating, the cased-hole interval is connected to the reservoir by a relatively

complex system of tunnels of different penetration depths, shapes, and orientations. The

optimization of a perforating job design requires establishing a correlation between the res-

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270180 3600 90

60

0

1

2

Fig. 2 Perforating gun with unwrapped surface illustrating the perforation pattern

Fig. 3 Asymmetrical gun

positioning inside the wellbore

leads to non-uniform clearance

and variable penetrations fordifferently oriented tunnels

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Fig. 4 Schematic of perforation

pattern with basic notationsOpen HoleDamage Zone

PerforationSpacing

PhasingAngle

PerforationLength

Crushed Zone

PerforationDiameter

ervoir flow into the perforated interval and the parameters of the perforated completion, such

as the number of perforations per length of the perforated interval, known as shot density perfoot, and the tunnel distribution over the wellbore surface, as well as the lengths and shapes

of perforation.

The flow performance of a production well is usually characterized by its productivity

index, which is defined as the ratio of the production rate to the applied drawdown pressure,

i.e., the difference between the far field formation pressure and the bottom hole pressure.

The main challenge of establishing the correlation between the geometrical parameters of a

perforated completion and its productivity index consists in solving a steady-state problem of

flow in porous medium in a rather complex 3D domain with an irregular producing boundary

represented by the open surface area of perforation tunnels.

The early approaches to this problem are discussed byBrooks(1997), who has created acorrelation for predicting the productivity of perforated completions and the guidelines for

gunselection basedon these correlations, using dimensional analysis and results of numerical

modeling. This correlation involves the average parameters of perforated completion such

as the shot density per foot, the distribution of shaped charges over the gun surface, and the

average tunnel length and diameter.

In order to estimate the productivity index of a perforated completion with variableparam-

eters of perforation tunnels within a gun section, numerical modeling may be required. The

numerical simulation, however, becomes a challenge for the perforated completions with

many tunnels due to the geometrical complexity of the flow domain. For this reason, thefirst approaches to numerical modeling of flows to perforated wells were focusing on regular

perforation patterns since the problem geometry can be significantly simplified if the element

of symmetry of the flow domain can be distinguished.

An example of perforation completion geometry for a regular perforation pattern is shown

in Fig. 4.It can be completely characterized by the phasing angle, the vertical perforation

spacing, and the size and shape of the identical tunnels. The tunnels are usually surrounded

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by a crushed zone with reduced permeability, which can also be represented by a cylinder.

The phasing angle and the perforation spacing determine the shot density.

Harris(1966) probably carried out the first numerical simulations of perforated comple-

tions for simple regular patterns of perforations and calculated the equivalent skin factors of

perforated completions, which could be used for calculating the productivity of the perfo-rated well.Hong(1975) has extended the numerical modeling of perforated completions for

a variety of regular perforation patterns, as well as the tunnel sizes and shapes. The damaged

zone around the wellbore with impaired permeability was also included in his model.

Locke(1981) has carried out a series of 3D FEM simulations of perforated completions

for a variety of realistic perforation patterns corresponding to different shot densities and

angular phasing of 0, 90, 120, and 180for the adjoining tunnels. Cylinders of equiva-lent lengths and diameters represented the tunnels. The numerical model also involved the

crushed zone around each perforation tunnel with reduced permeability. Nomograms were

created for calculating the productivity of perforated completions. This study confirmed that

the penetration (or perforation tunnel length) is substantially more important than the per-

foration diameter and also that, for the equivalent shot density, a spiral 90 shot phasingprovides the best productivity of the alternatives tested (0, 120, and 180).

This effort has been extended byTariq(1987) with more accurate FEM simulations,

accompanied by grid sensitivity analysis, and incorporation into the model of the non-Darcy

effect resulting from the higher velocity converging flow around each perforation tunnel.

The non-Darcy flow was found to be important for high rate gas wells. Using the same

approach,Tariq et al.(1989) have investigated the flow performance of perforated comple-

tions in the presence of natural fractures and laminations and demonstrated strong sensitivity

of productivity to the density and orientation of the formation of heterogeneities.Karakas and Tariq (1991) have generated an extensive data bank of FEM simulation

results representing various combinations of perforation parameters. Using this data bank

and a series of semi-analytical approximations, they have created correlations for the calcu-

lation of pseudo-skin factor and productivity index of perforated completions, which are still

in common use in the industry.

A comprehensive dedicated numerical modeling of perforated completions was car-

ried out byDogulu(1998). The approach developed, which is based on a finite-differ-

ence technique and adaptive grid generation, allows for a very detailed representation of

perforation completion geometry. It was found, however, to be extremely CPU intensive.

Discrepancies of the order of 20% to 30% with previous simulations and approximate solu-tions were observed for anisotropic formations with horizontal to vertical permeability ratio

of 10.

Brooks(2003) applied a boundary element method with distributed sources and sinks for

simulating reservoir flows into perforated completions and for estimating their productivity.

This technique, although being approximate, improved the productivity estimates in the pres-

ence of permeability anisotropy and a crushed zone. It is less computationally intensive than

the conventional finite-difference/finite-element methods. In this method, however, the total

number of unknown source strengths, distributed along the perforations and the borehole

surface, increases with the shot density especially for the complicated irregular perforationpatterns, andthis slows down thecalculations significantly andmay compromise their robust-

ness. Nevertheless this methodallows for the simulation of flows to very complex perforation

patterns, which are difficult to model with conventional techniques.

Recently, Hagoort (2007) has proposed an analytical model of steady-state flow to a single

perforation represented by an elongated prolate spheroid, which is similar to the model used

in this study. He has extended this model to a regular arrangement of perforations in a vertical

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Pay Zone

RepeatingPerforationPattern

SurroundingRock

SurroundingRock

Zero FluxBoundaries

SimulationDomain WithDistinguishedPerforationPattern

H h

Fig. 5 Perforated interval with distinguished simulation domain containing repeating perforation pattern

2rw

2re

p0

pw

h

p

c

-

+

eD

Fig. 6 Simulation domain with perforation pattern used for the construction of approximate solution

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Eliminating the velocity between Eqs. 2.1and2.2,one obtains the equation for the pore

pressure

2p

X2+

2p

Y2+ 1k

2p

Z2= 0, k=

kH

kV

(2.3)

Equation2.3has to be solved in a 3D domain Dsurrounding the perforated interval which

is shown in Fig. 6. This domain is restricted by an external cylindrical boundary e with

specified constant pressure on it p0and the two virtual top and bottom boundaries,+and

are maintained at the zero-flux condition. The internal surface consists of the surfaceof the wellbore casingc, also maintained at the zero-flux condition, and the surface of all

perforationsp, with the given wellbore pressure pw, on it.

Thus, the solution of Eq. 2.1has to satisfy the following set of boundary conditions:

p= p0 on e

p= pw on p

(n p) = 0 on +, , c, (2.4)

wherenis the unit normal vector at the surfaces+,, andc.If the solution for the problem (2.3) and(2.4) is found, the total production rate can be

obtained by integrating the velocity over the surfaces of all perforations within the simulation

domainD

Q=

P

(v n)d (2.5)

After that, the productivity index of the perforated interval can be calculated

= Qp

, p= p0 pw. (2.6)

This productivity index is usually compared to the productivity index of the openhole well-

bore interval(Economides and Boney 2000)

0=Q0

p= 2 hkH

log (re/rw). (2.7)

The productivity ratio

= /0 (2.8)is widely used for the comparison of different completions. Finding the productivity ratio for

the perforated completion, shown in Fig. 6,with arbitrary number, orientation, and size of

the tunnel, represented by elongated ellipsoids, is the main aim of this study. Dealing with

cylindrical perforation tunnels, we replace them by equivalent ellipsoids having the same

length and volume.

3 Construction of the Approximate Solution

3.1 Modeling Approach

The complexity of the above-formulated problem (mainly its geometry) makes it difficult to

solve even using advanced numerical techniques and powerful computers. For this reason, we

derive below an approximate solution based on additional assumptions and simplifications.

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unknown strengths. Then, the principle of superposition can be used to describe the pressure

generated by a series of such perforations at the far-field.

Let usconsideranarray ofnellipsoids with centers located at (xi , yi ,zi )1in , and havingsemi-axes

ai ,

bi ,

ci . The fundamental solution for this array of ellipsoids (Bateman

1959) is

p1= p0 +n

i=1i

i (wi )

i (0), i (w) =

w

dt(t+ ai ) (t+

bi ) (t+ ci ). (3.4)

Here,wi(i= 1, . . . , n) is given by the implicit equations

(x xi )2wi+ ai

+ (y yi )2

wi+ bi+ (z zi )

2

wi+ ci= 1, i= 1, . . . , n. (3.5)

The unknown source strengths i(i=1, . . . , n) have to be determined from the system oflinear equations

j+n

i= 1i= j

ii

wi j

i (0)

= pw p0, j= 1, . . . , n, (3.6)

where

wi j is the distance between the centres of the ellipsoids, which satisfy the cubic

equations

xj xi

2wi j+ ai

+yj yi

2wi j+ bi

+zj zi

2wi j+ ci

= 1, i, j= 1, . . . , n; i= j. (3.7)

When considering the interaction between the perforations, we have assumed as indicated

above that, from a distance, the effect of another perforation can be approximated by a point

source located at the centre of the ellipsoid. Indeed, on calculating the pressure perturbation,

induced by the ellipsoidiat the surface of the ellipsoid j , we have replaced it by the average

pressure found at the centre of ellipsoid j . The conditions of validity of these assumptions

have been investigated inChen and Atkinson(2001), and have found to give results within

10% of more elaborate semi-analytical methods.

3.4 Anisotropic Permeability and Oriented Perforations

In order to use the fundamental solutions shown above for deviated perforations in an aniso-

tropic formation, in which the pore pressure satisfies Eq. 2.3,one has to scale the coordi-

nates as

x= X, y= Y, z= kZ, k= kH/kV. (3.8)

This transformation reduces Eq. 2.3to the Laplace equation. It also affects the semi-axes ofthe ellipsoids representing the perforations. Since the long axes of the ellipsoids are directed

horizontally, the transformation (3.8)changes only their small vertical axes

ci .

Assuming that the semi-axes of nellipsoids after the transformation (3.9) are denoted

again as

ai ,

bi ,

ciand their centers are located in the points (xi ,yi ,zi ), one finds that

the fundamental solution is given by Eqs. 3.43.7.

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3.5 Correction Due to the Plane Boundaries

The correction pressure p2 (x,y,z)has to satisfy the Laplace equation inside the layer D

and compensate the fluxes induced by the pressure p1 (x, y,z)at its boundaries.

The problem of determining p2 (x, y,z) is significantly simplified by replacing the flow-ing ellipsoids by the point sources with their appropriate strengths, i.e., by replacing the

expression (3.4) everywhere by its asymptotic representation

p1n

i=1

2i

i (0)

1

w1/2i

as r=

x2 + y2 , (3.9)

wherewi x2 + y2 + (z zi )2 andr >>ri=

x2i+ y2i.This simplification can be justified if the thickness of the layerDand the distances from

the boundaries to the centers of the perforations are large compared to the lengths of theperforations. These conditions may not be satisfied in our situation, because some perfora-

tions can be located close to the plane boundaries. Bearing this in mind, the accuracy of the

approximate solution can be improved by choosing a layer of larger thickness so that most

of the ellipsoids can satisfy the above-mentioned requirement.

The approximate solution for the correction term is (see Appendix A)

p2 (x,y,z) =n

i=1

2i

i (0)F(i ,z,zi ), (3.10)

where the reference planez

=0 is located in the middle of the simulation interval

|z

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where

Ci (z) =ri

r2i + (z zi )23/2

+

0

K(t,z,zi )J0 (ri t) tdt. (3.13)

3.7 Summary of the Complete Solution

The complete approximate solution of the problem can be represented as

p(x, y,z) =n

i=1i

i (x,y,z)

i (0), (3.14)

where

i= i (wi ) + 2F(i ,z,zi ) 2r2wcos ( i )

hr

h/2h/2

Ci (z)dz. (3.15)

The unknownsi have to be determined from the requirement that the drawdown pressure

is the same for all perforations and equal to p0 pw. Due to the logarithmic behavior ofthe solution at infinity within the domain Dcontaining the simulation interval, i.e., when

r=x2 + y2

1/2 , the far-field pressure is specified at the virtual external boundaryr= re. This leads to the following system of linear equations fori

ni=1

i j i= p0 pw, j= 1, . . . , ni j= 1i (0)

i (xe,ye,ze) i

xj , yj ,zj

(3.16)Here,

xj , yj ,zj

are the centers of the ellipsoids and (xe,ye,ze)is the point at the virtual

formation boundary r= re. The far-field points (xe, ye,ze) may be chosen arbitrarily for anyequation of the system (3.16), because i (xe,ye,ze)const at large distances from thewellbore. Another option is to replace the terms i (xe,ye,ze) by their values, averaged over

the angle 0 and the formation thickness |z| h/2. This was the preferred choicein our calculations.

Some care needs to be taken for the calculation ofixj , yj ,zj

. Wheni= j , the term

i

wi j

/i (0)has to be replaced by 1, as has been done in Eq. 3.6.

4 Flow Efficiency Calculations

In order to find the total flow rate through the perforationsQone can use the far-field asymp-

totic behavior of the solution (3.14) at r >>h . Thepressure fieldsp1andp3do notcontribute

to the total flux at infinity and therefore, we have

Q= 2r h

kH

p2r

, r >>h . (4.1)

Using (3.10) and the asymptotic relationship F/r 2/(hr), one obtains

Q= 8 kH

k

, =n

i=1

i

i (0), k=

kH

kV. (4.2)

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Thus, theparameter is the main quantity, which has tobe determinedfor estimating the flow

performance of perforated completions. It has the dimension of pressure gradient multiplied

by flow area or pressure multiplied by length.

After the determination of, the productivity index and the productivity ratio =/0of the perforated completion can be found from(2.6) to(2.8). The productivity ratiois often represented in the equivalent form

= log (re/rw)log (re/rw) + S

. (4.3)

Here, Sis the pseudo-skin factor or, for short, skin, which is originally defined for an open-

hole completion as a pressure drop at the wellbore wall due to permeability damage during

drilling as

p S rp

r

r=rw = pw. (4.4)

This allows one to express the productivity index of an openhole interval as

= 2 hkHlog (re/rw) + S

(4.5)The equivalent skin is found for many typical situations, for example, for a partially pen-

etrating or fractured well, and also can be estimated from the field production data. If the

skin is equal to zero, the productivity matches the productivity of an openhole wellbore; a

positive skin indicates reduced productivity and a negative skin corresponds to the openhole

wellbore of larger radius. For this reason, the equivalent wellbore radius rwis sometimesused for the characterization of wellbore flow performance. It is defined as the radius of an

openhole wellbore, which has the equivalent productivity index, i.e.,

= 2 hkH log

re/rw

. (4.6)Comparing(4.6) with (2.7), one can express the effective wellbore radius through the skin

factor

rw=

rweS. (4.7)

Therefore, both parameters, Sandrw, can be used for the characterization of the flow effi-ciency of perforated completions.

In order to determine the skin, the production rate Qfrom the perforated completion for

the applied drawdown pressurephas to be found since

S= log

re

rw

+ 2 hkHp

Q. (4.8)

5 Examples

The model developed in the previous sections is now used to estimate the well productivity

for different geometrical configurationsof perforated completions. Following Brooks (1997),

the productivity ratio is plotted as a function of three dimensionless groups controlling the

hidden dependencies of the completion performance. They are determined by the dominant

variables: the perforation length (lpi), the shot density (N), the radius of the perforation

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tunnel (rpi), the permeability ratio (k), and the wellbore radius (rw). This dimensionless

grouping (see below) has been shown to capture the major physical phenomena involved and

gives a useful representation to assess the trade-offs between the different characteristics of

perforating guns.

5.1 The Dimensionless Groups

The flow rate Qfrom the reservoir to the perforated wellbore is a function of the dominant

variables, which can be grouped to form three dimensionless parameters. Since the forma-

tion is anisotropic, this means that any vertical dimension is multiplied by a factor

kand

therefore, the equivalent shot density is N/

k. The perforation tunnel is also represented

by an elliptical cross section of area proportional to rpi rpik, which is equivalent to acylindrical cross section of average radius rpi

1/4k . From these observations, three groups can

be defined

G1= rpi1/4k N/

k, G2= lpiN/

k, G3= rw/re (5.1)The first parameterG 1can be viewed as a dimensionless perforation radius. The second one

G2can be interpreted as a dimensionless depth of penetration of the tunnel length. The third

oneG 3is a dimensionless wellbore radius.Brooks(1997) combined these three parameters

into a single one instead of using the radius of the perforation tunnel rpiand its diameter

dpi= 2rpito give

G= 5/8k N3/2d1/2pi lpi. (5.2)

Brooks(1997) has also introduced the productivity efficiency, which is the dimensionless

parameter characterizing the productivity of the perforated completion, determined as the

ratio

= /. (5.3)Here, is the productivity index of the actual perforated completion and is the produc-tivity index of a virtual perforated completion at infinite shot density N. In fact, the limit

N corresponds to an open hole wellborehaving thediameter dwpmatching thedepth ofpenetration of perforation tunnels into the formation, i.e.,dwp

=2 rw + lp. In the absenceof permeability damage near the wellbore, the productivity index can be expressed as

=2 hkH

log

re/

rw + lp . (5.4)

Using 270 data sets generated with different modeling and simulation tools,Brooks(1997)

has plotted the productivity efficiencyversus the dimensionless parameter G, and hasfound that the relationship (G) for those data sets collapses to a single curve, which canbe approximated as

= 0.971 exp G4.4

0.38. (5.5)

5.2 The Input Data

In order to compare our solution with the approximation (5.5), we are using below exactly

the same data sets asBrooks(1997) corresponding to the spiral tunnel arrangements with

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a phasing angle of 45. The parameters of perforated completions are shown below in theimperial units as they have been stated in the original article byBrooks(1997).

Wellbore radius rw= 4.31inch

Reservoir thickness h= 12feetReservoir radius re= 660feetHorizontal permeability kH= 1, 000mdVertical permeability kH= 1, 000, 100, 20mdViscosity = 1 cpDrawdown pressure p pw= 3barPerforation length lpi= 3, 6, 9, 12, 15, 18inchPerforation tunnel diameter dpi

=0.31, 0.61, 0.92inch

Shots per meter N= 1, 2, 4, 8, 12spfPhasing angle = 45

5.3 Results of Calculations

We first plot the productivity ratio as a function of shot density in Fig. 7to show the diffi-

culty in understanding the trend in the data when perforating length, radius, and anisotropy

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

Shot density, spf

ProductivityRatio,

45 Phasing Angle

Fig. 7 Productivity ratioas a function of shot density N

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103

102

101

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

G

Produc

tivityEfficiency,

45 Phasing Angle

Brooks Correlation

Fig. 8 Productivity efficiencyversus the parameter Gfor the phasing angle 45

ratio are varied. The productivity efficiencyis then plotted in Fig. 8as a function of thedimensionless parameterG defined in(5.2). The color code used in Fig.8matches with the

color code in Fig. 7.The results corresponding to data sets with different shot density are

shown using the same color: blue for 1 spf, green for 2 spf, red for 4 spf, turquoise for 8 spf,

and violet for 12 spf, respectively. The clustering of the data proves that the dimensionless

parameter Ggives a better coherence to the results, but still is by no means perfect. There

is some significant scattering for the higher efficiencies. This scattering might be due to the

treatment of the boundary conditions and the oscillations they induced. The productivity effi-

ciencies obtained are close to the correlation(5.5) at higher shot densities, but significantly

different from the Brooks correlation (yellow points in Fig. 8)at lower shot densities. Our

hypothesis is that the cases with lower shot densities should be more difficult to simulate

using the semi-analytical approach ofKarakas and Tariq(1988). This approach is based on

distinguishing the elements of symmetry in flow patterns, which involve small numbers of

perforation tunnels, treating them separately, and then combining results together to charac-

terize an entire flow pattern. When the number of tunnels becomes small, especially if their

lengths are comparable to the borehole diameter, the flow pattern becomes more asymmetri-cal and sensitive to the presence of the wellbore. It requires careful treatment preserving all

its 3D features. This is a situation when oversimplified approximations may fail.

At the same time, the solutions obtained exhibit the behavior, which is very similar to that

predicted by the correlation (5.5):

An increase in shot density, length, and radius improves the productivity efficiency.

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103

102

101

100

101

102

5

0

5

10

15

20

25

30

35

40

45

G

Skin,

S

45 Phasing Angle

Fig. 9 Skin versusG

It is better advised to increase the shot density or the length of perforations rather thantheir diameter, if one wants to improve the productivity efficiency.

A strong anisotropy ratio may affect significantly the productivity efficiency.

The skin Sis plotted in Fig. 9versus the dimensionless parameterG. A good clustering of

the results with respect toG can be appreciated especially at high shot densities or large G .

Furthermore, the skin becomes negative above a value ofG= 3 indicating that the perforatedcompletion is then performing better than the openhole wellbore.

The example presented above illustrates the influence of the basic parameters of a perfo-

rated completion, such as the number of tunnels, their lengths, and diameters, on the flow

efficiency of a perforated completion. The parameter, which has been left behind so far, is the

phasing angle, which also should have some impact on the flow pattern and the productivity

of the perforated completion. It is also worth noting that the correlation (5.5) does not involve

the phasing angle at all although it is involved implicitly in the shot density for regular dis-

tributions of perforation over the surface of the perforated interval. Obviously, for large shot

densitiesanduniform distributions of tunnels over thewellboresurface, the impact of phasingangle on the productivity should be less pronounced. In contrast, for small shot densities and

asymmetrical distributions of tunnels, the effect of phasing angle on the productivity should

be stronger. The approach presented above allows for handling arbitrary shot densities and

arbitrary distributions of tunnels connecting the wellbore with the reservoir.

In order to demonstrate this capability and the impact of the phasing angle on the pro-

ductivity efficiency , the calculations have been repeated for the same geometrical

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103

102

101

100

101

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

G

Produ

ctivityEfficiency,

Zero Degree Phasing Angle

Brooks Correlation

Fig. 10 Productivity efficiency versus Gfor a zero degree phasing angle

parameters of perforated completions using the phasing angle equal to zero degree. This

means that all perforations are located on one side of the wellbore representing in some sense

the most asymmetrical and localized distribution of tunnels. Under these circumstances, one

can expect asymmetrical flow patterns and significant reductions in productivity efficiency.

The results are shown in Fig. 10using the same color code and they indeed indicate the

reduction in productivity efficiency pronounced at high shot densities. Our results overlapwith those predicted by the correlation(5.5) only for a small subset of simulated perforated

completions.

6 Summary

This work described a semi-analytical method to compute the productivity of perforated

completions for any given geometrical arrangement of perforation tunnels. In this model, the

perforations were approximated by equivalent elongated ellipsoids and their far-field pres-sures represented by point sources of unknown strength to be determined. We also indicated

the importance of a proper treatment of the no-flow boundaryconditions at the top and bottom

of the layered formation as well as on the wellbore surface. We then described the procedure

followed to calculate the unknown point source strength of each perforation.

The complete solution is finally used to calculate the productivity ratio, the productivity

efficiency, and the skin of a perforated interval as a function of the relevant dimensionless

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groups. We obtained relatively good agreement withBrooks(1997) for higher shot densities

and more uniform distributions of perforation over the wellbore surface. At the same time,

significant deviations from his correlation have been obtained at low shot densities and for

asymmetrical distributions of perforation tunnels around the wellbore corresponding to zero

phasing angles. We have demonstrated that our approach captures the net reduction in pro-ductivity efficiency for asymmetrical geometry of perforation patterns with high shot density

and the net gain in productivity efficiency for low shot density perforation patterns.

7 Appendix A: Correction due to Plane Boundaries

The correction term p2(x, y,z)has to compensate for the additional fluxes through the hor-

izontal boundaries of the layer D, which contains the representative pattern of perforations,

induced by the term p1(x,y,z). It also has to satisfy the Laplace equation inside the layer

D. Since we know that

wi= 2i+ (z zi )2 , 2i = r2 + r2i 2rricos ( i ) , (A.1)

then the equation 2p= 0 can be written as

1

i

i

i

p2

i

+

2p2

z2 = 0. (A.2)

The boundary conditions at boundariesz

= chof the layer D, according to Eq. 4.4, are

p2

z=

ni=1

2i

i (0)

(z zi )w

3/2i

, z= ch. (A.3)

EquationA.2and with the boundary conditions (A.3)can be solved by using the Hankel

transform defined as

p()= H(p (x) ; ) =

0

x p(x)J (x)dx

p(x)=

0

p ()J (x)d.(A.4)

The Hankel transform of order zero of the Laplace Eq.(A.2) expressed in cylindrical

coordinates thus becomes

H0

1

i

i

i

p

i

+

2p

z2

,

=

d2

dz2 2

p0 (,z) = 0. (A.5)

The transformed Eq. A.5has to be solved subject to the transformed boundary conditionsatz= ch. Omitting the constant coefficient 2i /i (0), the transform of each term of thesum in the right-hand side of Eq. A.3is given by the formula

0

i (z zi )2i+ (z zi )2

3/2Jo (i ) di= 1i ei , z= ch, (A.6)

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where we used the following result (Erdelyi et al. 1954):

0

1/2i J0 (i ) (i )

1/2

(z zi )2

+(i )

23/2di= (z zi )1 1/2e(zzi ). (A.7)

EquationA.5has a solution

p0 (,z) = A1ez + A2ez. (A.8)Its parametersA1andA2have to be determined from the boundary conditions atz= ch.

Since

p0(,z)z

= A1ez + A2ez (A.9)

using (A.6)for each component of the sum in the boundary condition(A.3), one obtainsA1ech + A2ech = e(chzi)A1ech + A2ech = e(ch+zi ). (A.10)

Solving this system of equation finally results in

p0(,z) =e2ch cosh [(z zi ) ] + cosh [(z + zi ) ]

sinh (2ch). (A.11)

The inverse Hankel transform then gives

p (i ,z) =

0

e2ch cosh [(z zi ) ] + cosh [(z + zi ) ]

sinh (2ch)

J0 (i ) d. (A.12)

Using superposition principal, the entire correction term p2can be represented as

p2(x, y,z) =n

i=1

2i

i (0)F(i ,z,zi ), (A.13)

where

2i = r2 + r2i 2rricos ( i ) , = arccos (x/r)

F(i ,z,zi ) =

0

K(,z,zi )J0 (i )

2

sinh (h)

d

K(,z,zi ) =eh cosh [(z zi ) ] + cosh [(z + zi ) ]

sinh (h).

(A.14)

The additional term 2/sinh (h ) is added to the integrand to fix the integral divergence at

0.

8 Appendix B: Correction due to Wellbore

The thirdcomponentp3(x,y,z) of theapproximate solution (3.1) is derived in this Appendix.

The pressure p3has to satisfy the Laplace equation inside the formation layer and zero-flux

conditions at itsplane boundaries. Thecorrection function of thepressurep3is to compensate

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for the additional fluxes through the wellbore surface, induced by the first two components

p1and p2, without creating an additional flux at the infinity.

Letusfirst calculate theflux through thewellboresurfacecreatedby thecombined pressure

field p12= p1 + p2. We shall assume again that the ellipsoids representing the perforationscan be replaced by the point sources that are located in their centers. Then, one has

p12=n

i=1

2i

i (0)

w1/2i +

0

K(t,z,zi )J0 (i t)

2

sinh (2cht)

dt

(B.1)

Differentiating (B.1) with respect tor, we obtain

p12

r=

ni=1

i

i (0)

w3/2i

wi

r+ 2

0

K(t,z,zi )J0 (i t)

i

rtdt

, (B.2)

where

wi= 2i+ (z zi )2 , 2i = r2 + r2i 2rricos ( i )wi /r= 2i /r= 2 [r ricos ( i )] .

For simplicity and for this calculation alone, we assume that rw


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p3/z= 0. (B.5)The pressure p3also has to be a solution of the Laplace equation in the remaining domain

p=1

r

r

r

p

r+ 1r2

2p

2+2p

z2 = 0. (B.6)Using Fourier series, the solution p3, we are looking for, can be represented in the form

p3 (r, ,z) =n

i=1

2icos ( i )i (0)

Ti (r,z), (B.7)

where

Ti (r,z)

=

A0

2r +

j=1

Ajcosj z

ch K1

jr

ch

+

j=0Bjsin

(2j+ 1)z

2ch

K1

(2j+ 1) r

2ch

.

(B.8)

Each term Ti (r,z)is the solution of Eq. B.6.It involves the modified Bessel function

K1 (x)and the unknown coefficients A0, Aj , and Bj , which have to be found from the

boundary conditions.

One can first verify that the conditions(B.5)are automatically satisfied. Then, by differ-

entiating(B.7)with respect torand matching (B.4), we get Ti

r

r=rw

= Gi (z) , |z|


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Transport in Porous Media Volume 80 Issue 2 2009 [Doi 10.1007%2Fs11242-009-9357-6] Colin Atkinson;...

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