1

Optimization of Horizontal Well Completion

Final Report by Yula Tang

Objective

The productivity of a horizontal welldepends on the reservoir flowcharacteristics, while the reservoir flowcharacteristics are functions of reservoirparameters and wellbore geometry as wellas the hydraulics of the wellbore. To obtaina comprehensive model for horizontal wellperformance, the influence of wellcompletion on both wellbore hydraulics andreservoir flow performance should be takeninto account. Therefore, the objectives ofthis study are as follows.1. Experimentally investigate the flow

behavior in horizontal wells with bothperforation completion and slotted-linercompletion. The experimental work isconducted to investigate the effects ofthe different completion geometry,densities and phasing on the flowbehavior in the horizontal well;

2. Based on the experimental study, awellbore flow model is developed whichcan be used for various completionscenarios;

3. Develop a reservoir performance modelwhich considers the effect of flowconvergence toward slots andperforations on the surface of the well;

4. Couple the wellbore hydraulics andreservoir models to build acomprehensive model that considers theinteraction between the horizontalwellbore and the reservoir through smallopenings on the surface of the well;

5. Develop efficient algorithms tonumerically evaluate the complexanalytical expressions;

6. Develop a user-friendly software forhorizontal well completion design thatcan be used to perform sensitivityanalysis and optimal well completiondesign;

7. Investigate various completion scenariosto develop completion guidelines foroptimzing the well performance.

Over the last two years, we have satisfiedthese objectives. These achievements areprovided to our member companies in theform of this final report and software ofHorizontal Well Completion Optimization(HORCOM).

Literature Survey

In a horizontal well, depending upon thecompletion method, fluid may enter thewellbore at various locations along the welllength. The pressure distribution in ahorizontal well can influence the wellcompletion and well profile design, as wellas having an impact on the productionbehavior of the well. Therefore, both thepressure drop versus flow behavior alongthe well and the relationship between thepressure drop along the well and the influxfrom the reservoir need to be understood.

The petroleum industry startedinvestigating horizontal wellbore hydraulicsin the late 1980’s. A new friction factor correlation for horizontal wellbore wasproposed by Asheim et al1 which includes

2

accelerational pressure losses due tocontinuous fluid influx along the wellbore.They assumed that the injected fluid entersthe main flow with no momentum in theaxial direction. Kloster2 performedexperimental work and concluded that thefriction factor versus Reynolds numberrelationship for perforated pipes with noinjection from the perforation does not showthe characteristics of regular pipe flow. Thefriction factor values were 25-70% higherthan those of regular commercial pipes. Healso observed that small injections throughperforations reduced the friction factor.

Yuan3 and Yuan et al4 studied the flowbehavior in perforated horizontal wells andslotted-liner completed horizontal wells. Byusing the principles of mass and momentumconservation, a general horizontal wellfriction factor expression was developed.Horizontal well friction factor correlationsfor limited cases of completion geometrywere developed by applying experimentaldata to the general friction factor expression.It was observed that the friction factor of aperforated pipe with fluid injection could beeither smaller or greater than that of asmooth pipe, depending on influx to mainflow rate ratios. Because the available dataconsider either single opening or limitedmultiple opening cases, the influence of theshape of the area of the opening was notthoroughly investigated. Yuan’s work, however, forms the basis of this extensionon wellbore hydraulics

In 1990, Dikken5 emphasized theimportance of wellbore pressure losses foran openhole horizontal well for the firsttime. He, however, used the assumption ofuniform specific productivity to couple thewellbore and reservoir flows. Thisassumption, in fact, neglects the influence ofwellbore hydraulics on the reservoirperformance. Therefore, it cannot predictthe correct flux and pressure drawdownalong the well length.

In 1993 and 1995, Ozkan & Sarica et al6,7

used the physical coupling conditions(pressure and flux continuity at the wellsurface) to obtain a solution to compute theopen-hole horizontal well performance.

In 1994, Yildiz & Ozkan8 studied theperformance of selectively completedhorizontal wells (i.e., only some segments ofthe well are open to flow with arbitrarydistribution of the open interval and skin).They derived a general Laplace spacesolution describing the transient pressureresponse. The flow rate distribution isobtained as a result of a matrix solution.They also derived the asymptotic solutionfor different periods of time. In their model,the wellbore pressure losses are neglected(assumption of infinite conductivity).

In 1990, Ahmed, Horne, and Brigham9

presented an analytical solution for flowinto a vertical well via perforations usingGreen's functions. This solution containsproducts and series of Bessel functions andtheir derivatives. An array of eigenvalues iscomputed from an implicit equation and areused in the computation of the solution.They failed to calculate the explicit equationfor the eigenvalues. Although theyconsidered the perforation as a surfacesource and performed coordinate transformsto express the integration for thecomplicated perforation geometry, they stilltreated the perforations as line sinks. For ourproject, the integration along perforationsurface on horizontal well is also difficult.So this method of coordinate transformmight be of special meaning as a reference.

Spivak and Horne10 studied the transientpressure response due to production with aslotted-liner completed vertical wellboreusing source function method in 1982(?).They modeled the slots as line sources offinite length. However, the simplificationthey used is not applicable for general slotdistributions.

3

Hazenberg and Panu11 investigated flowinto perforated drain tubes. The problemconsidered in their work bears similarities tothe horizontal well problem and haspotential of yielding a simplified solution.

In 1991, Landman12 studied theoptimization of perforation distribution forhorizontal wells. The model couplesDarcy’s flow for each perforation in an infinite reservoir with the pipe flow (1-Dmomentum equations). Thus the perforatedwell is treated like a pipe manifold with T-junctions representing the perforations alongthe wellbore. The authors claimed that theirmodel takes into account the wellborepressure drop and the effect of perforationdistribution. They, however, used a simpleapproximation for the reservoir flow andtheir wellbore pressure model does notconsider the effects of perforationdistribution on the flow pattern andmechanism. This paper still provides usefuldiscussion on the perforating optimizationstrategies.

In 1998, Yildiz and Ozkan13 presented a3-D analytical model for the analysis oftransient flow toward perforated verticalwells. This work based on their previous 2-D partially opening model. In their model,the perforations are presented as linesources. They applied Laplace transform totime and Fourier transform to x, y and zcoordinates. A pseudo-skin expression isderived from the long-time solution toestimate the inflow performance. Thetreatment for perforations gives us a goodreference to solve for the perforatedhorizontal wells.

14In 1999 , Ozkan, Yildiz and Raghavaninvestigated the transient pressure behaviorof perforated slant and horizontal wells anddiscussed the implications of perforations onthe analysis of pressure and derivativeresponses. The results presented are derivedfrom a 3D analytical model. It is shown thatconvergent flow into perforations

significantly influences the early time flowcharacteristics. Their model and 3-Dgeometrical treatment provides the basis forour long-time asymptotic solution ofperforated horizontal wells.

Experimental Study onHydraulics of CompletedHorizontal Wellbore*

1. Test Facility

An existing small scale Tulsa UniversityFluid Flow Projects (TUFFP) test facility(Fig. 1) was used to acquire data fordifferent horizontal well completiongeometries. The test facility is composed ofthree parts: a flow loop, test sections (Fig. 2)and an instrumentation console. The flowloop consists of the liquid handling system(water tank, and screw and centrifugalpumps) and metering and flow controlsections (turbine meters, temperaturetransducers, a pressure transducer andcontrol valves). The test section consists ofa perforated or slotted test pipe, 50 layers ofcloth to ensure uniform influx from theopenings, a 6-in. diameter casing housingand instruments to measure the pressuresand differential pressures. Water is used asthe testing fluid.

2. Tests

Ten new test sections were designed for theinvestigation of the effects ofslot/perforation density and phasing. Eachtest section is made up of a 10-ft long, 1-in.diameter horizontal pipe with a 4-ft long testsection. Experiments were conducted understeady state flow conditions with Reynoldsnumber ranging between 5,000 and 60,000.The following parameters are considered:

Perforation density and phasing.

* This part of work was finished by Weipeng Jang

4

Slot density and distribution.

Table 1 and Table 2 list the differentcombinations of the above parameters forperforated pipes and slotted liners,respectively. In total, 17 differentcombinations were available for the analysisof the effects of the completion geometry onthe horizontal well fluid behavior. 7 of the17 combinations, which are denoted by “X” in Table 1 and Table 2, were investigated byYuan (1997). The remaining 10combinations, which are denoted by “•” in tables, were investigated in this study.

3. Model Development for Apparent-Friction-Factor

In this study, the general model developedby Yuan et al. (1996) was adopted toanalyze the acquired data.

Consider an incompressible fluid flowingisothermally along a uniformly perforatedpipe of a cross-section A. The area of eachperforation is Ap. Fluid is injected throughthe perforations into the main flow streamuniformly as illustrated by Fig. 3. Themomentum balance for the control volumein the axial direction is

p1 A p 2 A w d x 2 2 (1)

2 A 1 A V V p Au u x r p n2 1

where p1 and ū1 are the pressure and averagevelocity at the inlet of the control volume,and p2 and ū2 are the pressure and averagevelocity at the exit. n is the number ofperforations along the distance Δx.

For the three terms on the left-hand sideof the above equation, we assume thataverage properties completely define theflow field. The first two terms on the righthand side of the equation use the averagevelocities by introducing momentumcorrection factors, β1 and β2, which aredefined by the following equation:

dAuAV A 2

21

(2)

where u and V are the velocity distributionand the average velocity in cross-section A,respectively.

The last term of Eq. (1) represents theacceleration of flow resulting from fluidinjection. When the injected fluid enters themain flow stream through the perforations,the streamlines change directions. Eachlocal mean velocity is tangent to thestreamlines and can be divided into twocomponents, Vr and Vx, as shown in Fig. 3.Fluid is transported into the main flow witha radial velocity component Vr, whileretaining some axial momentum fromvelocity component Vx. Vr is equal to Vp dueto continuity. βp is the momentum correctionfactor for the influx stream .

For multiple injection points, it isconvenient to use average properties. Theaverage velocity over Δx is ū and is defined as follows:

ū=( ū1+ ū2)/2 (3)

A mass balance for the control volume isgiven by

ū1A+nVpAp= ū2A (4)

The influx rate through each perforation is

qin=VpAp (5)

The total volumetric influx rate is

Qin=nVpAp (6)

Velocities ū1 and ū2 may be eliminatedby employing Eqs. (3) and (4). An apparentfriction factor, defined as the ratio of the netimposed external forces to the inertialforces, can be given by:

du

xpp

f T 2/)(

212

(7)

5

which is an average friction factor over alength Δx.

The wall friction factor fw is defined as

)/()8(2

uf ww (8)

Let

xn / (9)

4/)( 2 udQ (10)

uV x / (11)

where φ is perforation density.An expression for the apparent friction

factor can then be found by substituting Eqs.(6) through (11) into Eq. (1). Rearrangingand simplifying:

Qqn

Qqd

xdff

inp

inwT

2121

12

4

22

(12)

Let

QqnC in

pn 2121 4 (13)

Equation (12) then becomes

QqdC

xdff in

nwT

22 12 (14)

The second term on the right-hand sideof Eq. (14), 2d(β2-β1)/Δxis caused by achange in the velocity profile in the xdirection. No attempt has been made toevaluate this term in this study. However,this term is negligible in this project sincethe small rate injection will not affect thevelocity field significantly, except in thenear wall region. Using Blasius formulafw=a(NRe)b, we get

QqdCaNf in

nb

T 2Re(15)

where Cn, a, and b are determinedexperimentally for the different completionscenarios.

4. Results and Discussions

4.1 Multiple Slots Cases

A total number of 360 tests were conductedusing the four multiple slots test sections.Figures 4 to 7 show the variations inapparent friction factor with influx to mainflow rate ratios and Reynolds numbers forthe four test sections with multiple slotscompletions. Each figure is plotted asapparent friction factor fT vs. Reynoldsnumber NRe, and the different data seriesrepresent experimental results at differentinflux to main flow rate ratios. As we cansee from the figures, in most test sectionsthe fT is greater than the smooth pipe frictionfactor calculated from the Blasius formulafor all influx to main flow rate ratios. Whenthe influx to main flow rate ratio approacheszero, the fT vs. NRe curve will move closer tothe curve predicted by the Blasius formula.

Lubrication effects were found for thefirst test section when the flow rate ratio is1/1000. In all the cases, the fT decreasesconsiderably with the decreasing of influx tomain flow rate ratio at high flow rate ratiocases. However the decrease of the frictionfactor is negligible at very low influx/mainflow rate ratios. We can predict that thefriction factor will approach a constant atvery small influx over main flow ratios. Fora given flow rate ratio, fT decreases with theincreasing of Reynolds number. For a givencompletion density, fT is always the smallestwhen the phasing is 90o.

Applying regression analysis we get thefollowing correlations for a, b and Cn in Eq.(15):

6

2771.840651749.0611656.0 a (16)

000133823.07926.120191.165.4

41439.1198817.0 2

b

(17)

9954.120161188.025.2 nC (18)

Next we briefly discuss the effects of thecompletion phasing on the liquid behaviorin horizontal wells. As we can see fromFigure 8 and Figure 9, other parametersbeing equal, the decreasing of thecompletion phasing from 360o to 180o andthen to 90o decreases the total frictionfactor. The friction factor is smallest whenthe phasing is 90o. The possible reasons whythe completion phasing has such significanteffect on flow behavior in horizontal wellscan be: 1). When the phasing is smaller, say90o, the influx can be considered enteringfrom all sides, thus there is smaller twist(distortion) against the main stream velocityprofile and therefore there’s smaller pressure loss due to momentum change. 2).When the influx enters the main flow frommore than one direction, a larger area of theboundary layer is lubricated than if theinflux is entering from one direction (360o).The lubrication of the influx can lessen theextent of surface roughness introduced dueto completion.

The effect of slots density upon thepressure drop behavior in horizontal wells inthis study is quite straightforward: othercompletion parameters being equal, theapparent friction factor in general increaseswith the completion density mainly due toincreased influx introduced by the extraopenings (Figure 10). However this may notnecessarily be true when the influx overmain flow rate ratio is very small, as we willdiscuss in the multiple perforation section.

4.2 Multiple Perforation Cases

A total of 490 experimental tests areconducted on the six multiple perforationtest sections. The data acquisition and dataanalysis procedure for multiple perforationcases are the same as those of multiple slotscases. And the general trends of the pressuredrop behavior are the same except in thissection no lubrication effects were observed.

The following three equations areobtained through regression analysis toestimate a, b and Cn:

33.29.1 542.80800275189.024163.0

ea

(19)

e

b0513603.0447302.0

0001793990334316.025.0

5.0

2

(20)

24243.1

556613.000467386.09081.4

90235.0

eCn

(21)

As we mentioned early in the multiple-slots section, the completion phasing hassignificant effect upon the pressure dropbehavior in horizontal wells completed withmultiple slotted liners. The apparent frictionfactor usually drops as the phasingdecreases when the other parameters beingheld equal. The same thing is true inmultiple perforation cases (See Figures 11and 12). What we want to point out here isthe effect of phasing is insignificant oncethe completion density or the influx overmain flow rate ratio becomes too small.Under very small perforation densitysituations (the distance between twoneighboring openings is greater than 8 timesof the pipe diameter), the single perforation

7

modeling from Yuan’s study should be used to analyze the flow behavior in horizontalwells.

Experimental data are compared for thethree perforation densities when the influxto average main flow rate ratios equal to1/50 and 1/1000 (Figure 13 and Figure 14).Figure 13 shows that for influx to main flowrate ratio equal to 1/50, fT is higher for thehigher perforation density case. Figure 14shows that for influx to main flow rate ratioequal 1/1000, fT for the three perforationdensities are almost the same at lowReynolds number region while fT is slightlysmaller when the density is 20 shots/ft. Asdiscussed in Yuan’s study, at very small influx/main flow rate ratios, fT usually islower for high perforation density case. Oneprobable reason for this is that the third termon the right-hand side of Eq. 14 (influxcontribution to total apparent friction factor)is the dominant term at high influx to mainflow rate ratios, while the first term (wallfriction factor) is the dominant term at lowinflux to main flow rate ratios.

5. Evaluation and Comparison

In this section, the new apparent frictionfactor correlation for the multipleperforation cases is evaluated together withthe Asheim1 et al. model and the Ouyang16

et al. model against the experimental dataobtained in this study.

Figures 15, 16 and 17 give comparisonsamong the three correlations, and theBlasius formula for smooth pipe and theexperimental data of test section 6 for threeinflux over main flow rate ratios (the ratiosare 1/50, 1/500 and 1/1000 respectively).From the comparisons, it’s obvious that Ouyang et al. model almost always predictsthe smallest friction factor under turbulentflow regime. In their study, they claimedthat influx increases the friction factor forlaminar flow and reduces the friction factorfor turbulent flow. However in our study it

was found that inflow could reduce theapparent friction factor only when the influxto main flow rate ratio is very small. Andsince no consideration was given to theperforation distribution in their modeling,the model can not differentiate betweendifferent perforation distributions. TheAsheim et al. model in general gives closerprediction than the Ouyang et al. modelespecially when the influx from theperforated openings is small. However aswe can see in Figure 15, the Asheim et al.model over-predicts when the influx overmain flow rate ratio is high. Thisobservation is consistent with what Yuanhas observed when the Asheim et al. modelwas compared with the single perforationcorrelation.

Figure 18 shows the variations inprediction of pressure drop for differenthorizontal well hydraulics models using thehorizontal well data provided in Ouyang etal.’s study. For simplicity, we used average fluid properties in the calculation ofpressure drop. As we can see from the plot,Ouyang et al.’s model predicts the smallest pressure drop over the wellbore while thecorrelations we obtained in this studypredict the largest pressure drop.

Reservoir PerformanceModeling andComprehensive Model

1. Pressure Response and AsymptoticSolution for a Continuous PointSource

We derived our solution for the slabreservoir with sealed top and bottomboundaries.

1.1 Continuous Point-Source in Laplace Domain

8

We can derive the pressure responsep pt (x, y, z, xw , yw , zw , s) in a slab reservoirof thickness of h in Laplace domain for acontinuous point source extracting fluidwith a rate of Q at point of (xw, yw, zw) byusing image principle, where (x, y, z) is anylocation and s is the Laplace variable. Thisis illustrated in Fig. 19.

)}cos()cos(

)(2)({)(2.141

100

D

wD

D

D

n

nnDD

hz

nhz

n

RKsRKkh

sQp

(22a)where,

222 / Dn hns (22b)

222 )()( wDDwDDD yyxxR (22c)

p pt is a Fourier Bessel series. The infiniteseries in the solution is due to the imagesystem used to generate the effect of thesealed top and bottom boundaries.

1.2 Long-Time Asymptotic Solution for Pseudo-Radial Flow

We are not interested in the short-timesolution but concerned about the long-timesolution for our well completionoptimization problem. In addition, weconsider only pseudo-radial flow instead ofseeking solution in boundary-dominatedflow (pseudo-steady-state flow or steady-state flow) as shown in Fig. 20. The reasonfor this approach is as follows. Because theconvergence of flow toward the wellboreopenings will take place in the near vicinityof the well, the outer portions of thereservoir including the boundaries will notbe affected by the existence of the openings.Therefore, if we derive the pseudo-skin

expressions by comparing the transientpressure solutions of the open-holecompleted and slotted-liner completed orperforated horizontal wells, then the samepseudo-skin factors can be incorporated intothe bounded reservoir solutions for open-hole completed horizontal wells. This wouldrepresent the solution for a slotted-linercompleted or perforated horizontal well in abounded reservoir.

2. Pressure Response for a PerforatedHorizontal Well

2.1 Pressure Response for Single Perforation

(a). Perforation Inclination j0 or :

The geometry relationship between the j-th perforation and the i-th observation pointis illustrated in Fig. 21. The real 3-Dwellbore is considered. Assume the lengthof perforation is Lp,j, the inclination angle isj. The permeability anisotropy should beincluded to obtain the distorteddimensionless wellbore radius, perforationlength, and inclination angle as follows

jjz

rwwDj k

kl

RR 22 sincos (23a)

jjz

rPPDj k

kl

LL 22 sincos (23b)

]tan[tan 1'j

z

rj k

k (23c)

where l is the characteristic length and wechoose (Lh/2) as l. In order to use Eq. (22),we need to know RDi j , the dimensionlesshorizontal distance between an arbitrarypoint on the j-th Perforation and the i-thobservation point. For the triangle I’D’M’,

9

we use the law of cosine and get thedistance

2/1',

'2',

2' ]cos2[ ijijDDijDDijD rrrrR (24a)

where,

2/1'222 ]sin)2

()[( jjPD

jwDDjDiijD

LRxxr

(24b)

]sin)

2(

)([tan

'

1

jjPD

jwD

DjDiji L

R

xx

(24c)

To use Eq. (22), we also need to computethe dimensionless z coordinates for sourcepoint and for observation point. The z-coordinate for I-th observation point is

DWDwDi Rzz 0 (25a)

where

z

rwDw k

kl

RR 0 (25b)

The z-coordinate for an arbitrary point onthe j-th perforation is

)2

(coscot '''' jPDjDwjDjDwD

LRrzz

(25c)

In the following, we denote ji as thefollows

)cos()cos(

)(2)(

'1

00

D

D

D

Di

n

nnijDijDij

hzn

hz

n

RKsRK

(26a)

The pressure response for singleperforation can be obtained by integratingEq. (22) along the perforation length

'sin5.

sin5.'

'

'sin)(

D

L

Lj

jjPD

DjjD dr

Lsq

pjjPD

jjPD

(26b)

where jDq~ is the dimensionless flux throughthe j-th perforation

qlLq

q jPDjjD

)( (26c)

(b). Perforation Inclination j = 0 or :

The geometric relationship between the j-th vertical perforation and the i-th is muchsimpler than the case of inclined perforation

zD

L

Lj

jPD

DjjD d

Lsq

pjPD

jPD

5.

5.

)( (27a)

|| iDiDijD xxR (27b)

DWDwDi Rzz 0 (27c)

DwzDjPD

DwD RL

zz 0'

2 (27d)

2.2 Pressure Response for Multiple Perforationsand the Asymptotic Solution

For NP number of perforations, weobtain the total pressure response pD byusing superposition principle. Fig. 22 showsthe 3-D geometry for multiple perforations.The pressure response at any location as thesum of all the perforation sources is asfollows

21

DDfjD

NP

jD pppp

(28a)

10

where

'sin5.

sin5.0'

1

'

'

)(sin

)(D

L

LijD

jjPD

DjNP

jfD drsRK

Lsq

pjjPD

jjPD

(28b)

)]}cos(

)cos()(

[sin

)({2

'sin5.

sin5.

'

10

1'

1

'

'

D

Di

D

L

L D

DijD

n

n D

njjPD

DjNP

jfD

hz

n

drhznR

hnK

Lsq

p

jjPD

jjPD

(28c)

With long-time approximation we finallyobtain the asymptotic solution by takinginverse Laplace transform

21 DDD ppAp (29a)

where

'sin5.

sin5.'

11

'

'

)ln(sin D

L

LijD

jjPD

DjNP

jD drR

Lq

pjjPD

jjPD

(29b)

)]}cos(

)cos()(

[sin

{2

'sin5.

sin5.

'

10

1'

12

'

'

D

Di

D

L

L D

DijD

n

n D

njjPD

DjNP

jD

hz

n

drhznR

hnK

Lq

p

jjPD

jjPD

(29c)

]80709.0)[ln(21

DtA (29d)

For vertical perforations (’ = 0 or ), theterm of sin(’) in Eq. (29) will be replaced by 1, and RDjI, zD’, and ziD are computedwith Eqs. (27a)~(27c).

3. Pressure Response for Slotted-LinerCompleted Horizontal Wells

Similarly to the case of perforatingcompletion, we integrate the point-sinksolution pD, pt to obtain the solution for them-th slot with length lm and center at (xm, ym,zm). Using superposition principle toincorporate the effect of all slots (MS is thetotal number of slots on the wellbore), weobtain the total pressure response in Laplacedomain, which is obviously the function ofslot geometry and distribution. Fig. 23shows the multiple-slot geometry. For long-time asymptotic solution, we take theapproximate expression for K0(x) andevaluate the integrands for each slot.Finally, we get the total pressure responseon the wellbore (specifically at the top ofthe wellbore with yD = 0, zD = zwD+rwD ) inreal time domain by taking inverse Laplacetransform. The pressure response is asfollows

21 DDD ppAp (30a)

'2

2

22'

11

])(ln[

)(

D

lx

lx

mDDD

MS

m mD

mDDD

dxyxx

lqxp

mDmD

mDmD

(30b)

)]}cos()](cos[

])([

[{2)(

,0

2

2

'22'0

1 12

DmDWwD

lx

lx

DmDDDD

MS

m nmD

mDDD

znRzn

dxyxxLnK

lqxp

mDmD

mDmD

(30c)

where qmD is the dimensionless flux throughthe m-th slot

11

qlqq mm

mD (30d)

4. Discrete Form of Pressure Responsefor Perforated Horizontal Wells

A perforated horizontal well has much moreperforations than a vertical well (e.g., thereexist (1000 ft * 4 spf) = 4000 shots ofperforations on a wellbore of 1000 ft long).We discretize the wellbore length into 2Msegments with uniform flux in eachsegment. The segment length is equal to(Lh/2M). Let m0(I) be the starting sequentialnumber and m1(I) be the ending sequentialnumber of the perforations in the I-thsegment. For any location xDJ on thewellbore, the pressure response is

}{)(.2

}){()(

)(

)(2

2

1

)(

)(1

2

1

1

0

1

0

Im

Imm

M

ImD

Im

ImmmD

M

IDJD

IIq

IIqAxp

(31a)where,

'2sin

2sin

'1 ]ln[sin1

DDmJ

mL

mLmmPD

drRL

I

mPD

mPD

(31b)

)cos()]cos[

][sin1

''

2sin

2sin

0'1

2

'

'

JDD

DmDD

L

LJmD

DmmPDn

zhndrz

hn

RhnK

LI

mmPD

mmPD

(31c)

q (I )Note that mD is the dimensionless fluxthrough the m-th perforation on the I-thsegment (I=1,2,…,2M). We need to relate it

q (I )with hD , the flux in the I-th segment in

order to solve the coupling equation thatwill be discussed later

q (I )LqhD (I ) h h

q (32a)

Assume that there is MP number ofperforations in the I-th wellbore segment,and the total flow rate into the I-th segment,

)(~ Iqh , is

)2/()(

)(ML

MPlLIqIq

h

mPDmh

(32b)

Thus

MPMq

MMPqLq

ML

MPMP

qMLlLIq

qlLIq

q

hDhh

h

h

mPDmmPDmmD

2)2(1

2)2/()()()()(

.(32c)

In fact, MPM 2 is the total number ofperforations on the whole wellbore. Replace

)(IqmD with )(IqhD and we get

)()()(2

1II

M

IhDDJD FIqAxp

(33a)

where

}{2

1 )(

)(1

1

0

Im

ImmI I

MPM (33b)

}{2

2 )(

)(2

1

0

Im

ImmI I

MPMF (33c)

Notice that I2 includes the integration ofmodified Bessel function. The integrand isin the form of K0 (u) . We can useChebyshev polynomials to approximate themodified Bessel function of K0 with

12

Clenshaw’s recurrence formula for the summation. Finally we numerically solvethe integration with Chebyshev’s coefficients. In addition, I1 can beevaluated by deriving an accurateanalytical expression.

5. Discrete Form of Pressure Responsefor Slotted-Liner CompletedHorizontal Wells

Similarly to the perforated well, wedivide the wellbore length into 2M segmentswith each segment of length (Lh/2M). Forany location xDJ on the wellbore, thepressure response is

}{)(.2

}{)()(

)(

)( 12

2

1

)(

)(1

2

1

*

1

0

1

0

Im

Imm n

M

I mD

mD

Im

ImmmD

mDM

IDJD

Il

Iq

Il

IqAxp

,(34a)

where,

'22'2

2

1 ])(ln[ DmDDDJ

lx

lx

dxyxxI

mDmD

mDmD

(34b)

)cos()](cos[

])([2

2

'22'02

mDWDwD

lx

lx

DmDDDD

znrzn

dxyxxLnKI

mDmD

mDmD

. (34c)

Notice that qmD (I ) qmlm / q is used in theabove equations. Based on the same reasonas discussed in perforated wells, we need toinvert qmD (I ) into qhD (I ) . Assume that thereare NR rings of slots in each wellboresegment, and NS slots on each ring, asshown in Fig. 24. Thus the total number of

slots in one segment is (NR*NS), and thetotal flow rate into the I-th segment isqm (I )lm (NR NS ) . The flux on the I-thsegment (the length is Lh/2M) is

)2/()()(~

)(~ML

NSNRlIqIq

h

mmh

(35a)

Thus

)(2

)(1)2/(

)(1)2/(

)2/()()(

)(

NSNRMq

NSNRLML

qLq

NSNRLMLL

MLNSNR

qlIq

qlIqq

hD

h

hhh

h

hh

h

mm

mmmD

.

(35b)

In fact, 2M (NRNS ) is the total numberof slots on the whole wellbore. ReplaceqmD (I ) with qhD (I ) and we get

)()()(2

1

*II

M

IhDDJD FIqAxp

(36a)

where

}{2

1 )(

)(1

1

0

Im

ImmmDI I

NSNRMl (36b)

}{2

1.2)(

)( 12

1

0

Im

Imm nmDI I

NSNRMlF (36c)

Notice that I2 includes the integrationof modified Bessel function. The integrandis in the form of K ( 2 2

0 x a ) . We canuse Chebyshev polynomials to approximatethe modified Bessel function of K0 withClenshaw’s recurrence formula for the summation. Finally we numerically solvethe integration with Chebyshev’scoefficients. In addition, I1 can be

13

evaluated by deriving an analyticalexpression. However, such accuratecalculations are only adopted for the longslots (for instance, the partially completedopenhole can be treated as slotted linerwellbore with several long slots and longdistances between slots). For short slots,we can simply take mean value of theintegrand multiplying by the integrationinterval length to obtain approximate butfast results for the integration. Normally, itgives sufficiently accurate results.

6. Mechanical Skin

In the process of drilling andcompleting a horizontal well, the formationis usually damaged by mud filtration fluidor solid debris. Some dirt may also plugsome completion openings (perforations orslots). Any of these factors which result inchanges in the natural productivity may becategorized into the term of mechanicalskin factor, SF. In this model, the skinalong the wellbore may change fromlocation to location. We use the followingmodification to consider of the contributionof skin.

MJJSFqxpxp JhDDjDDjD

2...,,1),()()( ,

(37)

We perform this simple addition of theskin effect is based on our followingunderstanding. The skin only causes anadditional pressure drop to the wellbore,but does not affect the reservoir pressuredistribution. Furthermore, we incorporatethe effect of skin only onto the segmentwhere the skin exists. This is because thatskin only produces additional pressure dropon its own segment.

7. Coupling Procedure and NumericalSolution Algorithm

The wellbore hydraulics equation canbe expressed as

]2[

16)()(

'"

Re0 0

Re,,

'

DDhDtt

x x

DhD

ttDDDDwD

dxdxqfN

DxC

Nftxptp

D D

(38a)

where ChD and D are functions ofwellbore Reynolds’ number and friction factor

h

whD Lkh

rC41310395.7

, (38b)

FF fN

dNdf

ND ReRe

2Re 2 . (38c)

By dividing the wellbore into Msegments we obtain the discrete form ofthe above equation

}8

1)2

12({

168)()(

2,

,,

1

1

Re,,

MqD

qDMM

Ix

CCxNf

txptp

JhDJIhDIDJ

J

I

hDhD

DJttDDJDDwD

(39)

We have (2M +1) unknowns (pwD andqhD,,I, I=1,…,2M). We obtain 2M equations by evaluating Eq. (13) at xDJ, the center ofeach segment. An additional equation isthat the sum of the dimensionless flux is2M. Let Q1 = pwD, QI = qhD,,I-1(I=2,…,2M+1). The (2M+1) equation canbe written as

12...,,1),(),(12

MJJBIJGQM

II

(40a)

14

where G(I,J) is a function of fluxdistribution. Thus, this is a non-linearsystem. Let

F (Q) G(Q)Q B (40b)

where G(Q) is a square matrix withdimension of (2M+1), and B and Q are(2M+1)1 vectors.

We use Newton’s iteration method by computing Jacobian matrix evaluated atQ(k) (the k-th iteration) and calculateincremental the vector Q(k) by callingmatrix solver of LU decompositionsubroutine.

For computational implementation, weneed to construct an algorithm to cope withthe summation of the huge number ofperforations or slots. The geometricdistribution of the perforations is controlledby perforation length, perforation density,and phase angle. The geometricdistribution of the slots should becontrolled by inputting some limitedparameters such as the length of slot (ls),the distance between two adjacent slotrings (ee), slot array phasing (PHA), slotnumber of each concentrated slot array(NG), distance of adjacent slots in oneconcentrated slot array (de), and thestarting location of spacing slots (x0).These parameters are illustrated in Fig.?

8. Application of New Correlations forApparent-Friction Factor

The regression correlations for theapparent friction factor, fT, are obtainedfrom the experimental data. According tothe principle of modeling and similitude,the fT correlation from small-scale modelexperiment can be directly used to thesituation of large pipe diameter. This isbecause the regression equations areexpressed in the dimensionless form of

),*,( infRe Q

dNfunctionf luxpipeT , where,

NRe, the Reynolds number, is related todynamic similarity, dpipe*, is related to

geometry similarity, andQ

q luxinf is related

to kinematic similarity. With samedimensionless parameters, the fT given bythe model will be equal to thecorresponding fT for the prototype. Thus,we can substitute the actual value ofparameters into the model equations whenapplying them to the comprehensivecoupling equations.

q

However, the empirical correlationsfrom regression analysis are only validover the range of parameters covered bythe experiments. It will be risk toextrapolate beyond the range of parameters.For example, the multiple perforation pipetests were performed with density of 5, 10and 20 shots which may higher than theconventionally used perforation density(normally 1~8 shots per feet). Thus it isunwise to use the empirical relationship tothe cases with density less than 5 shots/ft.Of course, if the distance betweenneighboring perforations is larger than8*dpipe (e.g., 5 in. casing with density of 0.2spf, or one perforation controls 1/0.2 = 5 ft.= 60 in. > 8 dpipe =40 in.), the single-perforation model should be used. For mostof the cases (developing flow), theperforation distribution is denser than thesingle-perforation situation (developedflow), and sparser than the conditions of5~20 spf. In such cases, we need to figureout an approach to use the correlationsalternatively.

As we know that the inflow fluxaround the pipe affects the frictionsignificantly, we may assume that thefriction factor fT be the same when theinflow flux (mass rate) keeps the samevalue for a unit length of pipe. Based on

15

this idea, we infer that the fT be the same ifthe opening areas on the pipecircumference are the same in unit length.At the same time, we need to consider theaspect ratio (dpipe/Lpipe) which expresses thegeometric similarity. Using subscript “m” for model, we need equal aspect ratio

m

m

Ld

Ld (41a)

For example, dm = 1 in., d = 6 in., Lm : L =1 : 6.

The equivalent open-flow-area principle,for applying model correlations to actualsituation where perforation density is out ofthe testing range, is expressed as follows

LA

LA pp

m

mpmp .,, (41b)

or

pm

mp

pmp d

dAA

,

., (41c)

where,mp, = the transferred perforation density

which will be used in the model equation,spf;

p = the real perforation density, spf;

pA = the real single-perforation area, in.2;

mpA , = the single-perforation area used onthe model pipe, in.2;d = the real pipe diameter, ft.;dm = the model pipe diameter, ft.

For instance, dm = 1 in., d = 6 in., dp,m =1/8 in., dp = 3/4 in., p = 2 spf, then

)(126

26261

)81(

4

)86(

42

2

2

, spfmp

Thus, we transfer p = 2 spf to mp, = 12spf which is in the range of the testeddensity.

Meanwhile, the second term on theR.H.S of fT correlation (Eq. (15)),

QqdC inn /2 , includes the product ofd . For the same reason we should

substitute mmd into the calculation. Forabove example, we need to use mmd =(12 spf * 1/12 ft) = 1. The real value ofd should be avoided because we are

treating the formula apparently for thedata out of the range.

For slotted liner, the density valueseems in the normal range, and we canuse the actual parameter values in thecalculation of fT.

To summarize, our model combines thereservoir model, the wellbore hydraulics,the non-uniform mechanical skin, and therestricted entry (slots or perforations)together to obtain a correct picture of theflow characteristics for the purpose ofcompletion optimization. This isillustrated in Fig. 25.

Results and Discussion

1. Flux and Pressure dropDistribution Characteristics

We take the limiting case of slotted-linercompletion by setting slots distance closeto zero, and only one row of slots on thetop of the wellbore. This should representthe openhole performance.

Fig. 26a gives the comparison for flowrate and flux between the openhole and thelimiting case of slotted liner. Obviously,they match very well.

Fig. 26b indicates good match for DPand PI. The DP of the limited slotted lineris 47.5, and the PI is 632, while theopenhole well has pressure drop of 48.7

16

and PI of 607.7. The small discrepancy isdue to our neglecting the effect of no-flowwellbore surface in our model. The resultsshow that such approximation is acceptablefor the long-term behavior of completedhorizontal wells.

2. The Effect of Slot Length and SlotDistance, and Slot Density

We take the limited case of Fig. 26b asthe base case with slot length ls = 73 in.,and the distance between the adjacent slotse = 0.02 in.. We know it has DP of 47.5psi, and the PI of 632 b/d/psi..

Now increasing the distance to ee = 36.5in. by keeping ls = 73 in., we get largerpressure drop with DP = 51 psi and smallerproductivity of PI = 582 b/d/psi.. If wefurther reduce the length to ls = 36.5 in.and keep ee = 36.5 in., we get even largerpressure drop of DP = 56.7 psi., andsmaller productivity of PI = 528 b/d/psi..

In addition, if we reduce both length anddistance by taking ls = 2.5 in. and ee = 2.5in., we will get less pressure drop with DP= 47.4 psi, and increase productivity to PI= 632.3 psi.. Although the densities are thesame for the case of ls = 2.5 in. and ee =2.5 in., and for the case of ls = 36.5 in. andee = 36.5 in., the different combinations oflength and distance sizes give differentresults.

Thus, we find that both slot length andslot distance have significant effect on thepressure drop and productivity. Even forconstant density, smaller slots give higherPI.

We also find that in case of lowproductivity, flux is more uniform and lessskewed. In other words, low slot densityresults in higher pressure drawdown, andmore uniform flux distribution. Fig. (27a)and Fig. (27b) compare high density case(ls = 2.5, ee =2.5, PI = 632) with lowdensity case (ls = 0.5, ee = 5, PI = 364).

3. The Effect of Slot Phasing Angle andSlot Concentration

The slot phase angle (phasing) isdefined as the angle between two adjacentslot array (slot concentration) in one slotring, measured in degree. If the phasing iszero (one single slot array in one slot ring),we may call phasing of 360. One slot ringmay include slot arrays of number of360/PHA, while each slot array has NGconcentrated slots with the distance of debetween the adjacent concentrated slots.For example, we may have phasing of 90with 4 slot arrays (4=360/90), while eacharray includes 3 slots with distance of 0.5in. between the neighboring slots within thearray.

We have tested the effect of slotphasing on the PI and DP. We foundphasing has little effect on DP and PI. Forinstance, given slot length of ls = 1.5 in.and distance between neighboring slotrings of ee = 10 in., we get (1) PI = 423(DP=70.8) for PHA = 360, (2) PI = 417.5(DP = 71.9) for PHA = 120, and (3) PI =417.9 (DP = 71.8) for PHA=60.

In addition, slot concentration has littleeffect. For example, given PHA = 360, ls= 1.5 in, ee = 10 in., we get (1) PI = 423(DP=70.8) for NG=1 (one slot in onearray), (2) PI = 422 (DP = 71.1) for NG = 4and de = 1 in.(4 slots in one array withdistance of 1 in.).

The above results can be explained byanalyzing the model equations. Eq. (33)expresses that the pressure response on thereservoir side, pD , is function of 1(I) andF1(I), I = 1, …, 2M. 1(I) and F1(I) arefunctions of I1 and I2 (defined in Eqs.(31b) and (31c) ) respectively,

})({2

1 )(

)(1

1

0

Im

ImmmDI mI

NSNRMl , (33b)

17

})({2

1.2)(

)( 12

1

0

Im

Imm nmDI mI

NSNRMlF

(33c)

I1 and I2 are related to slot geometry(length lm, coordinates of xm, ym, and zm).ym, and zm are small quantities (smallwellbore diameter restrains ym and zm),while xm varies on large scale for the longhorizontal wellbore. The change of phasingwill only affect ym, and zm. Thus thedifference from the varying phasing will bevery small. Nevertheless, we may say thatphasing also changes the total number ofslots in one segment. Then, why does it notchange the system response? For example,let us say there are NS slots in one ring,and NR rings in one segment. Suppose wehave NS = 1 for phasing of 360, then wehave NS = 4 for phasing of 90. Thus, wehave more I1 and I2 term in the summationsof Eqs. (33b) and (33c). However, each ofthe extra terms is almost the same as theoriginal terms for the slots in the same ring(ym and zm result in little difference for theslots in one ring). Although the NSincreases in one ring, the summationdivided by NS (Eq. (33b) and (33c)) givesalmost the same results. This explains thenegligible effect of slot concentration inone slot array. In other words, the longwellbore makes the phasing andconcentration effects invisible. For veryshort wellbore length (vertical well), thephasing plays much more important rule.

4. The Effect of Perforating Parameters(Density, Phasing and Length)

The sensitivity analysis indicates thatperforation density has more significanteffect on the PI than that of phasing andperforation penetration length.Fig. 28 shows the effect of perforationdensity on PI. From the figure we find

that the perforation density has obviouseffect on PI before density reaches 1shot/ft. Beyond density of 1 shot/ft, theeffect becomes small.

Perforation penetrating depth hassmaller effect on horizontal wellperformance, as shown in Fig. 29.

Perforation phase angle has the leasteffect on productivity. The difference isvery small among the PI values fordifferent phasing. Under significantpermeability anisotropy, the phase hasslightly increasing effect. Phasing of 90is the best one, and phase of 360 is theworst one. In the middle are 45, 60, and180in the order from high to loweffects.

5. The Effect of Partial Completion

For slotted-liner or perfortingcompletion along the long horizontalwellbore, we may have some blind-pipesegments which insulate the flow fromthe reservoir. This scenario also need tobe simulated. The following are theresults obtained from our program.

Assume a slotted-liner completedhorizontal well. First, we space slots alongthe full length of the wellbore with ls =2.5in., and ee = 2.5 in.. We get PI = 632.2 (DP= 47.4). Then we divide the wellbore intoseven segments, and space slots on fourseparated segments out of the seven (twoend segments are open to reservoir flow).We get PI = 571 (DP = 52.5). Finally, wedivide the wellbore into six segments andspace slots on three separated segments outof the six. We get much smaller PI of 497(DP = 60.3). Fig. 30 shows the fluxdistribution characteristics as we expected,i.e, the flux distribution inside eachsegment is also a skewed U-shaped curve.We reach conclusion that both thepenetration ratio and the open segmentlocations affect the productivity.

18

Software Developmentwith Window GraphicalInterface

1. HORCOM --- a Borland C++ 4.0GUI Software

The model computing algorithm has beenwritten into a computer program. Afterpreliminary testing and running, we startedto develop a Window interface software thatprovides a user-friendly graphicalenvironment. The software is developed onthe Borland C++ Builder 4 which can easilyuse the Win32 GUI (graphical userinterface) and also for console C++application. We name the softwareHORCOM Version 1.0 that represents“HORizontal well COMpletion Optimization”. HORCOM can be used independently without the softwaredevelopment environment.

First, create a directory in your computerhard disk to operate the software, and copyHORCOM into your the directory. Then youare ready to use it.

2. Main Menu and SpeedbarDouble click the executable file

HORCOM, the window graphic interfacepops on the screen. A graphic icon with aconceptual picture of horizontal well andcolored text of “Horizontal Well Completion Optimization” appears in front of the gray background. On the top of thescreen are the design and analysis menuswhich can be chosen by mouse click. Belowthe top main window menu are sometoolbars which help to open files, edit, andview multiple windows.

We illustrate main menu and toolbar withthe examples of File, and Edit in thefollowing.

The menu File consists of drop-downmenu: New, Open, Close, Save, Save as,and Exit. The underscore letter means thatyou may use “Alt. plus letter” approach to activate the menu bar. For instance, youmay press “Alt” then press “F”, the Filemenu will be activated. Without releasingthe key of “Alt”, further press “O” will pop out the open dialogue frame. You may usemouse to directly point the menu and sub-menu. It is required that only text file can beoperated with the File menu. Click “Exit” under File menu will end the application ofHORCOM. Another alternate way is to clickthe toolbar to perform “Open” and “New” file operation.

Menu “Edit” is used to perform “Cut”, “Copy” and “Paste”. You can also click the toolbar for the same operation after youselect you text object.

3. Data Files for Input and OutputThe menu of “File” and “Edit” is mainly

used to perform data file operation. We havefour input data files: “Reservoir1.dat”, “WellborFluid1.dat”, “Slot1.dat” and “Perf1.dat”.

We also have some output data filessuch as “Ohor1.dat”, “Ohor2.dat”, and “Ohor3.dat” for open-hole completion. Forslot-liner completion output files, we add SLafter the filenames for openhole, such as“Ohor1_SL.dat”, “Ohor2_SL.dat”, and “Ohor3_SL.dat”. For perforation completion, we have files of“Ohor1_PERF.dat”, “Ohor2_ PERF.dat”, and “Ohor3_ PERF.dat”.

The menu of “Window” is used to perform such operation on the multiplewindow as “Cascade”, “Tile Horizontally”, ”, “Tile Vertically”, and “Minimize All”. You can also use the toolbar below the mainmenu.

4. Graphical Input Menu

19

We have four submenus under the mainmenu of “INPUT”.If you click Reservoir File under “Input”, a sub-window will be created for inputing thedata for reservoir characteristics. Use mouseto point to the text box or radial-group boxand input the values by entering thekeyboard. Clicking “OK” button will save your input to the file of “Reservoir1.dat” and exit this window.

Similarly, if you click “Perforating File” under the “Input” menu, you will be prompted to input related value by usingedit box, radial-group box and form. Therow number of the form will changed withyour input value of “Number of Wellbore Section”. You may use mouse to click and move the column and row of the form.

The same operation is need to submenuof “Slotted Liner File”. If you are not sure about the meaning of the slot parameters,you may click “Help” to open the picture of slotted-line completion geometry.

Similar operation is need for“WellboreFluid File”.

5. RUNUnder this menu, there are three

submenus for “Openhole Well”, “Slotted-Liner Completion” and “Perforated Completion”. By clicking you choice, the software automatically accomplishes thecomputation for the three kinds ofcompletion. You can watch some outputinformation on the screen that indicatewhere the program is running and whetheror not it has finished running.

6. OutputAfter running the program or if you have

already created output files from previousrun, you may perform the operation of themenu “Output”.

The Output menu includes threesubmenus: “Data”, “Lable” and “Plot”. At the beginning, the Output menu is dimmed(unable). Once you click “Open” under the

“File” menu, the Output menu is enabled.Then you can click “Data” and a form is presented for your to take a look at the data.You can even change some values of thedata. You may click “Label” to change the default axis labels of “x” and “y” to the axis labels you desire to use. Then you may clickthe “Plot” submenu to plot the figure. You can press “Print Screen” button on the topright of you keyboard. Then you can pasteyour picture to other places (e.g. paste toWord or Powerpoint”.

For checking the version of this product,you can click “Help” and click “About”, the brief information of the production willbe shown up with a TU logo.

Some Guideline forHorizontal Completion

1. Wellbore hydraulics plays animportant role in the wellperformance. The new experimentalcorrelations for friction factor fromthis study should be used foraccurate prediction and design ofhorizontal well completion. Whenthe control length by singleslot/perforation is larger than eighttimes the pipe diameter, the single-opening model should be used.Otherwise, the multiple-openingmodels should be used.

2. For slotted-liner or perforatingcompleted well, use phase angle of90. One reason is to obtain moreuniform flux distribution and lessflow convergence toward thewellbore. Another reason is toreduce the friction to the flow in thewellbore.

3. The slot-penetration ratio (slottedsection length over the total sectionlength) or perforation density hassignificant effect on the productivityof horizontal well. For slotted liner

20

completed well, the slot-penetrationratio should reach to 50%. Forperforated well, the perforationdensity should be larger than 0.4 spfbut less than 1.0 spf in order toobtain sufficiently large productivityand cost-effective operation.

4. Slot length has significant effect onthe productivity. Under fixed slot-penetration ratio, choosing slotlength as small as several inchesmight be a better practice thanchoosing longer slot length as largeas several feet.

5. Perforation penetration depth doesnot affect well flow performance.For perforation completion, we canuse small shaped-charge (e.g.,charge weight less than less than20g) to guarantee smallest damageto the casing mechanical strengthand integrity, and also controlpermeability reduction in theperforation crushed-zone, withoutsignificant loss of productivity.

References

1. Asheim, H., Kolnes, J., and Oudeman,P.: "A Flow Resistance Correlation forCompleted Wellbore," Journal ofPetroleum Science and Engineering, 8(1992), pp. 97-104

2. Kloster, J.: “Experimental Research on Flow Resistance in Perforated Pipe,” M.S. Thesis, Norwegian Institute ofTechjnology, 1990

3. Yuan, H: “Investigation of Single Phase Liquid Flow Behavior in a SinglePerforation Horizontal Well,” M.S. Thesis, The University of Tulsa, 1994.

4. Yuan, H: “Investigation of Single Phase Liquid Flow Behavior in Horizontal

Wells,” Ph.D. Dissertation, The University of Tulsa, 1997.

5. Dikken, B.J.: “Pressure Drop in Horizontal Wells and Its Effect onProduction Performance,” JPT (Nov.1990) 1426-1433.

6. Ozkan, E., Sarica, C., Haciislamoglu,M., and Raghavan, R.: “Effect of Conductivity on Horizontal WellPressure Behavior,” SPE AdvancedTechnology Series, Vol. 3, No. 1 (March1995) 85-94.

7. Ozkan, E., Sarica, C., Haciislamoglu,M., and Raghavan, R.: “The Influence of Pressure Drop along the Wellbore onHorizontal Well Productivity,” paper SPE 25502 presented at the SPEProduction Operation Symposium,Oklahoma City, OK, Mar. 21-23, 1993.

8. Yildiz, T., Ozkan, E.: “Transient Pressure Behavior of SelectivelyCompleted Horizontal Wells,” paper SPE 28388 presented at the SPE 69th

Annual Technical Conference andExhibition held in New Orleans, LA,U.S.A., Sept. 25-28, 1994.

9. Ahmed, G., Horne, R.N., and Brigham,W.E.: “Theorical Development of Flow into a Well through Perforations,” DOE/BC/14126-25 (1990).

10. Spivak, D., and Horne, R.N.: “Unsteady-State Pressure Response due toProduction with a Slotted LinerCompletion,” paper SPE 10785 presented at the 1982 SPE CaliforniaRegional Meeting, San Francisco, CA,March. 24-26, 1982.

11. Hazenberg, G., and Panu, U.S.:“Theoretical Analysis of Flow Rate into Perforated Drain Tubes,” Water

21

Resources Research, Vol.27, No.7,1411-1418 (July 1991).

12. Landman, M.J., Goldthorpe, W.H.:“Optimization of Perforation Distribution for Horizontal Wells,” paper SPE 23005 presented at the SPEAsia-Pacific Conference held in Perth,Western Australia, Nov. 4-7, 1991.

13. Yildiz, T., Ozkan, E.: “Pressure-Transient Analysis for PerforatedWells,” paper SPE 49138 prepared for presentation at the 1998 SPE AnnualTechnical Conference and Exhibitionheld in New Orleans, LA, U.S.A., Sept.27-30, 1998.

14. Ozkan, E., Yildiz, T., and Raghavan, R.:“Pressure-Transient Analysis ofPerforated Slant and Horizontal Wells,” paper SPE 56421 presented at the 1999SPE Annual Technical Conference andExhibition held in Houston, Texas, 3–6October 1999.

15. Ouyang, L., Arbabi, S. and Aziz, K.:“General Wellbore Flow Model for Horizontal, Vertical, and Slanted WellCompletions,” SPE 36608, presented at 1996 SPE Annual Technical Conferenceand Exhibition, Denver, Colorado, Oct.6-9, 1996.