. .,

.

.

Effect of Perforation Damage onWell ProductivityJ. A. Klotz, SP13-AIME,UnionOilCo,of California?. F. Krueger, SPE-AIME,UnionOilCo. of CaliforniaD. $. Pye, 5PE-AIME,Unionnil Co.of California

@

Introduction ●

Gun perforation has been used for more than 40 yearsfor generating a controlled flow channel between oiland gas reservoirs and the bore of an injection orproduction well. The first well reported to be gunperforated’ was a Union Oil Co. of California well inthe MontebeHo field, Los Angeles County, Calif., in1932. Since that time many types of special bulletsand jets have been int~oduced to improve the perfor-ating process, and we include both devices in theterm “gun perforating,”

Although gun perforating became widely acceptedas a practical completion method, engineers long sus-pected that well productivities should be better thanobserved. Experimental and field studies,:- 7 rangingover more than a 20-year period, exposed deficienciesin perforator design and perforating procedures thataccounted for the reduced productivities or injectiv-ities and lead to improved field results. Laboratoryobservations on linear, perforated cores demonstratedthat crushing and compaction of rock during perfor-ating substantially impaired the flow capacity of thehole.

Although this experimental work provided a betterunderstanding of the physical effects of perforatinginto rock, our ability to estimate how much oil, gas,or water should flow into a wellbore through perfor-ations formed under down-hole conditions has beenIimited by the simplifying assumptions required tohandle this problem. Because of the difficulty of the

problem, the early work’-” on the productivity ofperforated welis assumed flow through clean, un-damaged perforations. Recently, Bell et al,” inanalytical and experimental studies related ~erforationefficiencyfor a single damaged perforation in a radialsystem to that observed in a linear core, but no attemptwas made to estimate over-all well productivity.

The purpose of our paper is to extend the earlierwork on productivity of perforated completions bytaking into account the depth and severity of perme-ability damage caused by both the dril;ing and theperforating processes. We are now able to relate flowefficienciesand permeability damage in a laboratory-perforated linear core system to that in a practicalradial well system. To limit the length of the text andyet illustrate adequately the general combined effectof perforating and drilling damage, we have modeleda radial system with an optimum perforating conditionof 4 holes/ft. At this short density and a penetrationof 6 in., perforated well productivity approximatesopen-hole productivity. The actual numerical effectson well productivity will, of course, be different forother perforation patterns but the general effects ofperforation damage on productivity will be similar.Thus, regardless of perforation pattern, the trendsindicated in this work should be helpful in designingperforating jobs.

We accomplished this objective with a computer-ized finite element method; and, although our com-

To maximize pr~ductivity, perforations must penetrate substantially beyond the zone ofdrilling damage, and they must be of the highest possible quality. In a well with drillingdamage, a \ew deep perforations are more effective than many shallow ones; but withinthe limits of current technology and economics, severe perforation damage cannot beentirely overcome by increasing either shot density or penetration.

IOVEMBER,1974 1303

*“

puter model is not perfect, we beliel’e it representsa realistic step forward in estimating well productivitywhen the formation is damaged from the drilling andperforating processes.

Evaluation of Perforation DamagePublications of several investigators’-” have shownthat conventional perforating practices impair theproductivity and injectivity of perforations. Perfora-tions are never clean, at best; the act of perforatingcrushes the rock and forces the particles from the holearea into the surrounding formation. Under adverseconditions of too great wellbore pressure and presenceof drilling mud or dirty completion fluid, severe add-itional darrage may result,

As a result of this work, the industry adopted in1962 a standard test procedure, API RecommendedPractice No. 43 (RI? 43), for comparing the flowproperties of perforations. We shall describe brieflythe process and philosophy of this test because it formsa foundation for our present work.

The 1962 edition of RP 43 describes perforationeffectiveness in terms of 2 Well Flow Index (WFI)determined for flow through a perforation made intoa standard cylindrical Berea sandstone core undersimulated wellbore conditions. WFI was defined asthe ratio of the apparent permeability of the perforatedcore (kP)to the pemleability of the unpe.rfoiated corewith both ends open (kO),The WFI so defined wasmisleading in that the effect of perforation damagewas masked by the effect of penetration depth. Theindex name was a misnomer because it had no relationto down-hole weHproductivity in a radial system.

In recognition of these problems, API RP 43 wasrevised in 1971‘“to provide a measure of permeabilitydamage in the perforation. In the new procedure, theapparent permeability measured in a core with a real,

1!OL---J0“2 4 6

Perforation Depth, inchesFig. l-Calculated permeability ratio for ideal,

clean perforation in a 12-in. corecylinder (data from Ref. 13).

.damaged perforation (Q is compared mathematicallywith the permeability of a core with an ideal, cleanperforation of the same depth (ki). This new measureof perforator performance is called Core Flow Effi-ciency and is defined in RP 43 (197 1) as

For a clean, undamaged perforation, ki/ko canbe calculated by the finite element method to bedescribed later, as it depends upon the dimensions ofthe cylinder and of the perforation. RF 43 tabulatesthe theoretical permeability ratios, ki/kO, for severaltest core lengths, and Fig. 1 presents a graph of typicalvalues for a 12-in.-long core. Thus R~ 43 now pro-vides a measure of perforation qual:ty. CFE near 1,0indicates a relatively clean perforation; CFE < 1.0,a dirty, or damaged, perforation,

But how does CFE for a linear core relate to wellproductivity in a radial system? To answer this ques-tion a modification of our finite element model wasused to estimate flow through the same perforation ina radial well system.

The relationship between CFE and well produc-tivity depends upon a large number of parameters:well diameter, perforation diameter, perforation depth,shot density, severity and depth of perforation dam-age, severity zmd depth of formation damage fromdrilling or workover fluids, and well drainage radius.Because of length limitations for this paper, we willnot attempt to discuss all combinations of these fac-tors, but will present the results involving variationsof those we consider most influential: perforationdepth, formation damage from drilling or workoveras well as from perforating, and shot density. I-Iow-ever, before discussing the results, we shall describehow the calculations are made.

The Finite Element MethodThe heart of our work involves calculation of flowthrough a perforation in either the RP 43 core cylin-ders or in a well. The calculations were made usinga finite element method for estimating pressures andflow in permeab!e porous media.

The finite element method was first proposed foruse in problems involving analjjsis of stress and strainin structures such as steel frame buildings, aircraftframes, and offshore platforms. For such structureseach frame member constitutes an element and eachjoint between members a node. Equations can bewritten in matrix form for forces at each node andfor stress and strain in each element. Then, theproblem of determining the forces, stresses, andstrains can be solved by well known methods of matrixalgebra, This application is described by Willems andLucas.’4

After considerable success in structure analysis,the finite element method was extended to continuousmaterials, These applications and the theory behindthem are described by Zienkiewicz and Cheung.15Chapter 10 of their book outlines the finite elementtheory for “F]eld Problems — Heat Conduction,Seepage Flow, Etc.,” which has been used in our

1304 JOURNALOF PETROLEUMTECHNOLOGY

.

I

work.For this paper the finite element method consists

of the following procedure:A plane section of the reservoir rock or test

cylinder is difided into a number of four-sided finiteelemems. Th: elements can have almost any shape;but for this work, we use rectangular elements forline~r flow through a test cylinder and nearly paral-lelogram elements for radial flow into a well, Figs, 2and 3 illustrate the finite element grids used for thecylindrical core and the radial well system.

Junctions of the lines surrounding neighboringelements are serially numbered nodes, We used 600nodes and eIements for this work; but for clarity ofillustration, the figures are drawn for grids containingonly 200 nodes and elements. The geometry of thesystem is described by the position, i.e., the x and ycoordinate of each node. Each element is identifiedby means of the serial numbers of the nodes at itscorners. The program can handle a large number ofnodes and elements, limited only by dimension state-ment in the computer code. We used approximately600 nodes and 600 elements to describe each flowsystem.

Flow resistance through the permeable media isdescribed by assigning a permeability to each element;we used four different permeabilities:

1. Virgin undamaged reservoir rock permeability,k, = 1.0.

2. Permeability of the region around the wellboredamaged by drilling, 0.0< k, <1.0.

3. Permeability of the region around the perfora-tion damaged by perforating, 0.0< kv <1,0.

4. Permeability of the region damaged by bothdrilling and perforating, k,. For most of our work,k, = k, X k,,lG but for the work summarized in Figs.11 and 12, k, was varied independently.

The computer code works in the following reamer,First, it calculates coefficientsdescribing the flow con-ductivity of each element as it depends upon the shapeof the element (location of the corner nodes), and onthe permeability of the element. Then, these coefE-cients are stored in a 30 X 600 array that eventuallywill be a banded matrix describing the conductivity ofthe entire system,

Next, boundary conditions are used to modtiy a600-point vector that also will be part of the finalsolution. Our boundary conditions consisted of afixed pressure at the nodes along the input face ofthe test cylinder, or along the outer, input boundaryof the radial system, and a different fixed pressure atthe nodes lying on the surface of the perforation.

Next, the computer code solves the banded 30 X600 matrix along with the boundary condition vectorby means of a direct Gaussian solution to determinethe pressure at each of the nudes corresponding to itsposition, conductivity of each element, and specifiedboundary conditions. Finally, after pressurt% at eachnode have been determined, the computer code calcu-lates velocity and the direction of flow through eachelement in the system.

The finite element computer code that forms theheart of our calculation was written in the Civil

NOVEMBER,1974

Engineering Dept. at the U, of California at Berkeleyand is described in a paper by Taylor and Brown.17The basic mathematical technology in the code* hasnot been changed, but we adapted it to our particularproblem.

The arrangement of nodes and elements shown onFigs. 2 and 3 is calculated automatically by an auxili-ary gridding subroutine,

For the linear, cylindrical system, the griddingprogram accepts cylinder diameter, cylinder length,perforation diameter, perforation length, and dam-aged zone thicknesses as input data, Then it auto-matically draws a grid such as shown in Fig, 2,Elements and nodes are clustered near the tip of theperforation, where flow directions change rapidly andwhere pressure gradients are greatest.

Although the grid appears here as a two-dimen-sional, plane m-ray, the computer code considers itto be a two-dimensional, axisymmetric array with theaxis of the perforation, coincidental with the axis ofthe cylinder, as the axis of symmetry, Thus, eachelement is a three-dimensional ring whose rectangularcross-sectiori shows in Fig. 2. For the linear, cylin-drical system, this is an accurate representation.

For the radial system, the grid shown in Fig. 3is developed in a similar manner but an approxima-tion of well geometry is involved. Input data to the

*A Iktinz of Tavlor’s code can be obtained from the amhor% of

Rcoion of ;~ I k=kl=l.-OPw-foration5arnage-(1/2 inch) E$#av’

Typical

length

15 in

E&n

Pressure at Inlet IllNodes = + 10 I

Fig, 2—Finite.eiement grid for core cyiinder. Grid is ahaif section of cylinder with center

iine at perforation axis.

1305

I

auxiliary gridding code for this system include welldiameter, vertical and angular distance in the wellbetween perforations, and outer radius of the systemas well as perforation depth and diameter, and ciam-aged region dimensions.

The computer code considers that each perforationis centered on a portion of wellbore face whose areadepends upon well diameter and perforation spacing.For example, in a 6-in.-diameter wellbore, with 4shots/ft, spaced at 90° around the circumference ofthe well, each perforation is at the center of a boreface area 4.71 in. wide and 12 in. tall (56.5 sq in.),

I’he computer code makes the approximation thatthis area is a plane circle instead of a bent rectangle.The plane circle has the same area as the bent rec-tangle and is centered at the perforation axis. Forthis example, the circle has a radius of 4.24 in,Similarly at the outer edge of a 5-ft radial system,the cross-sectional area available for radial flowwould be a bent rectangle, 94.25 in. wide and 12 in,tall (1,131 sq in. total area). The computer codemaintains the assumption that this area is circular,Thus, the bent rectangle at 5-ft radius in a system withfour perforations per foot is represented by a circle18.98 in. in radius.

The net result is an axisymmetric flow system that

dPERFORATIONS

Region of Drilling I Td WELL

RadiuWell

IS from

t

Fig, 3—Finite.element grid for radial system around awell. The grid is a half section of a paraboloid with

center iine at perforation axis,

1306

is shaped somewhat like a truncated circular parabo-loid, Fig. 3 shows a section of this paraboloid cutthrough the axis of symmetry, the axis of the perfora-tion. The elements in this system are rings of nearlyparallelogram cross-section, except that elements inthe damaged zone around the perforation remainrectangular in order to maintain a damaged zone ofuniform thickness,

To obtain good flow definition with small elementsaround the perforation within the 600 node andelement limit imposed by our current computer code,we limited the radius of the radial finite elementsystem shown in Fig. 3 to 5 ft. For each perforationconfiguration, we then were able to calculate an effec-tive flow resistance for the 5-ft radial finite elementsystem and to translate this to a 660-ft radial wellsystem by means of the usual fcrrnula’S for flowthrough porous media with an effective discontinuityin permeability. The translation assur,les uniformpressure at the outer nodes of our radial system anduniform flow rates in the outer ring of elements,Uniform pressure was specified as a boundary condi-tion, and calculated flow rates in the outer elements,although not absolutely uniform, deviated by onlyabout 1 percent from an integrated average. Thus, wefelt secure in the translation from a 5-ft outer radiusto a 660-ft radius.

The computer code calculated the velocity anddirection of flow through each element. We consideredthat the summed flow through the single row ofelements at the input end of the cylinder, or thesingle row of elements at the outer input edge of theradial system, represented flow through the entiresystem. ‘Our calcu-latedflow rate was then the sum ofthe products: (flow velocity in each input element)X (annular area availab!e for flow in this element).

Figs. 4 and 5 show the flow distribution into aperforation in a linear core and in a radial system,respectively. A half-section of an 8-in.-long perfora-tion is shown schematically, and arrows indicate the~ercenta~e of flow into each of four 2-in.-long seg-.ments. In the linear system when the perforation isundamaged, almost 80 percent of the flOWis near thetip of the hole, However, when the permeability ofthe rock surrounding the perforation is damaged bythe perforating process, the added resistance forcesflow more toward the base of the perforation, andit is more evenly distributed over the entire surfaceof the perforation. Of course, total flow is less.

In the radial system, on the other hand, flow intothe perforation is distributed fairly uniformly overthe entire length of the perforation for both the un-damaged and damaged cases, and total flow at thetip of the undamaged perforation is only 9 percentgreater than at the tip of the damaged one.

ResultsWe investigated three relationships in this work:

1. The relationship between Core Flow Efficiency(CFE), as defined in API 22P 43, and a similar term,Well Flow Efficiency (WFE), that we defined for flowinto a perforated well, WFE is defined as the ratio offlow rate into a cased well through a real perforation

JOURNALOF PETROLEUMTECHNOLOGY

in a zone that has been damaged by perforating,drilling, or workover to the flOWrate into the samewell through a clean, ideal perforation of the samedepth in an undamaged zone. The term WFE for aradial system should not be confused with WFI fromRP 43, 1962 edition, which related to :; linear core.

2. The effect on WFE of a permeability-damagedregion around the wellbore, such as might be causedby the drilling or workover process when the perfora-tions are ideal and undamaged.

3. The effect on WFE of a permeability-damagedregion around the perforation, such as might becaused by the perforating process, in a formation with-out formation damage from drilling or workover.

Finally, we combined all these. effects to determinehow WFE is affected when both drilling damage andperforation damage are present at the same time.

Our calculations assume steady-state flow of anincompressible fluia and all single-phase perrneabili-ties. Thus, all permeabilities reported in this paperare, in effect, the permeability to whatever phase isflowing; all other phases are immobiie and permeabil-ity to the flowing phase does not change from placeto place in the system because of saturation changes.

Examples of the effects of various parameters onWFE in the following sections are given for a wellwith a 6-in.-diameter wellbore, perforated with four0.4-in.-diameter holes per foot, and with a drainageradius of 660 ft. This shot density was selectedbecause it is representative of common field practice;and, for comparative purposes, if data for a 6-in.perforation are selected, our results can be related toopen-hole well productivity in a formation withoutpermeability damage. However, we shall comparealso the effects of shot density in a damaged formationwith that in an undamaged formation.

Correlation of CFF, With WFEFig. 6 presents thv results of our calculations to

relate CFE for a 15-in. test core to WFE at a shotdensity of 4/ft. Correlations in this paper relate toCFES for a 15-in. test core because most publishedRP 43 data for commercial perforators are given forthis length, We should point out, however, that ourstudies showed that for a given penetration and dam-aged zone permeflbility CFE varies with test corelength because of geometrical effects; and, therefore,the numerical results would be somewhat different forother core lengths. However, these differences aresmall within the range of perforation depths specifiedin RP 43 for the various target Iengths and for theobserved CFE range for commercial perforators.

Both CFE and WFE are plotted against the pemle-ability, ks, of the damaged zone around the perfora-tion and for perforation depths ranging from 2 to 10in. The permeability, kt, in the damaged zone isnormalized and defined as the ratio of the permeabil-ity in the damaged region to the permeability of thevirgin rock. In this normalized system a damagedzone permeability of 1,0 is equivalent to no damage,For our studies, we assumed a ?4-in.-thick damagedzone. The assumed thickness is consistent with experi-mental observations that indicate values ranging from1/4 to s% in. in Berea cores, depending upon type of

NOVEMBE&1974

No Perforation Domage With Perforation Damogeks = 1.0 ks = 0.05

% Flow Through Each

[

T

—

2 inch Segment

‘2 v. \

4 % 8 \P,,l.a, otzon “

15 %

1

\

\

79%

% Flow ?hrough Each2 inch Segment

18 %

20%

25%

37%

Fig. 4-Distribution of flow into perforations in corecylinder for clean, ideal perforation and fordamaged perforation. Perforation diameter,

0.4 in.; damaged region, 1/2 in. thick.Outline is half section of a cylinder,

No Per foratmn Oamoge

k3= 1.0

D

7

% Flow Through Each2 snch Segment

I 9 “1.

*.. \

P*, Io#.t#o.

1

20%\

22%

\

39%

\\

With Perforot,cm Domage

k3=O05

‘T)

% flow ThroughEoch2 Inch Se9ment

23 “1.\

8“P., fo..,,m

23%

\\

24 “1.

L

\

\30%

“\

Fig. &Distribution of flow into perforations in a radialwell system for clean, ideal perforation and for damagedperforation. Well diameter, 6 in.; drainage radius, 660 ft;damaged zone, 1/2 in. thick; perforation diameter, 0.4 in.

4 shots/ft spaced at 90°. Outline is halfsection of a paraboloid.

1.0

.8 -g:~:.6 -e! “Cl* .-o~ .4 -

VLU

P’, 10,0 I,o”DeDlh

——. .-

— 2’”--—- .,4

——,. 8%> —.. —.- 10”~u

c. 4—~ .%= .:

;%” 6

8-

‘“ ‘o 1I I I I I I 1 I I 1

.03 .05.07 .1 .3 .5 .7 1.0Permeability

Fig. 6-Relationship between well tlow efficiency andcore flow efficiency. Well diameter, 6 in.; drainage radius,

660 ft 4 shot/ft spaced at 90°.

1307

gun. However, investigation of this parameter hasshown that the assumed thickness does not criticallyaffect the results.

Although for convenience we have related CFEvalues for Berea cores to WFE, the effects of differenttypes of rock material can be readily determined fromthe curves if experimental results of perforating inlinear cores are available. From experimentally deter-mined CFE values for particular rock types, thedamaged zone permeability can be estimated fromFig, 6. These values can then be used to interpolateWFE values in subsequent figures.

With the upper portion of Fig. 6 we can comparethe permeability damage caused by different perfo-rators. Published CFE values for most modem gunperforators range between 0.65 and 0.85, and fromthe figure we see that the damaged zone permeabilitycan range from about 7 to 35 percent of the undam-aged formation permeability. Our results comparewell with the calculated value of 10 to 20 percentof the undamaged zone permeability for a CFE of’0.75 reported by Bell et al.”; at this same CFE, Fig.6 indicates values ranging from 12 to 19 percent.

WFE and CFE are related as illustrated by thearrows on Fig. 6, For each CFE we determine adamaged permeability for the perforation and thenrelate this damage to a corresponding WFE. Forexample, a measured CFE of 0.7 in a 15-in. testcylinder with a 6-in. pel foration is the result of anormalized permeability of 0.085 in the assumed 1A-in.-thick damaged zone. In the example radial wellsystem with 4 holes/ft, the same damaged zone will,in turn, cause a WFE of 0.58, These results could betypical for some perforated completions shot underfavorable conditions (salt water in hole, pressure dropinto wellbore),

For CFE = 0,3, corresponding to a perforationmade under mud with pressure drop into the forma-tion, the normalized damaged zone permeability is0,013 and the corresponding WFE is 0.16.

The relationship between CFE and WFE is notsensitive to our assumption of a Y2 -in.-thick damagedregion. But if we had assumed a different thickness, 3the calculated intermediate values, the permeabilitiesof the d maged region, would be changed; for a givenCFE the permeabilities would be lower for a regionless than 1A in. thick and greater for a region morethan 1/2 in. thick. But the WFE/CFE relationship isessentially unchanged for a damaged zone thicknessranging between 0.2 and 0.7 in.

In the radial system, then, a WFE range of 0.50to 0.90 corresponds to the CFE range of 0.65 to 0.85observed for conventional perforators under the con-ditions set in RP 43. These WFE values are substan-tially higher than the values of 0.25 to 0.35 calculatedby Bell et al.” The major difference is probably asso-ciated with our shot density of 4/ft, compared withBell’s analysis for a single shot in a semi-inthitemedium.

From the correlations given in Fjg. 6 and referenceto relationships between open-hole productivity andperforation depth and density, we shall be able toestimate in subsequent sections how laboratory-

1308

measured perforator performance (CFE values) re-lates to well productivity when perforating and drill-ing conditions are known.

Effect of Perforation Depth in a ZoneWith Permeability DamageIn earlier work by others,’-” the effects of perfora-tion depth were investigaled for a virgin, undamagedformation, Now with our finite element model we areable to illustrate the effe,:ts of perforation depth cmthe productivity of a well that has been completed ina zone in which the permeability has been damagedby drilling, In the examp!e shown in Fig. 7, the per-forations are assumed tc be ideal and undamaged,and drilling damage extends for a radial distance of4 in. from the wellbore. Permeability of the damagedzone ranges between 5 and 100 percent of the virginreservoir permeability,

As mjght be expected, WFE is low when drillingdamage is seveie and the perforation does not pene-trate through the dama,ged zone, No appreciableimprovement occurs unti I the perforation penetratesthrough the damaged zone. However, the productivityis significantly reduced until the perforation extends40 or 50 percent bevond the region of driliing damage.

Inasmuch as simdar relationships will hold for per-forations in a deeply penetrating damaged zone, astrong effort should be made to avoid drilling or work-over damage that cannot be penetrated substantiallyby commercial gun perforators.

Combined Effects of PerforationDamage and Drilling DamageApplication of our radial model has been extendeda step further in Figs. 8 through 10 to include a studyof perforation effectiveness in a damaged radialsystem when the perforations are also damaged, Inthese figures it is assumed that perforating damagewas superimposed on the drilling damage; that is,k, = k, X k,. Inasmuch as the effect of damage dur-ing perforating on already damaged permeability isnot known, we shall also show similar results whenperforating damage is assumed to be independent ofdrilling damage.

Figs. 8,9, and 10 are the inverse of Fig, 7. WhereasFig. 7 demonstrates the effect of changes in perfora-tion depth in and through a 4-in. damaged zone, inthese figures the perforation depth is held constant at8 in, and the effect of formation permeability damageis shown for damage depths ranging from O to 24 in.The family of curves again shows the result~ fordamaged permeability ranging from 5 to 100 percentof virgin rock permeability,

The information given in Fig, 8 is derived for acompletion in which there is no perforation damageand therefore is comparable with the results in Fig. 7for the same depth of drilling damage and perforationdepth. As in Fig. 7, WFE is depressed stuix!antiallywhen a moderate to severe amount of drilling damageextends to, or beyond, the tip of the perforation; andit is not until penetraticm of the drilling damage islimited to about 50 percent of the perforation depththat a major improvement is noted in WFE.

JOURNALC)FPETROLEUMTECHNOLOGY

\ ● ✎

Figs. 9 and 10 include the effects of perforation4damage. In Fig. 9 we assume that the perforationprocess reduces the permeability in the V2-in.-thickregion around the perforation to 20 percent of thevirgin rock perrneability; in Fig, 10 we assume thatthe damaged permeability in this region is 5 percentof the original value. As shown in these figures, per-foration damage prevents attainment of undamagedproductivity values. For k, = 0.2, a common ckgreeof perforation damage, maximum WFE is only 80percent of the undamaged value; and for k:, = 0.05.severe perforation damage, maximum WFE is onlyabout 50 percent of the undamaged value, There islittle change in WFE as long as the drilling damagepenetrates beyond the perforation depth. As in thepreceding figures, WFE does not approach the maxi-mum value unless the depth of drilling damage canbe restricted to less than half the perforation depth.

The following example can be instructive, Fig. 10shows the WFE in a zone with drilling damage whenthe permeability of the perforation-damaged regionis 5 percent of the virgin permeability. Consider awell where drilling damage reduces permeability to

Oo&_i-~ I I I 1 I I

2 4 6 8 10 12 14 16 18 2

Depth of Penetration, inches

Fig. 7—Effect of ~erforation depth on WFE in well withdr~ling damage (k; < 1,0) when” perforations are urrdam.

aged (k, = 1,0). Well diameter, 6 in.; drainage radius660 ft; 4 shots/ft,

8 inch perforation depth, ka s 1.0

v

kz

1.0 1.0

>v — ,7

j .8 -u

z.4

~ .6 –

:ii ,4 - I

<3.2 _

.1

.05i

($~

Depth of Drilling Damage, inchesFfg. 8-Effect of drilling darnage on WFE when perforations

are undamaged,

NOVEMBER,1974

10 percent. Then, an S-in. perforation with 4 in. ofdrilling damage results in a WFE of about 34 percent.If the perforating process car, be improved so thatpermeability in the perforation-damaged zone is 20percent, then as shown on Fig. 9, the well will toleratea drilling damaged zone 8 in. deep (equal to theperforation depth) while maintaining a 34-percentproductivity ratio, If perforation damage can be elim-inated, then as shown on Fig, 8, the depth of drillingdamage can be twice the depth of the 8-in, perforationwhile maintaining a 34-percent productivity.

A major point of interest is the importance ofperforating performance on well productivity. Weconclude from this study thrit when there is no per-foration dan,age, it is possible to overcome the effectof drilling damage by substantial penetration of theperforation beyond the zone of drilling damage; hex<l-ever, when the perforation is damaged by the per-forating process, VVFEis limited to a maximum of0.8 when k. = 0.2 and to about 0.5 when k, = 0,05,even when the perforation extends well beyond thedrilling damage. On the other hand, when the perfora-tion does not adequately penetrate the drilling damage,

,8 inch,@rotion depth, k3 e 0.2

E

kz

1.0

.7

.4

Depth of Drilling Damage, inches

Fig. 9—Effect of drilling damage on WFE when perforationsare damaged (average performance, k:] = 0.2);

k4 = k3 X kz.

~“o~

I

IL

g ,2 -

Depth of Drilling Damage, inches

kz

1.0

.7

.4

k

Fig. lCzEffect of drilling damage on WFE when perfora.tiers are severely damaged (k3 = 0.05); k4 = ks x kz.

1309

1.0 - 8 inchdepth, k3 = 0,2

kz

~ .8 I 1.0al .7.-U

: .6 -.4

3

0 ,4 -L.1

z .2 - .05

oo~-Depth of Drilling Damage, inches

ilg. n-Effect of drilling damage on WFE when perfora.tions are damaged (average performance, ks = 0.2):

I(4 = k3; k4 # kz x k3.

~ .8 - 8 inchaJ.- perforation

depth, ka = 0.05~ .6 -u-! v“50 .4 -

c—5~ .2 -

‘j’~’,)’,\’;o’;4 I I

Depth of Drilling Damage, inches

Fig. 12—Effect of drililng damage on WFE when perfora.tions are severely damaged (ks = 0.05); ks = ks;

kd # k2 x k3.

WFE declines rapidly with increasing damage to aslow as 5 to 20 percent of the undamaged values.Experience has shown that in many formations mod-erate permeability damage ratios of 0.4 to 0.7 arecommon after drilling, and the perforation damagevalues used above are common for typical perforatorsand typical shooting conditions. Our curves indicatethat for this range of values WFE’S would range from0.3 to 0.7.

Because of the conditions selected for our examples,WFE values in our examples should approximate wellproductivity ratios relative to open hole. Thus ouranalysis indicates that commonly used drilling andperforating practices could result in restrictions in wellproductivities on the order of 30 to 70 percent ofundamaged open-hole productivities.

To relate these results tc observed perforator per-formance in laboratory, linear core systems, note thatthe 20-percent permeability ratio assumed in Fig. 9for the damaged zone around an 8-in. deep perfora-tion is associated with CFE = 0.75 to 0.85 (Fig. 6);a 5-percent permeability ratio is associated with CFE= 0.5 to 0.6, Both values can be easily obtained withcommercial perforators, depending upon perforatingconditions used.

As noted earlier in discussing Figs. 9 and 10, weassumed that when perforation damage occurs in azone previously damaged by drilling, the effectivedamaged permeability around the perforation, k,, isequal to k, X k,. In Figs. 11 and 12, both drillingdamage and perforation damage are again included;but k, is assumed to vary independently from k,, thepermeability after drilling damage, That is, k, = k,and k, is not a function of k,.

In view of the current lack of information onsecond-order damage, the reader may select thefamily of curves tha: best fits his own experience orinclination.

Effect of Thickness ofPwforation-Damaged ZoneIn the discussion of Fig. 6, we stated that the finalrelationship between CFE and WFE would be rela-tively independent of the thickness of perforationdamage. However, for a specific well, the well pro-duction efficiency obviously will depend upon thethickness and depth of the perforation damage. Thisrelationship is shown in Fig. 13, where the depth ofdamage around an 8-in, perforation is allowed tovary up to 3 in, Permeability in the damaged zoneranges between 5 and 100 percent of virgin rockpermeability.

These curves provide an interesting insight intoproductivity damage that may occur during work-over. The most severe damage to WFE occurs forthe first 1/2 in. of perforation damage. Beyond thatpoint WFE does not change much. Thus, if a dimworkover fluid is injected without fluid-loss controlthrough an undamaged perforation into a nondam-aged formation, the permeability damage producedaround the perforations can readily reduce well pro-ductivity as much as 50 percent, even with a limiteddegree of invasion. However, if reasonable precau-

‘0 1 2 3Thickness of Damage

Around the Perforation, inchesFig. 13—Effect of thickness of perforation-damaged zone

on WFE in well with no drilling damage.

1310 JOURNALOF PETROLEUMTECHNOLOG-f

I● tions are taken to use a relatively nondamaging fluid

with adequate fluid-loss control, the productivity re-duction can be limited to a minor amount.

If the perforation is already seriously ddmagedfrom the completion process, for example as indicatedby the curve labeled 0.1, fluid invasion that adds tothe depth of damage — say from 1/2 to 3 in. or more— reduces WFE a relatively small additional amountif reasonably good workover fluid is used.

From an over-all viewpoint we should keep in mindthat a workover treatment can create effectively acondition similar to perforating with insufficient pene-tration into a previously damaged zone. Therefore,previous relationships should be remembered regard-ing the productivity of wells in which perforations aredamaged in a zone with drilling damage also.

Some Examples of the Effect of Drilling andCompletion Conditions on Well Pro: ~~.ivityIn the Introduction, we pointed out that CFE valueswere, up to now, suitable only for comparisons as tothe quality of perforations produced by different guns.However, with the foregoing analysis, we are nowable to show how CFE values can be used practicallyto estimate how different completion practices canaffect well productivity. Although the values obtainedmay not be precisely accurate because of simplifyingassumptions, we believe the effects of the variousparameters are qualitatively correct and therefore ouranalysis provides a tool for critically evaluating theimportance of various operating practices. For ex-ample, we have seen that if poor drilling and comple-

tion fluids are used — that is, with high-fluid loss andformation damaging characteristics——inability topenetrate through the darnaged zone with existingperforators can result in extreme loss in well produc-tivity. On the other hand, even completions that areproducing in an undamaged manner because ofeffective, deep perforations, wi!l be easily damagedby invasion of a damaging fluid during workover,because after the job the perforation will be insidethe damaged region (see Fig. 7).

To illustrate more graphically the practical impli-cations of our studies, we have used our model toestimate Ihe productivity that could be expected froma typical well, completed with cemented casing andperforated with 4 shots/ft, compared with an assumedpotential productivity of 800 B/D in an undamagedopen hole. The results are given in Tables 1 and 2.We have assumed two different sets of drilling condi-tions: one with an ideal fluid that causes no formationdamage, the other with a fluid that damages formationpermeability, In the second case, we investigated theeffectsof both a moderately damaging fluid (k, = 0.7)and a severely damaging fluid (k~ = 0.1) with threedifferent invasion depths, 4, 8, and 12 in. In practicqof course, fluid invasion and the associated damagemay penetrate even deeper, depending upon drillingand completion conditions.

Results are given for two different perforationdepths, 4 and 8 in., and a range of CFE values from0.3 to 1.0. The 4-in. penetration corresponds to iesults

with certain small through-tubing guns, poor shotphasing, or shooting through multiple strings of pipe,

TABLE l—EFFECT OF PERFORATING CONDITIONS AND PERFORATION DEPTH ON PRODUCTIVITY OF A ‘WELLWITH AN OPEN-HOLE POTENTIAL OF 800 B/D

(No drilling damage, perforated with 4 holes/ft.)

Perforating ConditionsWell Productivity (B/D)at Perforation Depth of

CFE Perforator Fluid Pressu rc 4 in. 8 in.

0.3 Average - High solids, mud in hole + AP 115 1540.5 Poor Unfiltered salt water + Ap 253 3300.7 Average Filtered salt water + Ap 429 5690.8 Average FiItered salt water – AP 538 6890.9 Best Clean, nondamaging fluid, – Ap 653 792

best techniques available1.0 Ideal Clean, nondamaging - AP 768 856

-1-Ap = Wellbore pressure > formation pressure.- Ap = Wellbore pressure < formation pressure.

TABLE 2—EFFECT OF PERFORATING CONDITIONS AND DRILLING DAMAGE ON PRODUCTIVITY OF A WELL WITHAN OPEN-HOLE POTENTIAL OF 800 B;D

(S-in,.deep perfomtion, 4 holes/ft, formation damaged during drilling.)

Perforatimz ConditionsCFE Perforator “” ~luid Pressur=

0,3 Average High solids, mud + Ap0’5 Poor Unfiltered s?lt water + AI)0.8 Average FiItered salt water – Ap1,0 Ideal Clean, nondamaging – AP

Well Productivity (B/D)Formation Damaged to Depth of

4 in. 8 in. 12 in.kD/k. ko/k, kD/k.

0,1 _ _ _ _ —0.7 0.1 0,7 0.1 0.7Q~ 136 15 114 9 112

219 297 56 259 36 254576 661 247 615 162 601803 843 530 813 331 794

Note: k. permeability of damzged zone-K” permeability of undamaged zone = kz.

+ Ap = wellbore pressure > formation pressure,– Ap = wellbore pressure < formation pressure.

NOVEMBER,1974 1311

.

The 8-ii. depth is fairly typical for many well designedperforating jobs. Certain special guns can provideeven deeper penetrations; but with the conditionsselected for our example, the penetration values areadequate to illustrate the effects of perforating condi-tions and amount of formation damage.

The CFE values used cover common completionpractices ranging from poor to ideal, When a produc-ing formation is perforated under drifling fluid witha large pressure drop into the formation, earlier pub-lications’-” give permeability damage ratios of 0.5or less, which correspond to CFE values of about 0.3or less. The larger CFE values used in our examplecorrespond to improved perforators and perforatingconditions as described in the tables.

As shown in the tables, depending upon perforatingconditions and type of perforator, the productivity ofthis hypothetical 800 B/D well could range from aslow as 115 B/D to 856 B/D when no drilling damageis present, and from 9 to 843 B/D when drillingdamage is present.

1—K1.mz,K,v, o,, S P,.

1.4 ———-H.,,,,

. . ..-.. -. M. DL.w. !I 6 Mvsk.t—— rb..d & W.t,.a.

1.2

.2tiu_lA4~O 2 4 6 8 10 12 14 16

Perforation Penetration, inches

Fig. 14-Effect of shot density and penetration on produc-tivity ratio. Well diameter, 6 in.; drainage radius, 660 ft:

perforation diameter, 0.1 i%

o.-.

00+ -

FJo. of Shots Per FootFI . 15-Effect of perforation damage, as characterized by

fla oratoty CFE values or calculated k3 values, on WFE forgiven shot density Perforation depth, 6 in.

1312

This example illustrates the importance of carefulengineering of the completion to minimize formationdamage from drilling and to optimize perforatorselection. The prociuctivity values show clearly thatwhen drilling damage cannot be avoided, it is ex-tremely important to select a gun perforator with botha high CFE value and a penetration potential that willsubstantially exceed the depth of formation damage,particularly when the permeability reduction is severe.If this is done, the effect of formation damage canbe almost negated if the CFE value for the perforatoris 1.0; that is, if there is no damage in the perforationitself. However, with a CFE value of 0.8, typical ofmany commercial guns, the perforation damage pre-vents effectivebypassing of the drilling damage,

On the other hand, if the well can be drilled withlittle formation damage, the effect of perforationdepth is less critical, although it is still important toperforate in a manner that will provide high CFEvalues.

Relative Effects of Shot DensityAnd PenetrationPrevious investigators9,10 concluded that in ideal,undamaged radial flow systems shot density primarilygoverns the productivity of the well and is more im-portant than penetration. Four perforations 6 in. deepwere shown to provide well productivity equivalentto open hole, and several shallow perforations wereconcluded to be more effective than a single, deeplypenetrating one.

To compare the results from our finite elementmodel with the work of these earlier investigators, wecalculated the well productivity ratio

(productivity of a perforated wellproductivity of anopenhole )

for the same ideal, radial flow system (the perforatedwell is assumed io have no drilling or perforationdamage). The denominator of this ratio can be calcu-lated in our perforation model by assuming (1) aperforation cross-sectional area almost equal t‘ thesurface area of the wellbore segment and (2) an irdini-tesimal perforation depth. In Fig. 14 the results ofthis calculation are plotted along with publishedresults of Harrisll and of McDowell and Muskat.10The shape of our curves agrees well with the curvesof McDowell and Muskat, and the variation in actualvalues from point to point probably results from theirinability to exactly scale perforation diameter in thGh’electrolytic model. On the other hand, our curvesagree well with I-Iarns’ results at low penetrationvalues and at high-shot density but diverge at low-shotdensity. The simplifying assumptions that we makeare somewhat &Terent from his and therefore we can-not expect perfect agreemtint, For example, Harrisassumes a radially expanding perforation to conformwith his radial segment, whereas we assume a cylin-drical perforation in a truncated circular paraboloidmodel with cross-sectional areas equivalent to a radialsegment. Our truncated circular paraboloid modelcannot a“mountaccurately for flow around the casing.Because these effects are most important at ve~ low

JOURNALOF PETROLEUMTECHNOLOGY

perforation densities, our results are less accurate ata density of 1 shot/ft.

We carried these studies one step further and in-vestigated the effect of shot density in a perforatedradial system in which the perforations are damaged.As a measure of perforation damage, we usedlaboratory -measured CFE values for 15-in, coresand assumed again a 1/2-in.-deep damaged zone. Re-sults are given in Fig. 15 for 6-in,-deep perforations.The top curve labeled CFE = 1.0(k, = 1.0) corre-sponds to the results of previous investigations of anideal, undamaged completion. As previously, at adensity of 4 shots/ft the productivity ratio is 1,0.

We note now that, with the exception of perfora-tions with only moderate permeability damage (ks =0.46), corresponding to CFE = 0.9, it is not possibleto achieve open-hole productivity with a reasonableshot density, With a perforator and conditions char-acterized h, CFE = 0.9, about the best system pres-ently available, shot density must be increased toabout 7/ft to achieve open-hole productivity.

Corollary to the above, even neglecting the effectsof drilling damage, it is not possible within practicallimits to overcome poor perforating procedures byincreasing shot density. We have pointed out earlierthat use of over-balanced pressures and high-solids-contcnt fluids during perforating corresponds toCFE x 0.3 for a 6-in. perforation. At a perforationdensity of 4 holes/ft, this practice would result in awell productivityy ratio of about 0.35, according tothese curves. Doubling the shot density to 8/ ft in-creases the productivityy ratio only to about 0.5, com-pared with a value of 0,7 to 0.9 that could be attainedwith better perforating practice.

Fig. 16 illustrates the relative effects of shot den-sity and penetration depth on well productivity ratiowhen perforations are damaged in an otherwise un-damaged formation. Productivity ratios are plottedfor penetrations up to 18 in. and for shot densities of4/ft and 8/ft. The perforation-damaged zone is againassumed to be 1%-in.thick. Perforation damage values(k,) are indicated on the curves.

The curves in Fig, 16 show that an increase in shotdensity is more beneficial when the perforation isdamaged than when it is undamaged and that doub-ling perforation depth has about the same effect asdoubling shot density. However, for severely dam-aged perforations (ks = 0.05) a deeply penetratingperforation is more effective than an increase in shotdensity. For example, an increase in shot density from4/ft to 8/ft i~i a 4-in.-deep perforation raises thewell productivity ratio from 0.35 to 0.52, which stillcorresponds to a large restriction in well productivity;but an increase in perforation depth from 4 in. to 18in. at a shot density of 4/ft raises this ratio from 0.35to 0.79, thus overcoming a substantial portion of theeffectsof perforation damage. With the severely dam-aged perforations in this example, open-hole produc-tivity can be attained by increasing the combhtddeffects of shot density to 8/ft and perforation depthto 18 in. — or, as indicated by extrapolation of thecurve for k~ = 0.05, by increasing perforation depthto about 26 in. at 4 shots/ft.

In Fig. 17 we investigate the effect of shot density

and penetration when both drilling damage and per-foration damage are present in. the completed well.Commonly experienced damage factors are assumed:k, = 0.4 and k, = 0.2

The importance of perforation depth is highlightedagain, Two shots per foot 18 in. deep is more effectivethan 8 shots/ft 4 in. deep, In general, within practicallimits increasing shot density alone cannot overcomethe combined effects of permeability damage fromperforating and drilling or workover, Therefore,regardless of shot density, deeply penetrating perfora-tions that extend substantially beyond the permeabil-ity damage from drilling (or workover) are necessaryif the productivity of a damaged well is to approachthat of an undamaged, open-hole completion.

Our conclusions on the importance of per.-:rationin wells with permeability damage are contrary tothe results of previous work by other investigatorse-”who concluded that in ideal, undamaged completionsshot density is more important than perforation depth.

This work strongly emphasizes the importance ofcareful attention to completion practices to minimizethe depth and severity of formation damage fromdrilling, perforating, and workover,

ConclusionsThrough use of a finite element model, we have ex-tended previous studies of the productivity of a well

1 1 I I I I 1 ! 1 1

00 2 4 6 8 IO 12 14 14 18J

Perforation Penetration. inches

Fig. 16-Effect of perforation parameters on well produc-tivity in well with no drilling damage.

‘:’*6/ ,0 ,.,

,/ .,”/ ..:1’/ .,... -// ...””””/ ..../ ,..

..’”,..

Perforation Penetration, inches

Fig. 17—Effect of perforation parameter on well producetivity in well with moderate drillin[; damage and normal

perforation damage.

NOVEMEE~ 1974 1313

in art ideal radial system to include the effects ofpermeability damage from drilling, workover, andperforating. For illustrative purposes, this paper hasdealt specificallywith results for a perforation densityof 4 hcJes/ft in a symmetrical pattern, although themethod can be applied to other densities and patterns.Our analysis enables us to draw a number of generalconclusions about the effectivenessof gun perforating,but in the summary below it should be rememberedthat the numerical results apply specificallyto the shotdensity and pattern used.

1, Damage factors indicated in linear cores fromCore Flow Efficiencies, as determined according toAPI RP 43, second edition, can be related to Well

●

Flow Efficiencies fcr the model described.2. Permeability in a 95-in.-thick damaged zone re-

sulting from the perforating process ranges from about0.3 of the undamaged-formation permeability forgood perforating conditions to about 0.01 for adverseperforating conditions,

3, In a radial system with formation da”mageandno perforation damage, Well Flow Efficiency is sub-stantially reduced until the perforation penetrates sub-stantially beyond the damaged zone.

4. In a radial system with both formation damageand perforation damage, Well Flow Efficiency re-mains considerably below that for an undamaged sys-tem even when the perforation penetrates substantiallythrough the zone of formation damage.

5. The major effect of permeability damage arounda perforation occurs from the damage within the first%2 in, of the perforation,

6. Application of our study to a hypothetical per-forated completion indicates that productivity mayrange fronl as low as 5 to 90 percent of undemaged,open-hole productivity, depending upon the nature ofdrilling and perforating opemtions. Therefore, everyprecaution should be taken to avoid permeabilitydamage to the formation during drilling, workover,and perforating.

7. If formation damage is avoided during the drill-ing process, a perforation depth of 12 in. or more isrequired to overcome the loss in productivity fromdamaged perforations that is indicated for many com-mercial guns by standard API RP 43 tests (CFE -0.7-0.8). Increasing shot density from 4/ft to 8/ft hasabout the same effect as doubling penetration from 6in. to 12 in.

8. Perforation quality is more important thaneither shot density or penetration. The effect of severe

Original manuscrifM received in Society of Petroleum Engineersoffice Aug. 6, 1973. Revised manuscript received July 29, 1974.Paper (SPE 4654) was first presented at the SPE.AIME 48thAnnual Fall Meeting, held in Las Vegas, Nev., Sept. 30-Ott, 3,1973. @) Copyright 1974 American Institute of Mining, Metal.Iurgical, and Petroleum Engineers, Inc.

This paper will be printed In Transactions volume 257, which willcover 1974.

damage in the perforations cannot be overcome byincreasing either shot density or depth of penetrationwithin the limits of present-day technology and eco-nomics.

9. In completions with drilling (or workover) andperforating damage, a few deeply penetrating per-forations are more effective than many shallow per-forations.

AcknowledgmentWe are grateful to the Union Oil Co. of California forpermission to publish this paper. We acknowledgewith many thanks the efforts of M. L. Garrett whoperformed most of the computing work.

References1. History of Petrolewn Engineering, API Div. of Produc-

tion, Dallas, ( 1961).2. Oliphant, S. C., and Farris. R. F.: “A Study of Some

Factors Affecting Gun Perforating, ” Trans., Al ME( 1947) 170. 22 S-237.

3. Lewelling, ~1.: “Experimental Evaluation of Well Per-foration Methods as Applied to Hard Limestone;’Trans., AIME ( 1952) 195, 163-168.

4. Allen, T. O., and Atterbury, J. H,, Jr.: “Effectivenessof Gun Perforating,” Trans., AIME ( 1954) 201, 8-14,

5. Allen. T. O.. and WorzeL C. H.: “Productivity Method,. .> –,of Evaluating Gun Perforating,” Drill. and Prod. Prac.,API (1956) 112.

6. Krue~er, R. F.: “Join: Bullet and Jet Perforation Tests,Progre>s Report,” Dr;ll. and Prod. Pruc., API ( 1956)126.

7. Suman, G, O., Jr.: “Perforations — A Prime Sourceof Well Performance Problems,” J. Per. Tech. (April1972) 399-4i I.

8. Muskat, M.: “The Effect of Casing Perforation on WellProductivity,” TM/ns., AIPIE (1943) 151, 175-184.

9. Howard, R. A., and Watson, M, S., Jr.: “Relative Pro-ductivity of Perforated Casing—l,” Trans., AIME ( 1950)189, 179-182.

10. McDowell, J. M., and Muskat, M,: “The Effect on WellProductivity of Formation Penetratimi Beyond Per-forated Casing,” Trans., Al ME ( 1950) 189, 309-312.

11. Harris, M, H.: “The Effect of Perforating on Well Pro-ductively,” J. Per. Tee/I. (April 1966) 518-528; Tram.,AIME, 237.

12. Bell, W. T., Briege~, E. F., and Harrigan, J. W., Jr.:“Laboratory Flow Characteristics of Gun Perforations;J. Pet. TCC}I, (Sept. 1972) 1095-1103.

13. “API Recommended Practice: Standard Procedure forEvaluation of Well Perforators:’ API Div. of Produc-tion, RP 43, 2nd ed. (Nov. 1971).

]4. Willems, N., and Lucas, W. M,, Jr.: A4ufrix Analysis/or Srructmaf En~ineers, Prentice-Hall Inc., EnglewoodCliffs. N. J. (1968).

15. Zienkicwicz, O. C., and Cheun.g, Y. K.: The Finite Ele-ment Method in Structural ancf Continnam Mechunics,McGraw-Hill Publishing Co., Ltd., London ( 1967).

16. Krueger, R, F.: “An Evaluation of Well CompletionEtTectiveness,” API Paper 801-38P (May 9-10, 1962).

17. Taylor, R. L, and Brown, C. B.: “Darcy Flow Solu-tions with a Free Surface,” J. Hydrmlics Div. Proc.,ASCE ( March 1967) 93, IHYDJ 25.

18. Muskat, M.: Physical Principles oj Oil Production,McGraw-Hilt Book Co., Inc., N,Y. (1945) Eq. 6, 244.

tJPT

1314 JOURNAL OF PETROLEUM TECHNOLOGY