Factors Influencing Optimum Ball Sealer PerformanceROBERT W. BROWN

JUNIOR MEMBER AIMEGEORGE H. NEill

RAYMOND G. LOPERMEMBERS AIME

THE WESTERN CO.DALLAS, TEX.FORT WORTH, TEX.

WESTON, W. VA.

FACTORS AFFECTING CONTACT WITHPERFORATION

(1) (2) (3) (4)t ~ + Ud ~ ~F,t6+ tt t ~ t t t ~i/! J9 !l\ \ F.)~ \) !fY + ~ Y'v {(; ~~ + ~ f

Inerlial forc(' Drag I'orce Holding I"Olce (Tmwaling I'orce

lJnsealingBalance of F. and f'1/

SealingBalafl(~e of f'" a IIII f',

Ball sealers are injected into well treating fluids for thepurpose of contacting and sealing those perforations whichare accepting the fluid flow. The efficiency of the sealersis primarily influenced by: (1) the velocity of the ballsdown the pipe, and (2) the fluid velocity through the per-forations. To divert the sealer to the perforation, the iner-tial force of the ball must be overcome by the drag forcecreated by the fluid velocity through the perforation.BALL VELOCITY

The final, or stable, velocity of the ball is the sum ofthe actual fall velocity and the fluid velocity. This will be

nlay not be exactly true, it has been verified within prac-tical field limitations.3 The discharge coefficient would notvary significantly even if the perforation size or configura-tion varied pronouncedly. The approach of assuming uni-form perforation size is not as strained as it might firstappear. Normally, a perforator contains charges equallyspaced with the same number in each vertical (or horizon-tal) plane. If the perforating gun is not centralized, theholes in one plane will be smaller than average, while theholes in opposing planes will tend to be larger thanaverage. The net result is that the two effects tend tobalance.

INTRODUCTION

The perforation sealing proces's has been proven highlysuccessful and economical since it was introduced to theoil and gas industry in early 1956.1 Since it has proved sosuccessful in multistage fracture treatments at a 'muchlower cost' than conventional packers, the ball sealer proc-ess has become a byword in well completions.:! Not onlyhas it changed the design of fracture treatments, it haschanged the concepts of selectively perforating and hasgreatly aided the success of the single-point and limited-entry techniques.:l

Yet with all the advantages, there are still facets of ballsealer behavior that are not widely recognized. These mustbe known and understood to better design and executethe optimum ball sealer treatment.

The down-hole behavior of ball sealers is influenced bya number of factors including an "inertial force", a "dragforce" and a "holding force". These are considered in twophases-the initial seating of the ball and the forces tend-ing to unseat the ball after contact has been made (seeFig. 1).

This paper presents a theoretical study of these factorsand formulas for predicting their influence. The authorsacknowledge the fact that the flow equations presentedherein describe the behavior of Newtonian fluids and donot cover the so-called power-law fluids. The assumptionsmade, however, are reasonable and well within practicallimitations of the conditions encountered during field ap-plication. For example, it is assumed that the perforationsare round with a discharge coefficient of 0.82. While this

1L\BSTRACT

All facets of ball sealer behavior must be known andunderstood to design for their optimum use in well treat-/nents.

The down-hole factors including the inertial forces, dragforces and holding forces have been discussed. Equationshave been developed and presented for these forces. Theuse of these equations in this process will contribute to the/rLost efficient use of an economical and proven ball sealerprocess.

The procedure presented has been used to successfullyanalyze field results indicating problems involving the useof ball sealers. These problenls have been isolated and cor-rected in future treat/nents by using the procedure pre-sented to revise the perforating progranz, the injection rate,the design of ball sealer stages and/or the type and sizeof ball sealers. The procedure presented also indicates thatdislodgenlent of ball sealers occurs priJnarily in ultra slinz-hole completions.

Original manuscript received in Society of Petroleum Engineers officeJan. 3, 196,3. Revised manuscript received March 18, 1963.

lReferences given at end of paper.FIG. I-SIMPLIFIED SKETCH OF THE BASIC FORCES GOVERNING

BALL SEALER EFFICIENCY.

450 JOURNAL OF PETROI.. EUM TECHNOLOGY

tledbetterTypewritten Text

tledbetterTypewritten TextSPE 553

or

used as the initial velocity of the ball sealer as it entersthe perforated interval of the casing.

V.r == Vi +Va (1 )The actual fall velocity is computed by first establishingthe theoretical fall velocity (terminal velocity) and cor-recting this value for prevailing well conditions.

The terminal fall velocity of the ball through the treat-ing fluid is primarily influenced by the density and diam-eter of the ball sealer and the fluid density.

V, = 1.89.../(PB-~I)D . (2), PI D

The drag coefficient fD varies with the Reynolds number.This relationship is shown in Fig. 2. Since V s and fDareboth unknown, a trial-and-error solution is suggested.

The trial-and-error procedure of solving for V s can beeliminated by combining the equations for fD and Re andeliminating the term V s.4, 5

log f/) = -2 log Re+loge(PII-~::, gD3PJ)when

Re == 1,

f == (_4 (pn - PI) g D:JpI )D 3p.,'2 ,

when

The

(3)

fD=-=I,R == (4 (plI~PI) P1 D3 g)lhe 3p.,'2 ,Re = 1.46XlO' CPlI-P~PfD'f' (4)intersection of the straight line through these two

points with the plot of fD vs Re (Fig. 2) will give thevalue of Re and fD incorporating the value of Vs. V smay then be calculated from the basic Re equation.

Re = 7.72X 10' ( D:,PJ ).or

V s == 1.295 X 10-4 DRe

p., (5)PI

Correcting V s for wall effects of the treating pipe string,in turbulent flow the equation becomes

V" = V. (I +~/d) (6)where

d == 4 (hydraulic radius) == D (' - D, andVa ==Vs in laminar flow.

The fluid velocity V i is expressed simply as

V;=13.5( Q/A,) =17.2( Q/Dc') . (7)INERTIAL FORCE AS A FACTOR OF BALL VELOCITY

In order to divert the ball from its vertical path downthe casing and to permit the ball to contact a perforation,the vertical component of ball velocity must be reducedto zero. The inertial force which must be overcome toeffect this velocity reduction is represented by the equation.

F/S=; (V/-O").The effects of a perforation on flow in the pipe are dis-sipated over a distance (5) of about 1 to 2.5 pipe diam-eters fron1 the perforation:; The lower value of one pipediameter, or Dc was selected to represent extrceme cone-ditions. The selection of the lower value results in ahigher inertial force which must be overcome to changethe direction of ball travel.

1010 10REYNOLDS NUMBER 'I R -~e - M

FIG. 2-DRAG COEFFICIENT (!D) VS REYNOLDS NUMBER (Re).

10

\

\~ \~ \I ~ \~

-

-

--

-----

\~ ~I

-

5 2 5 :3 5 4 5 ~ 50.1 10

.5

a:::~ 5u

Since V f is the initial velocity at the entrance of the per-forated interval, and substituting V f == Vi + Va, the cor-rected inertial force becomes

FORCES ACTING ON THE BALLON THE PERFORATION

and

The force tending to remove the ball from the perfora-tion is created by the fluid drag upon the exposed portionof the ball. The force tending to hold the ball on theperforation is proportional to both the area of the per-foration and the pressure differential across the ball (orperforation). In order to dislodge the ball from the per-foration, the fluid drag on the ball must exceed the ver-tical vector of the holding force.

FORCE TENDING, TO UNSEAT THE BALLNeglecting the unexposed area of the ball inside the

perforation, the existing fluid drag tending to unseat theball is

F" = 5028X10''(fvPJ V/D')

near the casing center, and in turbulent t10w the ball iscontinually moving with a horizontal component. Thisindicates that for laminar flow when 'Pn 2 F[ the ball willhit, and for turbulent flow the ball should hit when F n == F I ,or perhaps some degree less.

(9)

( 10)

QzVi == 17.2-D~c n

F[ = 3.52X 103e:') (V/) . (8)As the ball traverses a perforated interval, V f and F I

decrease because of the decrease in flow caused by a lossof injected volume into the upper perforations. Providedthe perforations are at least one diameter (Dc) apart,each perforation will behave as a single isolated channelwith respect to inertia. The following calculations canthen be made. Correcting Vi for the effects of decreasingflow,

-:l PB D3 [ 17 2 Qz V] -F I == 3.52X10 -D- . -D.2 + (l c (" n

Eq. 10 permits calculation of F I at any point within theperforated interval. Eq. 8 calculates the maximum inertiaand would apply for single plane perforating or for in-stances where the flow per perforation is sufficient to divertthe ball regardless of its point within the interval.

(17)

( 16)

DRAG FORCE ON THE BALL VSPERFORATION FLOW VELOCITY

To calculate the drag force created on the ball by thefluid flow through a perforation, it is assumed that thehorizontal component of fluid velocity varies fron1 zero,a distance of Dc away, to the maximum velocity VII'through the perforation. The velocity used in the equa-tions is the average of these values, or V p j2. It is furtherassumed that the relative drag velocity between the fluidand the ball is in a horizontal plane.

The velocity of fluid through each perforation is shownrelative to total injection rate.Qc

V p == 17.2 D 2D (11)n }9 rl

The drag on the ball tending to divert it from the cen-ter of the flow channel toward the perforation is shownin the following equation.

nl A .jFJ) == CdT nV-R ,

( In Pf D2 Q~c)

Fn == 0.391 n~D/C/ (12)The drag coefficient If) varies with the Reynolds number asshown in Fig. 2. The Reynolds number is calculated fromthe expression

Re==6~65XIO,"1( D~9~). (13). nIL p II

When P J) is greater than FI, the ball will divert fromthe central flow channel to contact the perforation. Thisis based on the assumption that the force component ofthe velocity acts or dissipates over a distance of one cas-ing diameter (D c) and that the ball enters the force sys-tem a vertical and horizontal distance of Dc from the per-foration. This develops the theory that the net velocitytending to pull the ball into the perforation varies fromo at the distance D (; to V p at the perforation with an aver-age drag velocity of V p j2. (The drag force calculated byEq. 12 represents the "exact" pull that would be appliedto the ball at the casing center line.) With this in mindit is suggested that, if the ball remains at the extremecasing wall throughout the entire interval of Dc, it isprobable that Pn would not divert the ball. However, itis probable that the ball would never remain at this dis-tance; e.g., in laminar flow the ball should remain at or

Re = 7.72X 10 (D:iPI ).Assuming no deformation of the ball against the perfora-tion, the following equations correct for the area insidethe perforation.

F ==5.28XlO-:11 V. 2 [D 2 _D 2,fJ D11(D2_D2)1J:I]u . npf l 180 + 7r p

( 14)and

Re = 7.72X10" V~~[D'_ ~~~ + ~P(D'-D/)%r(15 )

h . D pwere SIne 8 == v.

FORCE TENDING TO HOLD THE BALLThe force holding the ball on the perforation is created

by the differential pressure across the perforation. This willbe at a minimum the instant of sealing and will increasetherefrom due to subsequent bleed-off of fracture fluidsto the matrix of the formation. Under normal conditionsthe change in holding force due to this bleed-off is nominaland is neglected in the follo\ving analysis.

The holding force is represented by the followingequation.

PII == : DfI".! !SP p

and

f:lP" = 1~9( VJ"~- V,'+ kV,.')where

1k == C/ - 1 = 0.062,

since, for practical purposes, C v can be estirnated(j,7 toequal 0.97.Then,

(1.062 Q('2 Q('2 )~Pfi = 1_99 PI -".!D Ie ".!- - --D~ .

n II .(/ CAt the instant of impact of the ball against the perforation.the holding-force equation becon1es

?(1.062 Q('~ Q/)F H == 1.56 pfDp - ---:FD etC:! -D4 n 1J d ('452 JOURNAL OF PETROLEUM TECHNOLOGY

To remove the ball from the perforation during the treat-ment, the drag force F u must exceed the vertical vectorof F ll Then,

F~u.'> D p F./ (D'2_D/)l/:l H,

or

1.56ptD p:: (1.062 Q/ Q/) (18)F U >(D'2_D

p'2)% n2D

p4C

rl'2 -Jj: .

Eq. 14 permits calculation of existing fluid drag and Eq.18 determines the fluid drag required to unseat the ball.If the existing F u, (Eq. 14) is less than that required todislodge the ball (Eq. 18), the ball will remain seated.

Actually, to insure that the ball remains seated once ithits a perforation, FJ[ must exceed F u +F[ which intro-duces a correction for the event that the seated ball isstruck by a second ball traveling down the casing. Theprobability of one ball hitting another is relatively low;however, this possibility should be recognized.

SELECTION OF BALL SIZEThe proper selection of the ball size is important due

to the effect of ball diameter on the afore-mentionedequations. Several sizes are available in several composi-tions and combinations of materials.

Based on test data with 90 durometer solid rubber balls,the following relationship was established for the condi-tions where D '2.1.25 D p and the bottom-hole treating tem-perature is within the range of 60 to 200F. The equationcalculates the maximum pressure differential which theball will withstand for one hour (under the describedconditions) without extruding through the perforation.

(D" D ")1 /6.Pl'=4 -;;l',L "[1800-15(T-llO)] . (19)

This equation can be rearranged to calculate the minimum-size baH required to withstand a given pressure differentialagainst a given size perforation at the designated bottom-hole treating temperature.

'2 ~ D p '2 tiP]I'2 +D '2 (20D - 16 [1800-15(T-I10)f ]/. .)

The factor 1,800 is characteristic of a particular rubber for-mulation, as is the temperature factor 15 (T - 110). Thesevalues are determined empirically for a given rubber com-pound. Similar factors are available for other types ofrubber, but only one is presented here for brevity.

CONCLUSIONS

Having considered the factors affecting the performanceof a ball sealer, it is possible to select the optimum condi-tions required for a successful ball sealer treatment. Theseconditions include size and type of ball, perforation sizeand number, injection rate and type of fluid. In addition,some insight can be derived regarding the size and numberof ball stages required to obtain the desired diversion oftreating fluid. These factors have provided a useful aidto the design and operation of successful ball sealer treat-Inents. It should be recognized that the equations are, ingeneral, conservative; in isolated instances, a successfuljob could result where a failure is predicted.

Ball sealer treatments performed in the field have beensuccessfully analyzed with these equations. Tteatmentswhich had been considered failures were systematicallyinspected and the cause of failure determined. On thenine wells considered, this design proced~re did not failto locate the problem area. In nearly all instances theinertial force on the ball was greater than' the drag force,thus causing the ball to either 'miss the perforations com-pletely or, at best, catch only the lowermost perforations.

APRIL, 1963

Ball sealer performance was significantly Improved on fol-lowing treatments by either: (1) changing the perforatingprogram, (2) increasing injection rate (and, thus, "catchdrag"), (3 ) changing size and/or type ball (minimizeinertial force), or (4) improving ball stage design (recog-nizing that only lowermost perforations were being sealed).It should be recognized that, in many cases, increasingthe injection rate alone will not improve the catch effi-ciency. While this practice will increase the catch drag, italso increases the ball inertia.

In only a few of the cases considered was there indi-cation that the ball becan1e dislodged from the perforationonce it had "hit". This occurrence was further limited toultra slim-hole con1pletions.

Thus, we have a proven process for designing and pre-dicting ball sealer performance.

NOMENCLATURE

V f == final vertical ball velocity, ft/secVi == fluid velocity in casing, ft/secVa == corrected terminal fall velocity, ft/secV s == terminal fall velocity of ball, ft/secpn == specific gravity of ballPt == specific gravity of fluidD == diameter of ball, in.ff) == drag coefficient

Re == Reynolds numberI), == fluid viscosity, cpg == gravitational constant, lb mass/lb force/sec'2d == equivalent diameter, in.

Q(' == casing injection rate, bbl/minA(. == cross-sectional area of casing, sq in.F 1 == inertial force of ball, lb forceS == distance over which F 1 is reduced to zero, in.

In == mass ball, lb force sec'2/ftz == perforation number (lowermost == No.1,

consider only those taking fluid)n == total number of open perforations existing

at a given timeC el == discharge coefficient of perforationF D == drag force tending to divert ball to perforation,

lb forceV R == relative velocity between fluid and ball, ft/secA B == cross-sectional area of ballF u == drag force tending to unseat ball, lh forceDc == diameter of casing, in.e == angle representing ball area within perforation,

degreesV]) == fluid velocity through perforation, ft/sec.k == constant

C v == coefficient of velocity for perforation dischargeD.Pp == pressure differential across perforation, psiF Il == force tending to hold ball on perforation,

lb forceT == temperature, OF

D]I == diameter of perforation, in.

ACKNOWLEDGMENT

I 1he authors wish to extend their appreciation to themanagement of The Western Co. for permission to pub-lish this information.

453

REFERENCES

1. Kastrop, J. E.: "Newest Aid to Multi-Stage Fracturing", Pet.Eng. (Dec., 1956).

2. Neill, George H., Bro,vn, Robert Wade and Silnmons, CharI.e!"M.: "An Inexpensive Method of Multiple Fracturing", D'rill.and Prod. Prac. API (1957).

3. Brown, Robert Wade and Loper, Raymond G.: "StimulationTreatment Selectivity Through Perforation Ball Sealer Tech-nology", Pet. Engr. (June, 1959).

4. Brown, G. G., et al: Unit Operations, John Wiley & Sons, Inc.,N. Y. (1950).

5. Gray, Kenneth E.: "The Cutting Capacity of Air at Pressure!"Above Atmospheric", Trans., AIME (1958) 213, 180.

6. Perry, John H.: Chemical Engineers Handbook, IVlcGraw-HillBook Co., Inc.N. Y. (1950).

7. Daugherty, R. L.: Hydraulics, l\1cGraw-Hill Book Co., Inc.N. Y. (1937) 118-126.

APPENDIX

EXAMPLE PROBLEMThe following example problem will aid in understand-

ing the foregoing equations and discussion. This examplewill be worked on the basis of the following assumptions:rn == J.28, Pi == 1.07, fl- == 1.0, D == 0.875, D(' == 5.0 (ID),Q == 30, n == 50, D]) == 0.375.

The first step is to calculate the final velocity (Vi) ofthe ball. This is acconlplished as follows. When Re == 1,

til = 2.13Xl0' [~1.28~~I~072 ~0.875)3J~~2~](3)

3.84X 10iWhen Iv == 1,

Re == (3.84X 10i) % (4)== 6.2 X 10::.

A straight line drawn through these two points (on Fig. 2)intersects the If) vs Re curve at a Re equal to 1 Xl 0 1Use of this value in Eq. 5 permits calculation of the ter-minal velocity for laminar flow.

Vs

== 1 295X 104 (10,000) ( 1) (5). . (0.875) ( 1.07 )

1.38 ft/sec.At the assumed injection rate of 30 bbl/min, the flow inthe casing would be turbulent. Correcting the terminalvelocity for turbulent conditions,

1Va == 1.38 1+0.875/4.125 == 1.135 ft/sec (6)

Since Vi is the sum of Va and Vi' it remains to deter-nline Vi'

Vi = 17.2(~~)= 20.6 ft/sec; (7)then,

VI == 21.735 ft/sec (1 )This velocity is used to calculate F I which is the inertialforce of the ball.

F1 = 3.52X 10-'( (1.28) ~0.875)') (21.735)'. (8)== 0.284 lb force.

This is the maximum inertial force and is not correctedfor the decrease in flow rate through the perforated inter-val. Since the assumed conditions indicate that the dragforce will be relatively large, this force will be computedprior to correcting Fl. If F D is greater than 0.284 lbforce, it will not be necessary to consider the reducedinertia.

The drag force tending to seat the ball is calculatedfrom Eqs. 13 and 14.

454,

I (0.875) (1.07) (30)Re == 6.65XI0 (50)(1)(0.375)2(0.82) (13)

3.24X 105 ;hence, 1/)==0.2 from Fig. 2. This value is used in Eq. 12.

,( (0.2) (1.07) (0.875)2(30)2)F n == (0.391) (50)2(0.375)4(0.82)2 (12)

== 1.74 lb force.Since F D is greater than FI, the ball will contact the per-foration-probably the uppermost. It will not be neces-sary to correct F I since F D is significantly larger in mag-nitude. Under these conditions, the ball sealers would have(statistically) 100 per cent efficiency.

It has been established that the balls will "hit" the per-foration. The next step is to ascertain that the ball willremain se~ted once it hits. Arbitrarily, the force tendingto unseat the ball will be calculated first. Eqs. 14 and 15are provided for this purpose.

Re = 7.72XIO:(21.7\(1.07)[(0.875),~(0.875)2(25.40) (0.375)(0 8752-O.375:!) %]1/2

180 + 3.14 .(15 )

== 17.8 X lOt (0.873) == 1.55 X 10";from Fig. 2,

If) == 0.47.Then, using this value in Eq. 14,

Fu == 5.28 X 10-:; (0.47) (1.07) (21.7) :!(0.873):!( 14)

== 0.934 lb force.The holding force tending to overcome this force is cal-culated from Eq. 17.

,,( (1.062) (30) 2F II == (1.56) ( 1.07) (0.375) - (50) :! (0.375 ) 4 ( O.82)~

(30):!)- ---- (17)(5)1 .6.38 lb force.

The holding force F II is a normal force while the F u cal-culated above is a vertical force. The vertical vector ofFJ[ which resists motion can be calculated from Eq. 18.

(0.375)FIl == (0.8752-0.3752)%(6.38) (18)

== 3.03 lb force.Since F u is less than the vertical vector of F JI , the ballwill not be dislodged from the perforation. (It should berecognized that an extreme case was selected. That is, themaximum unseating force was equated to the minimumholding force. These conditions depict a situation wherethe top perforation is sealed and the ball is exposed to thefull casing flow rate. The actual unseating force can becalculated at any perforation through use of the z/n cor-rection factor. Likewise the existing holding force can bedetermined at any time by reducing n as perforationsare sealed.) The ball would remain seated even if it werestruck by a second ball traveling down the casing. This isverified because FJ[ is greater than the sum of F u +Fl'

The selection of ball size will not be necessary since:( 1) the treatment can be performed as assumed, (2) the0.875-in. balls are standard and (3) experience has estab-lished that the 0.875-in. balls will not extrude throughthe 0.375-in. perforations.

Careful consideration of this example will reveal theimportance of the many parameters influencing a success-ful ball sealer treatment. Familiarity with these equationswill enable the engineer to design successful ball sealertreatments under virtually any conditions. ***

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