SOCIETY OF PETROLEUM ENGINEERS OF AIME Fidelity Union Building D9.llas, Tex.

PAPER NUMBER 1257 - G

THIS IS A PREPRINT --- SUBJECT TO CORRECTION

Hydraulic Losses

How to Determine How to Use By

Robert Wade Brown, Junior Member AIME, and Marion A. Whitfield, Associate Member AIME

The Western Co., Midland, Tex.

ABSTRACT

The factors which affect the treatment cost of an oil or gas well are numerous. These fac-tors, hydraulic hp, treating fluid, well e~uipment, etc., may be determined and controlled prior to the treatment. An engineering design of a treatment can reduce costs by properly using hydraulic hp, but unless the treatment is engi-neered properly too much or too little hp may be used.

Better wells will result if each productive zone is treated. Until recently, the treatment of each zone re~uired expensive bridge plugs, packers or other special well e~uipment. Ball sealer treatments can be designed which will effectively stimulate each zone and reduce com-pletion costs.

INTRODUCTION

There are many factors which must be con-sidered when acidizing or fracturing an oil or gas well - what well conditions are optimum, which treating fluids should be used, what rate of injection is best, how much hydraulic hp should be used, etc. If these factors are not considered and controlled as accurately as possi-ble, a poor well may result and completion costs may be higher than necessary.

The demand for increasing injection rates on oil and gas well treatments has necessitated the use of greater ~uantities of hydraulic hp. The utilization of increased hydraulic hp is usually accompanied by increased treatment costs. These increased treatment costs have resulted in a de-sire for more efficient treatment design -- an optimum treatment with minimum hydraulic hp. A study was made to determine how unnecessary hp losses could be determined and eliminated. As a direct result of this investigation, well treat-ments are being engineered to produce maximum re-sults for the lowest possible cost. It is the purpose of this paper to present the methods for References and illustrations at end of paper.

engineering the hydraulic hp re~uirements for any given treatment.

THEORY

Hydraulic hp [Hhp] is directly proportional to pump rate [Q] and pump discharge pressure [p] as shown in E~. 1.

Hhp = QxP 40.8 . . . [1]

The pump discharge pressure is the surface treating pressure of the formation at the pre-vailing injection rate. The discharge pressure includes then:

[1] Surface injection pressure of formation or instantaneous shut-down pressure [SIP]

[2] The friction pressure [~f] [3] Any pressure differential caused by

restricted flow-perforations, packers, etc. [~ ] p To obtain a maximum discharge rate with a

given ~uantity of hp, the discharge pressure should be at a minimum. During well treatments the minimum pump discharge pressure that can be encountered is essentially that pressure corre-sponding to the surface inject.ion pressure of the formation. This pressure is IJOssible only when the injection rate is very lmT. As the injection rate increases an additional pressure - friction pressure - is encountered. Tne pump discharge pressure is the sum of the surface injection pres-sure [which is essentially a constant for a given formation] and this friction pressure. Friction pressure is directly proportional to the s~uare of the injection rate. For treatments performed through perforated intervals another pressure may be encountered: namely, the pressure differen-tial caused by restricted flow through the per-forations. The magnitude of this pressure loss is also proportional to the injection rate although indirectly. The magnitude of friction pressure plus the corresponding injection rate must be

ade~uately balanced in order to dictate the opti-mum Hhp necessary for the given situation.

2 HYDRAULIC LOSSES - HOW TO DETERMINE - HOW TO USE 1257-G

The solution of E~. 1 re~uires that two of the three variables [Hhp, Q or p] be known or given. In many cases a particular injection rate is desired leaving the discharge pressure and re~uired Hhp to be determined. In this situation the discharge pressure is calculated and by use of E~. 1 the injection rate and discharge pres-sure are converted into Hhp. In calculating discharge pressure both the surface injection pressure of the formation and the pipe friction must be determined. Surface injection pressure is usually determined from experience but may be estimated from charts similar to Fig. 1. [Sub-tract the hydrostatic head from the bottom-hole injection pressure to obtain the surface injec-tion pressure.] In order to determine pipe friction, e~uations were derived or modified to represent oilfield conditions. These e~uations so obtained were essentially the e~uations for calculating friction pressure in tubular gOOdsl , the e~uations for calculating annular friction pressures2 , and e~uations for the calculation of pressure differentials across perforations. 3 Friction pressures and "perforation differen-tials" will be taken from curves based upon these calculations. These curves and other variations of the hp-pressure-rate relationship will be dis-cussed in conjunction with the following examples.

HOW TO DESIGN CONVENTIONAL TUBING OR CASING-TYPE TREA'lMENTS

In designing this type treatment it is neces sary for us to know what friction pressures are developed under the given conditions. Experience has indicated that the best method is to deter-mine the friction pressure at various rates of flow under identical conditions and plot the results on a graph. This allows the visual com-parison of friction pressure vs rates. Friction values for various rates, fluids, and pipe strings are depicted in Figs. 2, 3, 4, 5, 6, 7. The curves were calculated by use of the e~uationsl :

fQ2 Lp D5

NRe = 1.326x 105 ~

iJ.D

[2]

[3]

Surface irijection pressure of the formation must be determined with the pipe loaded with the frac fluid. The surface injection pressure can be determined from a previous treatment on the well in ~uestion, experience within the area, etc. If the surface injection pressure [instan-taneous shut-down pressure] is available for a fluid other than that designated for the treat-ment being designed, the pressure reading should be corrected for the difference in hydrostatic head between the two fluids. Heavier fluids, greater hydrostatic heads, provide for a lower surface treating pressure and vice versa. Fig. 8 is included to allow for the rapid determination

of different hydrostatic conditions. For example consider the following case:

Depth = 5,000 ft Treatillj Fluid

sand/gal = 27.5 API Refined oil w/2 lb

Surface Injection Pressure from previous treatment = 2,500 psi w/15 per cent HCl

Hydrostatic head 15 per cent = 2,335 [Refer to Fig. 8. 15 per cent HCl has a spe-cific gravity of approx. 1.075. Follow dotted line up graph to line marked "Fluid head only" - Pivot at this 1Utersection and read H H/ 1000 ft off left scale. Read 467 psi per 1000 ft. Multiply 467 by 5, since depth is 5000 ft and obtain 2,335.]

Hydrostatic head frac oil = 2265 [Refer to Fig. 8. On gravity scale find 27.5 API. Proceed vertically up to 27.5 API line to curve labeled 2.0 lb sand/gal. Pivot at the intersection and read H H/1000 ft off left scale. Read 453 pSi/1000 ft. Multiply the 453 psi by 5 to obtain total head or 2265 psi.

Corrected Surface Injection Pressure = 2,500 + 70 psi

= 2,570 psi

All the factors necessary for an engineered treatment are now available or known. The best method of presenting the data has been found to be by way of curves similar to Fig. 9. Fig. 9 is a graphical representation of E~. 1. Using this graph and the surface injection pressure, the treatment may now be designed. [Since for the time being we are not considering annular type or ball sealer treatments, we need only to add to the factors already known the diameter of the casing string and/or tubing string.] On the scale [Fig. 9] marked "surface treating pressure" locate 2,570 psi [at "0" injection rate] and mark. As the injection rate increases another pressure -friction pressure - becomes significant. Assume that 5-1/2, 17.0 lb, J-55 casing [maximum pressure 3700 psi] is set with a 2-in. tubing string. The 2-in. friction is determined by using Fig. 2 and the 5-1/2-in. friction loss by use of Fig. 6. First consider the 5-1/2-in. string since the faster rate is desirable provided the well can be treated below pipe maximum. Note from Fig. 6 the following:

Injection Rate - Q Q = 10 BPM Q .. 20 " Q .. 30 " Q = 40 "

Diff. Press - Friction -&f

40 pSi/1000 ft 145 psijlOOO ft 300 pSi/1000 ft 475 pSi/1000 ft

The total friction losses are then the &f/1000 ft values multiplied by 5, or

1257-G ROBERT WADE BROWN and MARION A. WHITFIELD 3

Q = 10 BPM Q 20 " = Q 30 " = Q 40 " =

&f &f &f &f

=

= =

=

200 psi 725 psi

1500 psi 2375 psi

Since the friction pressure plus the surface in-jection pressure equals the surface treating pressure under the specified conditions, the two values are added. For example:

Rate & SIP Treating Pressure 0 0 2570 2570 psi

10 200 2570 2770 psi 20 725 2570 3295 psi 30 1500 2570 4070 psi

Plot the ratEs and corresponding treating pressure on Fig. 9 and connect the points by way of a smooth curve. It may now be determined that the well can be treated under the assumed condi-tions at 3400 psi and 21.5 BPM with 2400 BliP [or 2400 x 75 = 1,800 Rhp] by following the dotted lines on Fig. 9 and l-eading the coordinates. [Originate from the point of intersection of the surface treating pressure curve and hp curve nearest to 3700 psi. Since the BHP curves are drawn for even multiples of 600 it may be neces-sary to interpolate between any two curves on occasion. In order to facilitate this operation the curve for 600 BHP is included.] By use of the finished curve, it is possible to see what additional injection rate and what pressure in-crease accompanies each additional 600 BliP pump. For example, in going from 2400 BHP to 3600 BHP the rate increase is only from 21.5 to 28 BPM or 6.5 BPM for 1200 BliP. The treating pressure in-creased from 3400 psi to 3900 psi.

If the treatment design exceeds casing pressure maximum, the design process may be re-peated using a frac fluid that produces less &fl 1000 ft or redesigned using tubing and packer methods for the treatment.

Other variations of this type desi~ proced-ure can be found in prior publications. Fig. 10 is a typical example illustrating the overall accuracy of this method.

HOW TO DESIGN ANNULAR-TYPE TREA'1MENTS

The same data is necessary for designing this type treatment as was required for conven-tional tubular conduit treatments. The only difference is that in this case annular friction losses are required and the accurate determina-tion of annular friction losses is more difficul~ A change in casing weight, tubing dimensions re-maining constant, will significantly alter the annular friction loss. Figs. 11, 12, 13, 14 de-pict annular friction losses based on the equa-tion2 :

&f 23.8 rcD! - ~)- (n! ~)2 ] -1 t: IN DlfD2 .. [ 4 ]

pQ

These equations are based on an equivalent pipe diameter equal to the hydraulic radius. [Classically the equivalent diameter is equal to four times the hydraulic radius in the standard Reynold's number vs friction factor plot5 .] The authority for using the hydraulic radius here is the agreement between the values so calculated and the experimental data available. If these curves prove to be low as additional data becomes available, the following equations, based on four times the hydraulic radius, should be used:

& = 95.176 fQ2 Lp f(D! - D~) - ( nI. - ~)2l-1 (Dl i" D2 ) L LN D1/~ J

. . . . . . [6]

. . . . . . [7] Annular type treatments are usually consid-

ered when for some reason the operator does not wish to "trip" the tubing and yet desires an in-jection rate higher than that provided by tubing alone. When this condition exists there are two choices:

1. Treat via annulus alone 2. Treat through annulus and tubing simul-

taneously.

For choice one a curve identical to that depicted by Fig. 9 is used except annular friction losses [Figs. 11, 12, 13, 14] at various rates are determined and added to the surface injection pressure. The curve so produced [Fig. 15] indi-cates the BliP required to develop a certain in-jection rate and treating pressure. The condi-tions under which the curve was drawn are given on Fig. 15.

Choice two is designed by adding a curve for the tubing to Fig. 15. This is accomplished by using the method outlined for tubular conduits. The curve so produced shows the hp required to develop additional rates by injection through the tubing. [Note that hp is available in units of 600 BliP. If the full multiple of 600 is not used then hp is definitely being wasted.] For example:

The well designed in Fig. 15 can be treated through the annulus alone with two pumps. The treating pressure would be 3450 psi at 10.5 BPM.

By adding a third pump two factors are noted:

1. If the pump is added on the tubing the total rate of injection into the formation is the 105 BPM annular rate plus approximately 3.5 BPM tubing rate or 14 BPM overall. Under these con-ditions the casing treating pressure remains at 3450 psi and the tubing treating pressure is about 5,300 psi.

4 HYDRAULIC LOSSES - HOW TO DETERMINE - HOW TO USE 1257-G

2. If the additional pump is put on the annulus, the total injection rate becomes 14.7 BPM and the treating pressure increases to about )800 psi.

Since the casing maximum was assumed to be 3,700 psi the third pump would go on the tubing. The total injection rate so obtained is five per cent less than that obtained if the third pump had been added on the annulus. In many cases this difference is even more pronounced.

TREA'lMENTS PERFOlMED USJNG BALL SEALERS

The experience and data collected through the use of the e~uations and charts previously discussed has proved to be very helpful in de-signing ball sealer treatments. Unfortunately J a large number of the ball sealer treatments in the past have been performed mainly by guess work. However, the method described below will allow the accurate determination of the number of stages of ball sealers to use and the number of balls to use in each stage. The ball sealer design techni~ue will result in each perforation being treated -- each zone receiving the desired amount of treating fluid.

For simplicity this discussion will be limited to treatments in cased wells which are void of tubing; however, the same theory is applicable for annular type treatments.

Assuming that Fig. 9 or a similar curve has been drawn, the only other pressure differential

re~uired is that pressure differential re~uired to inject a given volume of fluid through a given diameter perforation. Fig. 16 illustrates this relationship and is based on calculations ob-tained by use of the e~uation3:

p (v~ - v~) ...... [8] 149.4

Fig. 17 will be used to design the follow-ing ball sealer treatment. [Note the basic curve is identical to Fig. 9.] Working from Fig. 9 we need only to determine the additional pressure caused by perforations restriction to obtain the treating pressure curve. Assuming 3/8-in. per-forations, refer to Fig. 16 and note:

Perf. Injection Rate - Qp Qp = 1/2 BPM per perf. Qp = 1 BPM per perf. QPp = 1-1/2 BPM per perf.

Perf. Press. Diff. - N'p

N'p = 50 psi N'p = 160 psi N'p = 360 psi

These pressure differentials should then be added to the treating pressure of 3400 psi [2400 BHP], the points marked, and curves drawn through these points parallel to the base curve. [Note Fig. 17.] The points are observed at 3450 , 3560, and 3760 psi [neglecting the decrease in friction pressure caused by the decreased injection rate.]

Since these points represent a certain rate per perforation and the overall injection rate is known, the number of perforations taking fluid at a given time is known. For example: if N'p = 50 psi, the rate per perforation [Qp] is 1/2 BPM. At an overall rate of 21 BPM, 21 -:- 1/2 or 42 perforations are open. This also corresponds to 2Q where Q is the overall rate. If N' = 160 psi, the rate per perforation 1 BPM. At an overall rate of 20 BPM, 20/1 or 20 perforations are tak-ing fluid. This corresponds to Q perforations where Q is the overall rate at 3560 psi [2400 BHP]. The flow through perforation lines on Fig. 17 are related to Q in order to prevent further calculations in the event that, for some reason, the BHP available is altered during the course of a treatment. By relating the number of perfora-tions open to Q it is only necessary to find the Q at a given pressure and multiply by the correct proportionality constant.

Using Fig. 17 [identical to Fig. 9 except for perforations] and assuming there are 78, 3/8-in. perforations shot in two intervals within the same zone consider the following example:

The well has been broken down, a rate has been established and sand is on the formation. Note that the treating pressure is 3450 [Service Engineer's gauge]. Refer to Fig. 17 and observe that 3450 intersects the 2400 BHP curve on the perforation line marked 2Q. Q at this point is 21 BPM. Th~re are 42 perforations open and 42 balls should be injected [neglecting ball effi-ciency]. Suppose that after this stage of balls "hit" the pressure increases to 3550 psi. Once again refer to the design curve [Fig. 17] and note that 3550 intersects the BHP curve on the Q perforation open curve. Q is 20.5j hence, there are 21 perforations open. The second stage of balls should consist of 21 balls. After this second stage of bails "hit" the pressure in-creased to 3775. This pressure corresponds to the Q/l.5 or 2Q/3 curve. At this point Q is 19.5 and there are 13 perforations taking fluid. There are only 15 perforations that have not been sealed. Of this 15, 13 are open. The two remaining per-forations would not warrant a stage of fracj therefore, the treatment is complete at this point.

Figs. 18 and 19 are typical perforation ball sealer design treatments. The treatment of the well in Fig. 18 followed very closely the pre-treatment design expectations. As expected, no pressure increase was noted until the fourth stage of ball sealers had blocked. The pressure in-creased 150 psi, meaning that 28 or Q holes were taking fluid. The fifth stage resulted in an additional increase of 200 psi, meaning that 16 or 2Q/3 holes were taking fluid. A final stage of 16 bails was used. The results from the well indi-cated that all zones were successfully treated. The well is a top allowable producer, flowing 184 BOPD, three and one-half months after treatment.

1257-G ROBERI:' WADE BROWN and MARION A. WHITFIELD 5

Another example as shown in Fig. 19 also closely followed the pre-treatment chart design. The first pressure increase was noted following the third stage of ball sealers. Pressure in-creases and breaks of 600 psi were noted on the third, fourth, fifth and sixth stages of sealers. The well is a top allowable producer, flowing 119 BOPD three months after treatment.

A careful analysis of the treatment pres-sures will often permit a prediction that one zone may not have been treated or that two or more zones are communicating. Predictions of this type have been confirmed by testing each zone with packers. other data and variations to the design procedure can be found in prior publi-cations5 .

CONCLUSIONS

It is believed that an awareness and under-standing of the factors affecting an acidizing or fracturing treatment is essential for optimum economical well completions. The effect each of these factors, well conditions, Hhp, perforations etc., has upon the treatment can be determined. Field experience has proved that these factors can be used to give completions at less cost. The challenge to each of us is to see that serv-ice engineers and production people are trained and use all the available data for better well completions.

ACKNOWLEDGMENT

The authors greatly appreciate the assis-tance and data furnished for this paper by the various oil producing companies. Appreciation is also extended to The Western Co. for the privilege of publishing these data.

NOMENCLATURE

Hhp Q P SIP

l',Pf f L P D I-L NRe HH BHP Dl

~ l',Pp

Vp

- Hydraulic horsepower - Injection rate, BPM - Pressure, psi - Surface injection pressure or instantan-

eous shut-down pressure, psi - Friction pressure, psi - Fanning friction factor - Length or depth, ft - Specific gravity, gros/cc - Diameter, in. - Viscosity, cps - Reynold's number - Hydrostatic head - Brake horsepower - Casing diameter, in. [ID] - Tubing diameter, in. [OD] - Pressure differential across perforation,

psi - Fluid velocity through perforations,

ft/sec. - Fluid velocity in caSing, ft/sec. - Injection rate per perforation, BPM

REFERENCES

1. Brown, Robert Wade: "How to Calculate Fric-tion Losses During Well Treatments", Pet. Engr., Oct., 1956. --

2. Br'CiWIi, Robert Wade: "Annular Friction Losses Between Tubing and Casing", Pet. Engr., Aug., 1958.

3. Brown, Robert Wade, and Gilbert, Bruce: "Pressure Drop Across Perforations Can Be Computed", Pet. Engr., Sept., 1957.

4. Brown, Robert Wade, and Neill, George H.: "Hydraulic Horsepower Requirements for Well Treatments", Pet. Engr., March, 1957.

5. Perry, John H.: Chemical Engineers Handbook, McGraw-Hill Book Co., New York [1950] p. 381-382.

6. Brown, Robert Wade, and Loper, Raymond G.: "The Latest JI.dvancements in Ball Sealer Technology", Pet. Engr., June, 1959.

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