Tubing Movement Calculation Montrose v1.0 03.05.2003

Montrose Engineering Department

TUBING MOVEMENT CALCULATIONS

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TUBING MOVEMENT CALCULATIONS .................................................................... 1 INTRODUCTION ........................................................................................................ 3 THE BASIC EFFECTS................................................................................................ 4 A. PISTON EFFECT.................................................................................................. 5 1. 2. 3. 4. 5.

PISTON FORCE - F1 ....................................................................................................................5 PISTON FORCE STEP-BY-STEP CALCULATION PROCEDURES............................................7 PISTON LENGTH CHANGE - ∆L1 ................................................................................................9 PISTON LENGTH CHANGE STEP-BY-STEP CALCULATION PROCEDURES .........................9 PISTON EFFECT FORMULAS.....................................................................................................9

B. BUCKLING EFFECT........................................................................................... 11 1. 2. 3.

UNDERSTANDING THE BUCKLING EFFECT ..........................................................................11 BUCKLING EFFECT STEP-BY-STEP CALCULATION PROCEDURES: ..................................13 BUCKLING EFFECT FORMULAS: .............................................................................................16

C. BALLOONING EFFECT...................................................................................... 19 1. 2. 3.

BALLOONING FORCE (F3) STEP-BY-STEP CALCULATION PROCEDURES.........................20 BALLOONING LENGTH CHANGE (∆L3) CALCULATION PROCEDURES: ..............................22 BALLOONING EFFECT FORMULAS.........................................................................................24

D. TEMPERATURE EFFECT.................................................................................. 27 1. 2. 3. 4.

UNDERSTANDING THE TEMPERATURE EFFECT .................................................................28 TEMPERATURE FORCE (F4) STEP-BY-STEP CALCULATION PROCEDURES.....................30 TEMPERATURE LENGTH CHANGE (∆L4) STEP-BY-STEP CALCULATION PROCEDURES 31 TEMPERATURE EFFECT FORMULAS: ....................................................................................32

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INTRODUCTION The most important aspect when evaluating a packer installation is the determination of the length and force changes due to varying pressures and temperatures. When the magnitude (size) and direction of these length and force changes have been calculated, this information can then be used to aid in the proper packer selection, to determine if tubing damage will occur, and to determine the proper “spacing out” procedure for the packer. This chapter deals with the effects that changing well conditions (temperatures and pressures) will have on the packer installation as it is installed.

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THE BASIC EFFECTS When either the temperature, the tubing pressure, or the annular pressure is changed in a packer installation, conditions are created which will cause the tubing string to change its length. The tubing string will either shorten or elongate. If the tubing string is not permitted to change in length (i.e., latched at the packer) forces are generated on both the packer and the wellhead because these length changes are kept from occurring. There are four different effects that create these length and force changes. All of these effects must be combined together to get the total effect for the packer installation. The four effects are: A.

Piston Effect

B.

Buckling Effect

C.

Ballooning Effect

D.

Temperature Effect

The piston effect, buckling effect, and ballooning effect result from pressure changes in the system. The temperature effect is related only to temperature change and is not affected by pressure changes. While some of the effects are related to each other, each must be calculated independently. Each effect will have a magnitude (size value) and a direction. Once each effect is known, they are combined to obtain the total effect. The decision to add or subtract when combining is based on the direction that each effect acts. The approach used to evaluate packer installation problems will depend on the type of tubing-to packer hook-up being considered. There are three different possibilities that exist. The packer may permit free motion (stung through tubing), limited motion (landed tubing), or no motion (latched tubing). If the total effect acts in the direction in which a packer will allow motion, then the packer installation is evaluated by calculating the length changes that will occur. If the packer system will not permit length changes in the direction of the total effect, then the packer installation is evaluated by calculating the force changes. Before evaluating a total packer installation, each effect must be examined individually to determine why it occurs and how its magnitude and direction are calculated.

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A.

PISTON EFFECT

The piston effect is the result of pressure changes inside the tubing string and pressure changes in the casing annulus. The pressure changes inside the tubing string act on the difference in the areas between the packer valve area (Ap) and the tubing I.D. area (Ai). Pressure changes in the annulus act on the difference in the areas between the packer valve area (Ap) and the tubing O.D. area (Ao). The result of the piston effect is a force up or down on the end of the tubing string. Because the piston effect acts only on the bottom of the tubing, it is often referred to as the end area effect. If the tubing is free to move with respect to the packer, the piston effect will result in a length change of the tubing. If the tubing is not free to move with respect to the packer, the piston effect will result in a force change on the packer.

1.

PISTON FORCE - F1

In every packer installation, forces “A” and “B” exist as shown in the Figure 1. These forces exist even before the packer is set. The result of forces “A” and “B” is called the buoyant effect of the fluid on the tubing if the packer is not set. As the tubing and packer are being run to depth in the well, forces “A” and “B” are increasing and causing the tubing to shorten. Although the tubing is being shortened by forces “A” and “B”, the packer-to-tubing relationship is not affected since the packer is not set. However, once the packer is set, the effect of a change in tubing pressure or a change in casing pressure is a change in the forces “A” and “B”. These changes in forces “A” and “B”, after the packer is set will also affect the packer-to-tubing relationship.

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Figure 1 When determining the piston force, always consider the packer-to-tubing relationship as balanced when the packer is set. Consider any changes after the packer is fixed (set) as having an effect on the packer-to-tubing relationship. Figure 2 illustrates this point by showing a packer ready to be set after a heavy fluid was displaced from the tubing with a lighter fluid. While setting the packer, a pressure is required to be held on the tubing to prevent the heavy fluid from flowing back into the tubing. Before the packer is set, there is a force “A” and a force “B” acting on the tubing, causing its length to change. These forces do not affect the packer-to-tubing relationship since the packer is still free to move up or down in the wellbore as the tubing length changes. However, once the packer has been set, any change made, such as bleeding off the tubing pressure, results in a change in the forces “A” and “B”. These changes will now affect the packer-to-tubing relationship since the packer is no longer free to move with the tubing.

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Figure 2

2.

PISTON FORCE STEP-BY-STEP CALCULATION PROCEDURES

The following step-by-step procedure is used to calculate the force change due to piston effect: ANNULUS CALCULATIONS Step 1

Calculate the total pressure in the casing annulus at the packer that existed when the packer was set or when the seals were stung in and located. (Po initial)

Step 2

Calculate the total pressure in the casing annulus at the packer that will exist for the condition being analysed. (Po final)

Step 3

Calculate the change in the total annular pressure at the packer (∆Po) by subtracting the initial total annular pressure (Po initial) found in (Step 1) from the final total annular pressure (Po final) found in (Step 2).

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Step 4

Subtract the tubing outside area (Ao) from either the packer valve area (Ap), if the packer has a valve, or the packer seal bore area (Ap), if the packer has a seal bore. If the tubing O.D. is larger than either the packer valve diameter or the packer seal bore diameter (SB), this quantity will be negative.

Step 5

Calculate the change in force “A” by multiplying the change in the total annular pressure (∆Po) found in (Step 3) times the difference in the areas found in (Step 4). BE SURE to keep the signs of the numbers correct, since they determine the direction in which the change in force “A” is acting. A negative number means that the change in force “A” is acting upward. If there was no change in the casing pressure between the initial condition and the condition being analysed, the change in force “A” is zero.

TUBING CALCULATIONS Step 6

Calculate the total pressure inside the tubing at the packer that existed when the packer was set or when the seals were stung in and located. (Pi initial)

Step 7

Calculate the total tubing pressure at the packer that will exist for the condition being analysed. (Pi final)

Step 8

Calculate the change in the total tubing pressure at the packer (∆Pi) by subtracting the initial total tubing pressure (Pi initial) found in (Step 6) from the final total tubing pressure (Pi final) found in (Step 7).

Step 9

Subtract the tubing inside area (Ai) from either the packer valve area (Ap) or the packer seal bore area (Ap). If the tubing I.D. is larger than either the packer valve diameter or the packer seal bore diameter (SB), this quantity will be negative.

Step 10

Calculate the change in force “B” by multiplying the change in tubing pressure (∆Pi) found in (Step 8) times the difference in the areas found in (Step 9). KEEP TRACK of the signs of the numbers since they determine the direction in which the change in the force “B” is acting. A positive number means the change in force “B” is acting upward and a negative number means the change in force “B” is acting downward. If there was no change in the tubing pressure between the initial condition and the condition being analysed, the change in force “B” is zero.

PISTON FORCE CALCULATIONS Step 11 - Calculate the piston force “F1” by subtracting the change in the force “B” found in (Step 10) from the change in the force “A” found in (Step 5). The piston force units will be pounds (lbs). BE SURE to keep the signs correct in order to know in which direction “F1” is acting. If “F1” is negative, it acts upward on the bottom of the tubing (tension on the packer) and if “F1” is

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positive, it acts downward on the bottom of the tubing (compression on the packer).

PISTON LENGTH CHANGE - ∆L1

3.

If the piston force “F1” (calculated in Step 11 above) acts in a direction in which the packer permits motion, it will cause a length change (∆L1) in the tubing to occur. The procedure used to calculate the length change (∆L1) is the same for both the shortening and the lengthening (elongating) of the tubing.

4.

PISTON LENGTH CHANGE STEP-BY-STEP CALCULATION PROCEDURES

The step-by-step procedure used to find the length change ∆L1 due to the piston force is: Step 1

Determine the piston force (F1). (This force is calculated in (Step 11) of the preceding step-by-step procedure).

Step 2

Divide the piston force (F1) (E = 30,000,000 psi for steel)

Step 3

Multiply the value found in (Step 2) by the length of the tubing string (L) in inches. (The length in inches equals the length in feet times 12)

Step 4

Find the tubing cross-sectional area (As) in inches2.

Step 5

Calculate the length change due to the piston effect, ∆L1, by dividing the value found in (Step 3) by the tubing cross-sectional area found in (Step 4).

5.

by

the

modulus

of

elasticity,

E:

PISTON EFFECT FORMULAS

Both of the preceding step-by-step procedures may be written in equations. They are as follows: Piston Force (F1):

[

] [

]

F1 = (A p − A o )× (Po final − Po initial ) − (A p − A i )× (Pi final − Pi initial )

Writing this in a more condensed form, it becomes:

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[

] [

]

F1 = (A p − A o )× (ÄPo ) − (A p − A i )× (ÄPi )

Piston Length Change (∆L1): ÄL 1 =

(L ) × {[(A p − A o )× (ÄPo )] − [(A p − A i )× (ÄPi )]} (E ) × (A s )

Writing this in terms of the piston force F1, it becomes: ÄL 1 =

F1 L × E As

The terms in the above equations are defined as follows: AI

= Area of the tubing I.D. (in2)

Ao

= Area of the tubing O.D. (in2)

Ap

= Packer valve area or packer seal bore area (in2)

As

= Cross-sectional area of the tubing (in2)

E

= Modulus of elasticity (30,000,000psi for steel)

F1

= Force change due to the piston effect (lbs)

L

= Length of the tubing string in inches (in)

∆L1

= Length change of the tubing string in inches due to the piston force F1 (in)

∆Pi

= Change in the total tubing pressure at the packer (psi) ∆Pi = Pi final - Pi inital

Pi final

= Total tubing pressure at the packer that will exist for the condition being analysed (psi)

Pi inital

= Total tubing pressure at the packer that existed when the packer was set or when the seals were stung in and located (psi)

∆Po

= Change in the total annular pressure at the packer (psi) ∆Po = Po final – Po inital

Po final

= Total annular pressure at the packer that will exist for the condition being analysed (psi)

Po inital

= Total annular pressure at the packer that existed when the packer was set or when the seals were stung in and located (psi)

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B.

BUCKLING EFFECT

The buckling effect is perhaps the most difficult to understand of all the effects. One reason for this difficulty may be the fact that buckling is caused by two different force distributions. They are a compressive force on the end of the tubing and a force distribution acting across the tubing wall. A compressive force acting on the end of the tubing is one of the force distributions that will cause tubing to buckle. When a compressive force is applied to a long limber tubing string, it is easy to visualize that the force will cause the tubing to buckle. An example of this is tubing that is stacked in the derrick. The stacked tubing will buckle or bow out due to its own weight. The other force distribution causing the tubing to buckle is more difficult to visualize. Tubing will also buckle due to a force distribution created by a larger pressure inside the tubing than the pressure on the outside of the tubing. The pressure inside the tubing creates a force distribution that acts on the inside area of the tubing while the outside pressure creates a force distribution that acts on the outside area of the tubing. Since the pressure inside the tubing is higher than the pressure outside the tubing, these force distributions will produce burst stresses in the tubing. The wall thickness of tubing used in the oilfield will vary along the length of the tubing. Tubing cannot be manufactured in the lengths required for use in oil wells without having wall thickness variations. Since the wall thickness is not constant, the burst stresses can not equally distribute themselves in the tubing. The unequal burst stress distribution will cause the tubing to buckle.

1.

UNDERSTANDING THE BUCKLING EFFECT

Before the procedure for calculating the buckling effect is explained, the following statements about the buckling effect must be understood. a.

Buckled tubing is tubing that is bowed from its original straight up and down condition (See Figure 3). In a buckling condition, the tubing will continue to bow out until it contacts the casing wall. When this contact is made, the tubing will begin to coil. This coiling of the tubing is referred to as “corkscrewing” the tubing. As shown in Figure 4, corkscrewed tubing is a form of buckled tubing. As long as the stresses in the tubing produced from buckling do not exceed the yield strength of the tubing, the tubing will return to its original shape when the force causing the buckling is removed. When the stresses due to buckling exceed the yield strength of the tubing, permanent corkscrewing, as shown in Figure 5 will take place. When the tubing is permanently cork-screwed, the tubing will not return to its original shape when the force causing the buckling is removed.

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Figure 3

Figure 4

Figure 5

b.

Buckling can only shorten the length of the tubing. Buckling cannot cause an increase in the tubing length. Buckling due to pressure is only capable of exerting a negligible force on a packer.

c.

Buckling due to pressure cannot occur if the final pressure outside the tubing is greater than the final pressure inside the tubing.

d.

A tubing string can buckle even if the entire string is in tension. The buckling results from the unequal stress distribution in the tubing wall (See Figure 6).

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Figure 6

Figure 7

e.

The buckling effect is most severe at the bottom of the tubing string (near the packer). It decreases in severity the farther up the hole you go (as shown in Figure 7). There is normally a point in the tubing string, above which buckling will not occur. This point is called the neutral point. Below the neutral point the tubing is buckled and above this point there is no buckling.

f.

There are many factors that affect buckling. The factors that have the most influence on the amount of buckling which will occur are: the amount of radial clearance between the tubing O.D. and the casing I.D. (r), the magnitude of the pressure differential from the tubing I.D. to the tubing O.D., and the size of the packer seal diameter (SB). These factors have a direct effect on buckling. That means, as any of these factors increase, it will cause the length change due to buckling to increase.

2.

BUCKLING EFFECT STEP-BY-STEP CALCULATION PROCEDURES:

After you understand the reasons for buckling and the statements regarding how to determine if buckling will occur, the following step-by-step procedures can be used to calculate the length change due to buckling. TUBING CALCULATIONS: Page 13 of 33 Tubing Movement Calculations Manual Rev01.doc

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Step 1

Calculate the total pressure inside the tubing at the packer that existed when the packer was set or when the seals were stung in and located (Pi initial).

Step 2

Calculate the total pressure inside the tubing at the packer that will exist for the condition being analyzed (Pi final)

Step 3

Subtract the initial total tubing pressure (Pi initial) found in (Step 1) from the final total tubing pressure (Pi final) found in (Step 2). This result is the change in the total tubing pressure (∆Pi).

ANNULUS CALCULATIONS: Step 4

Calculate the total annular pressure at the packer that existed when the packer was set or when the seals were stung in and located (Po initial)

Step 5

Calculate the total annular pressure at the packer that will exist for the condition being analyzed (Po final)

Step 6

Subtract the initial total annular pressure (Po initial) found in (Step 4) from the final total annular pressure (Po final)found in (Step 5). This result is the change in the total annular pressure (∆Po).

BUCKLING CALCULATIONS: Step 7

Subtract the change in the total annular pressure (∆Po).found in (Step 6) from the change in the total tubing pressure (∆Pi).found in (Step 3). This subtracting must result in a positive answer or buckling will not occur. That means, the length change due to buckling (∆L2) is zero and (Steps 8 through 23) can be skipped.

Step 8

Square the result found in (Step 7). (Remember to square a number, multiply it times itself).

Step 9

Calculate the area of the packer seal bore (Ap) in square inches.

Step 10

Square the packer seal bore area found in (Step 9).

Step 11

Determine the radial clearance (r) between the tubing O.D. and the casing I.D. in inches (See Figure VI-8). The radial clearance is found by subtracting the tubing O.D. from the casing I.D., then dividing the quantity found by (2.0).

r=

(Ca sin g ID − Tubing OD ) 2

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Figure 8

Step 12

Square the radial clearance found in (Step 11).

Step 13

Divide the square of the pressure change difference (Step 8) by the modulus of elasticity, (E). (E = 30,000,000 psi for steel)

Step 14

Multiply the square of the radial clearance found in (Step 12) times the square of the packer seal bore area found in (Step 10) times the value found in (Step 13).

Step 15

Calculate the moment of inertia (I) for the tubing size being used. 4 4 ð (OD ) − (ID ) I= 64

Step 16

Calculate the “Tubing Weight Factors,” W s, W i, and W o,

[

Wi

]

= Weight of the fluid displaced in the tubing per unit length (lbs/in) W I (lbs/in) = 0.0034 x (Tbg ID)2 x (Fin. Tbg. Fluid Wt. in lbs/gal)

W

= Weight of the final outside fluid displaced per unit length (lbs/in) W o (lbs/in) = 0.0034 x (Tbg OD)2 x (Fin. Ann. Fluid Wt. in lbs/gal)

W s = Weight of the tubing in lbs per inch (lbs/in) Add W s and W i together. Then, subtract W o from this quantity. This result is the adjusted weight of the tubing in pounds per inch. Adjusted Tbg Wt (lbs/in)= W s (lbs/in) + W i (lbs/in) - W o W0 (lbs/in) Step 17

Multiply the adjusted tubing weight found in (Step 16) times the moment of inertia (I) found in (Step 15) times (-8.0).

Step 18

Calculate the length change due to buckling, (∆L2) by dividing the result found in (Step 14) by the result found in (Step 17).

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The answer is the length in inches the tubing will shorten as a result of buckling. The length change due to buckling will always be a negative number. NEUTRAL POINT CALCULATIONS; Step 19

Subtract Po final (Step 5) from Pi final (Step 2), then multiply this value by Ap (Step 9).

Step 20

Calculate the neutral point by dividing the result found in (Step 19) by the result found in (Step 16). The value obtained represents the distance in inches from the packer to the neutral point. (Remember, the neutral point is the point in the tubing string below which buckling exists and above which there is no buckling.)

If the distance from the packer to the neutral point calculated in (Step 20) is larger than the entire length of the tubing string in inches (L), the length change due to buckling (∆L2) is different than what was calculated in (Step 18) and it must be corrected. The length change due to buckling is corrected by completing (Steps 21 through 23). If the distance from the packer to the neutral point is less than the entire length of the tubing string in inches (L), the length change due to buckling (∆L2) calculated in (Step 18) is correct and the (Steps 21 through 23) will be skipped. When the distance from the packer to the neutral point is larger than the length of the tubing string, the entire length of the tubing string is buckled and the corrected length change due to buckling (∆L2’) must be calculated. Step 21

Divide the length of the tubing in inches (L) by the result found in (Step 20).

Step 22

Subtract the result found in (Step 21) from (2.0).

Step 23

Calculate the corrected length change due to buckling (∆L2’) by multiplying the result found in (Step 22) times the result found in (Step 21) times the length change due to buckling (∆L2) found in (Step 18). The value obtained is in inches.

3.

BUCKLING EFFECT FORMULAS:

All the preceding step-by-step procedures for calculating the length change due to buckling may be written as the following formulas: Length Change Due to Buckling (∆L2):

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If Po final is greater than Pi final, ∆L2 = 0. That is, there is 0 inches of buckling due to pressure.

(r )2 × (A p )2 [(Pi final − Pi inital ) − (Po final − Po inital )]2 ÄL 2 = (− 8 )(E )(I)(Ws + Wi − Wo ) This may be rewritten as:

(r )2 × (A p )2 (ÄPi − ÄPo )2 ÄL 2 = (− 8 )(E )(I)(Ws + Wi − Wo ) Buckling Factor = (A p )(ÄPi − ÄPo ) (This is only used if buckling factor tables are available) Length from the Packer to the Neutral Point in Inches (n): n=

(A )× (P p

i final

− Po final )

(Ws + Wi − Wo )

If the value calculated for the length of the tubing from the packer to the neutral point (n) is greater than the total tubing length in inches (L), the corrected length change due to buckling (∆L2’) must be calculated. Corrected Length Change Due to Buckling (∆L2’): (only calculate ∆L2’ if (n) is greater than (L)) ÄL 2 ' =

(r )2 × (A p )2 (ÄPi − ÄPo )2  (L )  (L ) × 2− × (− 8 )(E )(I)(Ws + Wi − Wo )  (n)  (n)

This may be rewritten as: ÄL 2 ' = (ÄL 2 ) ×

(L ) × 2 − (L ) (n)  (n)

(The corrected length change (∆L2’) will always be less than the originally calculated length change.)

The terms in the preceding equations are defined as follows: Page 17 of 33 Tubing Movement Calculations Manual Rev01.doc

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Ap

= Packer valve area or packer seal bore area (in2)

E

= Modulus of elasticity (30,000,000psi for steel)

I

4 = Moment of inertia of the tubing (in ).

[

ð (OD ) − (ID ) I= 64 4

4

]

L

= Length of the tubing (in)

∆L2

= Length change in inches due to buckling (in)

∆L2’

= Length change in inches due to buckling when the neutral point is above the top of the tubing. (in)

n

= The distance from the packer to the neutral point in inches. It. is also called the length of buckled tubing. (in)

∆Pi

= Change in the total tubing pressure at the packer (psi) ∆Pi = Pi final - Pi inital

Pi final

= Total tubing pressure at the packer that will exist for the condition being analysed (psi)

Pi inital

= Total tubing pressure at the packer that existed when the packer was set or when the seals were stung in and located (psi)

∆Po

= Change in the total annular pressure at the packer (psi) ∆Po = Po final – Po inital

Po final

= Total annular pressure at the packer that will exist for the condition being analysed (psi)

Po inital

Total annular pressure at the packer that existed when the packer was set or when the seals were stung in and located (psi)

r

= Radial clearance between the casing I.D. and the tubing O.D. (in) r=

Wi

Ca sin g ID − Tubing OD 2

= Weight of the fluid displaced in the tubing per unit length (lbs/in) W I (lbs/in) = 0.0034 x (Tbg ID)2 x (Fin. Tbg. Fluid Wt. in lbs/gal)

Wo

= Weight of the final outside fluid displaced per unit length (lbs/in) W o (lbs/in) = 0.0034 x (Tbg OD)2 x (Fin. Ann. Fluid Wt. in lbs/gal)

Ws Note:

= Weight of the tubing in lbs per inch (lbs/in) The above calculations did not solve for a force due to buckling, since buckling due to pressure can only exert a negligible force.

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C.

BALLOONING EFFECT

The third effect which must be considered is the ballooning effect. When pressure is applied to the inside of a tubing string, the pressure differential from the inside tubing to the outside tubing creates forces which try to burst the tubing. These burst forces cause the tubing to swell as shown in Figure 9. As the tubing swells, its length becomes shorter, if it is free to move. If the tubing is held from moving it creates a tension force on the packer. If the pressure differential is reversed by applying a higher pressure on the outside of the tubing than on the inside of the tubing, forces are created which will try to collapse it (see Figure 9). As the tubing tries to collapse, its length becomes longer if it is free to move. If the tubing is restrained from moving, it creates a compression force on the packer. The shortening of the tubing due to burst forces is called ballooning. The lengthening of the tubing due to collapse forces is called reverse ballooning. The effect of ballooning is directly related to the area over which the pressure acts. Since the area outside of the tubing is larger than the inside area, the effect of reverse ballooning is slightly larger than that of ballooning.

Figure 9 Unlike the piston and buckling effects, the ballooning effect occurs throughout the entire length of the tubing. Since the ballooning effect occurs throughout the tubing string, the calculations for the ballooning are based on the changes in the average Page 19 of 33 Tubing Movement Calculations Manual Rev01.doc

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pressures on both the inside and the outside of the tubing. Because the average pressure is based on the surface pressure plus the pressure at the packer, increasing the bottom hole pressure by changing the fluid gradient would only have half the effect of making the same change by applying added surface pressure. It is possible for well conditions to affect the average pressure both inside and outside the tubing, so the ballooning and reverse ballooning effects are calculated together. Ballooning can cause either a force or a length change depending on the tubing ability to move at the packer.

1.

BALLOONING FORCE (F3) STEP-BY-STEP CALCULATION PROCEDURES

The following is a step-by-step procedure for determining the force changes (F3) due to ballooning. TUBING CALCULATIONS: Step 1

Determine the surface pressure inside the tubing that existed when the packer was set or when the seals were stung in and located. (Initial Applied Tubing Pressure)

Step 2

Calculate the total pressure inside the tubing at the packer that existed when the packer was set or when the seals were stung in and located. (Pi initial)

Step 3

Calculate the initial average pressure inside the tubing by adding the initial applied tubing pressure found in (Step 1) to the initial total tubing pressure at the packer (Pi initial) found in (Step 2) and then divide this value by 2.

Initial Avg Tbg Pr ess =

(Initial Applied Tbg Pr ess + Pi initial ) 2

Step 4

Determine the surface pressure inside the tubing that will exist for the condition being analyzed. (Final Applied Tubing Pressure).

Step 5

Calculate the total pressure inside the tubing at the packer that will exist for the condition being analyzed. (Pi final)

Step 6

-

Calculate the final average pressure inside of the tubing by adding to the final applied tubing pressure found in (Step 4) to the final total tubing pressure at the packer (Pi final) found in (Step 5), and then divide this value by 2.

Final Avg Tbg Pr ess =

(Final Applied Tbg Pr ess + Pi final ) 2

Step 7 - Calculate the change in average tubing pressure (∆Pia) by subtracting the initial average tubing pressure found in (Step 3) from the final average tubing pressure found in (Step 6). ∆Pia = (Final Avg Tbg Press) — (Initial Avg Tbg Press) Page 20 of 33 Tubing Movement Calculations Manual Rev01.doc

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ANNULUS CALCULATIONS: Step 8

Determine the surface pressure in the annulus that existed when the packer was set or when the seals were stung in and located. (Initial Applied Annular Pressure)

Step 9 - Calculate the total annular pressure at the packer that existed when the packer was set or when the seals were stung in and located. (Po initial) Step 10 - Calculate the initial average pressure in the annulus by adding the initial applied annular pressure found in (Step 8) to the initial annular pressure at the packer (Po initial) found in (Step 9) and then divide this value by 2.

Initial Avg Annular Pr ess =

(Initial Applied Annular Pr ess + Po initial ) 2

Step 11

Determine the surface pressure in the annulus that exists for the condition being analyzed. (Final Applied Annular Pressure)

Step 12

Calculate the total final annular pressure at the packer that exists for the condition being analyzed. (Po final)

Step 13 - Calculate the final average pressure in the annulus by adding the final applied annular pressure found in (Step 11) to the final total annular pressure at the packer (Po final) found in (Step 12), and then divide this value by 2.

Final Avg Annular Pr ess =

Step 14

(Final Applied Annular Pr ess + Po final ) 2

Calculate the change in average annular pressure, (∆Poa), by subtracting the initial average annular pressure found in (Step 10) from the final average annular pressure found in (Step 13).

ÄPoa = (Final Avg Annular Pr ess ) − (Init Avg Annular Pr ess ) BALLOONING EFFECT (F3) CALCULATIONS: Step 15

Find the area of the inside of the tubing. (Ai)

Step 16

Multiply the inside area of the tubing (Ai) found in (Step 15) times the change in average tubing pressure, (∆Pia), found in (Step 7).

Step 17

Find the area of the outside of the tubing. (Ao) Page 21 of 33

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Step 18

Multiply the outside area of the tubing, (Ao), found in (Step 17) times the change in average annulus pressure, (∆Poa),’ found in (Step 14).

Step 19

Subtract the results of (Step 16) from the results of (Step 18).

Step 20

Calculate the force change due to ballooning, (F3), multiplying the result of (Step 19) by (.6). If (F3) is a negative number, it will result in packer tension. If (F3) is a. positive number, it will result in packer compression.

2.

BALLOONING LENGTH CHANGE (∆ ∆L3) CALCULATION PROCEDURES:

If the tubing is held from moving, it will impose a force on the packer. If the tubing is free to move, it will change its length. (Steps 1 through 14) of the following step-by-step procedures for calculating the ballooning length change, (∆L3) are the same as (Steps 1 through 14) of the preceding step-by-step procedures for calculating the ballooning force change, (F3). The step-by-step procedures for calculating the ballooning length change, (∆L3), are as follows: TUBING CALCULATIONS: Step 1

Determine the surface pressure inside the tubing that existed when the packer was set or when the seals were stung in and located. (Initial Applied Tubing Pressure)

Step 2

Calculate the total pressure inside the tubing at the packer that existed when the packer was set or when the seals were stung in and located. (Pi initial)

Step 3

Calculate the initial average pressure inside the tubing by adding the initial applied tubing pressure found in (Step 1) to the initial total tubing pressure at the packer (Pi initial) found in (Step 2) and then divide this value by 2.

Initial Avg Tbg Pr ess =

(Initial Applied Tbg Pr ess + Pi initial ) 2

Step 4

Determine the surface pressure inside the tubing that will exist for the condition being analyzed. (Final Applied Tubing Pressure).

Step 5

Calculate the total pressure inside the tubing at the packer that will exist for the condition being analyzed. (Pi final)

Step 6

Calculate the final average pressure inside of the tubing by adding to the final applied tubing pressure found in (Step 4) to the final total tubing

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pressure at the packer (Pi final) found in (Step 5), and then divide this value by 2.

Final Avg Tbg Pr ess =

Step 7

(Final Applied Tbg Pr ess + Pi final ) 2

Calculate the change in average tubing pressure (∆Pia) by subtracting the initial average tubing pressure found in (Step 3) from the final average tubing pressure found in (Step 6).

∆Pia = (Final Avg Tbg Press) — (Initial Avg Tbg Press)

ANNULUS CALCULATIONS: Step 8

Determine the surface pressure in the annulus that existed when the packer was set or when the seals were stung in and located. (Initial Applied Annular Pressure)

Step 9

Calculate the total annular pressure at the packer that existed when the packer was set or when the seals were stung in and located. (Po initial)

Step 10

Calculate the initial average pressure in the annulus by adding the initial applied annular pressure found in (Step 8) to the initial annular pressure at the packer (Po initial) found in (Step 9) and then divide this value by 2.

Initital Avg Annular Pr ess =

(Initital Applied Annular Pr ess + Po initial ) 2

Step 11

Determine the surface pressure in the annulus that exists for the condition being analyzed. (Final Applied Annular Pressure)

Step 12

Calculate the total final annular pressure at the packer that exists for the condition being analyzed. (Po final)

Step 13

Calculate the final average pressure in the annulus by adding the final applied annular pressure found in (Step 11) to the final total annular pressure at the packer (Po final) found in (Step 12), and then divide this value by 2.

Final Avg Annular Pr ess =

(Final Applied Annular Pr ess + Po final ) 2

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Step 14

Calculate the change in average annular pressure, (∆Poa), by subtracting the initial average annular pressure found in (Step 10) from the final average annular pressure found in (Step 13).

BALLOONING EFFECT (∆L3) CALCULATIONS: Step 15 - Find the outside diameter of the tubing. Step 16 - Find the inside diameter of the tubing. Step 17 - Find the ratio, (R), of the tubing O.D. to the tubing I.D. by dividing the tubing outside diameter found in (Step 15) by the tubing inside diameter found in (Step 16). The ratio, R, will always be a number greater than 1.

R=

Tubing OD Tubing ID

Step 18

Square the ratio R.

Step 19

Multiply the change in average annular pressure (∆Poa) found in (Step 14) by R2 found in (Step 18).

Step 20

Subtract the change in average tubing pressure, (∆Pia), found in (Step 7) from the result of (Step 19).

Step 21

Subtract 1.0 from the result found in (Step 18).

Step 22

Divide the result found in (Step 20) by the result found in (Step 21).

Step 23

Calculate the length of the tubing string in inches.

Step 24

Multiply the length of the tubing in inches found in (Step 23) by (-0.2) and then divide this quantity by 10,000,000 psi.

Step 25

Calculate the length change due to ballooning in inches, (∆L3), by multiplying the result found in (Step 22) by the result found in (Step 24). If ∆L3 is a negative number, the tubing will shorten. If ∆L3 is a positive number, the tubing will elongate.

3.

BALLOONING EFFECT FORMULAS

The preceding step-by-step procedures for calculating the ballooning effects may be written as the following formulas: BALLOONING FORCE (F3):

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F3 = 0.6[(ÄPoa A o ) − (ÄPia A i )]

CHANGE IN THE TUBING LENGTH, (∆L3), DUE TO THE BALLOONING FORCE: 2  0.2 × L   R ÄPoa − ÄPia  ÄL 3 =  ×    R2 − 1 10,000,000   

This may be rewritten as: 2  0.2 × L   R ÄPoa − ÄPia  ÄL 3 =  ×   7  R2 − 1 1× 10   

The terms used in the formulas are defined as follows: Ai

= Area of the tubing ID (in2)

Ao

= Area of the tubing OD (in2)

F3

= Force change due to the ballooning effect. (lbs)

L

= Length of the tubing in inches (in)

∆L3

= Length change of the tubing string in inches due to the ballooning force. (F3) (in)

∆Pia

= Change in the average tubing pressure. (psi)

Where:  (Final Applied Tbg Pr ess + Pi final )  (Initial Applied Tbg Pr ess + Pi initial ) ÄPia =  −  2 2     Pi final

= Total tubing pressure at the packer that will exist for the condition being analysed (psi)

Pi inital

= Total tubing pressure at the packer that existed when the packer was set or when the seals were stung in and located (psi)

∆Poa

= Change in the average annular pressure (psi)

Where:

 (Final Applied Annular Pr ess + Po final )  (Initial Applied Annular Pr ess + Po initial ) ÄPoa =   − 2 2    

Po final

= Total annular pressure at the packer that will exist for the condition being analysed (psi)

Po inital

Total annular pressure at the packer that existed when the packer was set or when the seals were stung in and located (psi)

R

= Ratio of the tubing O.D. to the tubing I.D.

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R=

Tubing OD Tubing ID

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D.

TEMPERATURE EFFECT

The fourth basic effect is the temperature effect. The temperature effect is the only one of the four basic effects that is not pressure related. The length and force changes due to the temperature effect are only functions of the change in the average temperature of the tubing. When an object is heated, it will grow in size. On the other hand, if an object is cooled, it will shrink in size. (See Figure 10) A couple of examples to show the effect of changing an object’s temperature follow: When removing a pulley from a shaft, the pulley can be heated causing it to expand. This expansion (or growth) will allow easy removal of the pulley from the shaft. Another example is dipping a person’s hand in cold water so that a ring can be removed. The cold water will shrink the person’s hand allowing the ring to be removed easier. These same principles of expansion and contraction also hold true when the average tubing temperature is increased or decreased. When an average tubing temperature is decreased (by injecting cool fluids), it will either shorten in length if the tubing is free to move or it will create tension force on the packer if the tubing is restrained from moving. When the average tubing temperature is increased, (by either injecting hot fluids or producing hot fluids), it will either cause the tubing to elongate if it is free to move or it will create a compressive force on the packer if it is held from moving. In many packer installations the temperature effect will be the largest of the four basic effects.

Figure 10

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1.

UNDERSTANDING THE TEMPERATURE EFFECT

Since the temperature change occurs over the entire length of the tubing, the change in the average tubing temperature must be used to determine the magnitude and direction of the temperature effect. To find the average temperature of the tubing. string, both the surface temperature and the bottom hole temperature must be known. The average tubing temperature is found by using the following formula: Hole Temp( F )) ( ) (Surface Temp( F) + Bottom 2 o

Average Tubing Temp o F =

o

Sometimes these tubing temperatures are not known for all the different well conditions. In these instances, assumptions must be made which will allow the temperature effect to be calculated. All the assumptions made should, therefore, be conservative to prevent possible equipment failure. The following is a list of conservative assumptions that will, in the absence of more accurate data, allow the calculation for the maximum temperature effect. If more accurate data is available, it should be used instead of the following statements. a.

The initial surface temperature of the tubing can be assumed to be 70°F. This temperature is measured 15 feet to 30 feet below the wellhead and remains constant regardless of the temperature of the surrounding air. However, the actual surface temperature should be available from the oil company who drilled the well. The initial surface temperature does vary with geographical location.

b.

When the bottom hole temperature (BHT) is not known, it is assumed to be the surface temperature, (approximately 70°F, as found in the previous statement), plus 1.6°F for every 100 feet of the true vertical depth of the well. Shown in the form of an equation this calculation becomes:

( )

1.6 o F × True Vert Depth(ft ) BHT o F = Surface Temp o F +   100(ft )  

( )

( )

This would make the bottom hole temperature of a 10,000 foot non-deviated well

equal to 230°F. The “1.6” number in the above equation is the geothermal gradient. It usually has units of °F per 100 ft (°F/100 ft). The geothermal gradient varies from area to area and from field to field. The actual geothermal gradient should be available from the oil company who drilled the well. The geothermal gradient for a particular location may, however, be determined if there is a well near the location that has a known bottom hole temperature. The bottom hole temperature equation may be re-arranged as follows to find the geothermal gradient (geoth grad). Page 28 of 33 Tubing Movement Calculations Manual Rev01.doc

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( )

( )

BHT o F − Surface Temp o F  Geoth Grad o F / 100ft = 100(ft ) ×   True Vert Depth(ft )  

(

)

When working with the geothermal gradient, it is assumed the temperature from the surface to the true vertical depth will increase uniformly. c.

The tubing temperature is assumed to be the same temperature as either the fluid that surrounds it, if the well is static, or the fluid that passes through it, if there is fluid movement (injection or production).

d.

The temperature of the injected fluid that is not heated is assumed to be the same as the current air temperature at the wellsite.

e.

When injecting fluids, assume that the entire length of the tubing is cooled or heated to the temperature of the injected fluid.

f.

When producing fluids, the entire length of the tubing can be assumed to be the same temperature as the initial bottom hole temperature.

g.

In a dual packer installation, the primary string and the secondary string are treated separately. The previous temperature statements about the injection, producing, or static conditions are applied to both the primary string and secondary string separately.

By making these assumptions, the temperature effect magnitude and direction can be calculated. When accurate well data is available, it should be used instead of the assumptions presented. Two important points about the temperature effect should be kept in mind when equipment is installed downhole. The first point is that the temperature effect is not felt immediately at the packer. When pressure changes occur, their effect is felt immediately at the packer while the temperature effect can require anywhere from several. minutes to several hours to change. However, it is normally assumed that the temperature effect does occur immediately. This assumption allows the temperature effect to be added to the pressure effects so that all the effects can be considered at one time. In some situations this assumption can create problems resulting in equipment failure if it is required that the temperature effect occurs immediately in order for the installation to operate properly. The second point is that in injection situations, the temperature of the injected fluid will vary with time as a result of climatic changes. When an installation is planned where the injection temperatures will vary, the average temperature calculation must be based on the worst case of the injection temperature. Cold winter nights have been responsible for packer failures. When the temperature drops at night, the injection fluid temperature also decreases (See Statement d on page 29) causing a tension force or a tubing shrinkage which may not have been considered.

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2.

TEMPERATURE FORCE (F4) STEP-BY-STEP CALCULATION PROCEDURES

The following step-by-step procedure will show how to determine the temperature effect force change (F4).

TEMPERATURE CALCULATIONS: Step 1

Find the initial surface temperature in °F when the packer was set or when the seals were stung in and located. If this temperature is unknown, use 70°F. (See Statement a, page 28).

Step 2

Find the initial bottom hole temperature in °F when the packer was set or when the seals were stung in and located. If this temperature is unknown, use 70°F plus 1.6°F per every 100 feet of true vertical depth to the bottom of the well. (See Statement b, page 28).

Step 3

Calculate the initial average tubing temperature in °F by adding the initial surface temperature (Step 1) to the initial bottom hole temperature (Step 2), and then divide this quantity by 2. Bottom Hole Temp( F)) ( ) (Initial Surface Temp( F) + Initial 2 o

Initial Average Tubing Temp o F =

o

Step 4

Find the final surface temperature in °F of the tubing. If this temperature is unknown, use the temperature of the injected fluid for injection wells or use the initial bottom hole temperature for producing wells. (See Statement e, page 29).

Step 5

Find the final bottom hole temperature in °F of the tubing. If this temperature is unknown, use the temperature of the injected fluid for injection wells or use the initial bottom hole temperature for production wells. (See Statement e, Page 29).

Step 6

Calculate the final average tubing temperature in °F by adding the final surface temperature (Step 4) to the final bottom hole temperature (Step 5) and then divide this value by 2. Bottom Hole Temp( F)) ( ) (Final Surface Temp( F) + Final 2

Final Average Tubing Temp o F =

Step 7

o

o

Calculate the change in average tubing temperature (∆T), in °F by subtracting the initial average tubing temperature (Step 3) from the final average tubing temperature (Step 6). If the change in average temperature is a negative number, the tubing will create a tension force on the packer.

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If the change in average temperature is positive, the tubing will create a compressive force on the packer. Chg in Avg Tbg Temp (∆T) = (Fin Avg Tbg Temp) - (Int Avg Tbg Temp) FORCE CALCULATIONS: Step 8

Find the cross sectional area of the tubing (As) in square inches.

Step 9

Calculate the force change due to temperature effect, (F4), in pounds by multiplying (207) times the cross sectional area, (As), (Step 8) times the change in average temperature, (∆T), (Step 7). Remember, a negative temperature force (F4) means a tension force on the packer and a positive temperature force means a compressive force on the packer.

3.

TEMPERATURE LENGTH CHANGE (∆ ∆L4) STEP-BY-STEP CALCULATION PROCEDURES

The first seven steps for calculating the length change (∆L4) due to temperature changes are the same as the first seven steps for calculating the force change, (F4). These steps are repeated here for convenience.

TEMPERATURE CALCULATIONS: Step 1

Find the initial surface temperature in °F when the packer was set or when the seals were stung in and located. If this temperature is unknown, use 70°F. (See Statement a, page 28).

Step 2

Find the initial bottom hole temperature in °F when the packer was set or when the seals were stung in and located. If this temperature is unknown, use 70°F plus 1.6°F per every 100 feet of true vertical depth to the bottom of the well. (See Statement b, page 28).

Step 3

Calculate the initial average tubing temperature in °F by adding the initial surface temperature (Step 1) to the initial bottom hole temperature (Step 2), and then divide this quantity by 2. Bottom Hole Temp( F)) ( ) (Initial Surface Temp( F) + Initial 2 o

Initial Average Tubing Temp o F =

Step 4

o

Find the final surface temperature in °F of the tubing. If this temperature is unknown, use the temperature of the injected fluid for injection wells or use

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the initial bottom hole temperature for producing wells. (See Statement e, page 151). Step 5

Find the final bottom hole temperature in °F of the tubing. If this temperature is unknown, use the temperature of the injected fluid for injection wells or use the initial bottom hole temperature for production wells. (See Statement e, Page 151).

Step 6

Calculate the final average tubing temperature in °F by adding the final surface temperature (Step 4) to the final bottom hole temperature (Step 5) and then divide this value by 2. Bottom Hole Temp( F)) ( ) (Final Surface Temp( F) + Final 2 o

Final Average Tubing Temp o F =

Step 7

o

Calculate the change in average tubing temperature (∆T), in °F by subtracting the initial average tubing temperature (Step 3) from the final average tubing temperature (Step 6). If the change in average temperature is a negative number, the tubing will create a tension force on the packer. If the change in average temperature is positive, the tubing will create a compressive force on the packer.

Chg in Avg Tbg Temp (∆T) = (Fin Avg Tbg Temp) - Int Avg Tbg Temp) LENGTH CHANGE CALCULATIONS: Step 8

Calculate the length of the tubing in inches (L).

Step 9

Calculate the length change due to this temperature effect, (∆L4), in inches, by multiplying (.0000069) times the tubing length in inches (Step 8) times the change in average tubing temperature (Step 7). Remember, a negative temperature length change, (∆L4), means the tubing shrinks and a positive temperature length change means the tubing elongates.

4.

TEMPERATURE EFFECT FORMULAS:

The preceding step-by-step procedures can be written in the following equation form: TEMPERATURE FORCE (F4): F4 = (207) (As) (∆T)

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Change in the Tubing Length, (∆L4): (∆L4) =(L) (β) (∆T) The terms used in the formulas are defined as: As

2 = Tubing cross sectional area (in )

F4

= Force change in pounds due to the temperature effect (lbs)

L

= Length of the tubing in inches (in)

∆L4

= Length change in inches due to the temperature effect (in)

∆T

= Change in the average tubing temperature (°F) Chg in Avg Tbg Temp (∆T) = [Fin Avg Tbg Temp (°F)] - [Int Avg Tbg Temp (°F)]

Where: Bottom Hole Temp( F)) ( ) (Final Surface Temp( F) + Final 2 o

Final Average Tubing Temp o F =

o

Bottom Hole Temp( F)) ( ) (Initial Surface Temp( F) + Initial 2 o

Initial Average Tubing Temp o F =

β

= The coefficient of thermal expansion (in/in/°F) (For Steel β =0.0000069 in/in/°F)

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o