EXPONENTIAL TEMPERATURE PROFILE IMPROVES PRODUCTION TUBING ELONGATION ESTIMATE

D. Ram Babu
Oil India Ltd.
Duliajan, India

The use of an exponential form of the temperature profile in producing wells will increase the estimate of tubing elongation and make the estimate more accurate.

Tubing elongation is currently often calculated from temperature changes estimated from linear temperature profiles.

ELONGATION

Well tubing undergoes elongation or contraction depending on the pressure and temperature differences between the completion and production stages. A quantitative estimate of the tubing length change is very important for preventing release of packers, opening equalizing valves, determining the seal-assembly length of permanent packers, and minimizing rod wear in pumping wells by keeping the tubing under tension during pumping.

Four effects that influence length changes are:1 2

1. Hooke's law

2. Buckling

3. Radial pressures and fluid flow

4. Temperature.

Of the four effects, elongation because of temperature changes was observed to be significant. Elongation is commonly estimated by considering linear temperature profiles during all types of operations.3-5

Ramey6 analyzed the problem of well bore heat transfer and suggested an exponential form for temperature profiles in flowing and injection wells. Subsequent field studies7 confirmed the validity of Ramey's model.

In this article, the tubing elongation calculated from the linear temperature profile model and Ramey's model are compared. The analysis shows that Ramey's model is closer to field conditions and predicts elongations as much as two times more than the linear temperature profile model.

ELONGATION COMPARISON

Linear and exponential forms of temperature profiles are mathematically represented by Equations 1 and 2. A producing oil well's gradients are shown in Fig. 1. In both cases, the geothermal temperature, TG, is assumed to be linear and given by Equation 3.

T1 and T2 are the fluid temperatures as predicted by the two models at a distance L from the bottom of a vertical well with a total depth Z. Gr is the geothermal gradient and TI is the bottom hole temperature. The b is the flowing temperature gradient and A is the relaxation distance.7 8

If the entire tubing is assumed to be at the geothermal temperature during completion and production, the change in temperature _T experienced by an element of differential length dL, at distance L, is given by Equations 4 and 5. These equations are obtained by subtracting Equation 3 from Equations 1 and 2.

At the surface, both models should give the same flowing temperature, Equations 6 and 7. The dimensionless depth is defined in Equation 8.

If de1 and de2 are the differential elongations in the element caused by the temperature profiles represented by Equations 1 and 2, then by definition of the co-efficient of thermal expansion, a, Equations 9 and 10 can be written.

Equations 11 and 12 can be obtained by substituting Equations 4 and 5 in Equations 9 and 10, respectively, and integrating with the boundary conditions of L = 0 and e1 = e2 = 0. Note that Equation 11 is the same as that used by Lubinski, et at.,2 for calculating elongation due to temperature changes.

Equation 13 is obtained by dividing Equation 12 by Equation 11 and substituting Equation 7.

APPLICATIONS

In Fig. 2, e2/e1 is plotted as a function of Z/A using Equation 13. The ratio increases with dimensionless depth and for very large values of Z/A, e2/e1 approaches 2.

To accurately estimate tubing elongation due to temperature changes, Equation 12 should be used in all practical situations.

For an injection well having a fluid temperature approximately equal to the surface geothermal temperature, the above analysis can also be applied. In this case because the tubing temperature is lower than the geothermal gradient, the tubing contracts.

To apply this analysis, it is important to estimate the temperature profile prior to landing the tubing. During a workover operation, the temperature of the tubing continuously changes because of different operations such as circulation or running-in.

Under such circumstances, tension or slack-off applied on the tubing, based on this elongation analysis, may lead to unsatisfactory results if the temperature profile of the tubing is significantly different from the geothermal gradient.

Therefore, it is always advisable that the temperature profile of the tubing be allowed to approach thermal equilibrium with the neighboring formation. This can be done by allowing the string to sit in the well for several hours.

EXAMPLE

Assume that a well is being completed as a clean oil producer with the following information:

• Oil rate = 500 b/d

• Gas/oil ratio (GOR) = 500 scf/bbl

• Gas gravity = 0.65 (Air = 1.0)

• API = 30

• Water = nil

• Total depth = 10,000 ft

• Tubing ID = 2.441 in.

• Geothermal gradient (GT) = 0.012 F./ft

• Coefficient of thermal expansion (a) = 6.9 x 10-6 degree F.-1

The relaxation distance A can be estimated from the correlation, Equation 14.9 In the equation, the empirical constants C1-C6 are 0.0149, 0.5253, 2.9303, -0.2904, 0.2608, and 4.4146, respectively.

From Equation 14, A is estimated to be 727 ft. Then from Equation 12, e2 is calculated to be 6.7 in. or 1.8 times greater than e1 from Fig. 2.

ACKNOWLEDGMENT

The author is grateful to the management of Oil India Ltd. for allowing this article to be published.

REFERENCES

1. Lubinski, A., Althouse, W.S., and Logan, J.L., "Helical Buckling of Tubing Sealed in Packers," JPT, June 1962, pp. 655-70.

2. Lubinski, A., and Blenkarn, K.A., "Buckling of Tubing in Pumping Wells, Its Effects and Means for Controlling It," Pet. Trans. AIME, 1957, pp. 73-88.

3. Hammerlindl, D.J., "Packer-to-Tubing Forces for Intermediate Packers," JPT, March 1980, pp. 515-27.

4. Matson, B.G., Whitfield, M.A., and Dysart, G.R., "Computer Calculations of Pressure and Temperature Effects on Length of Tubular Goods During Deep Well Stimulation," JPT, April 1967, pp. 551-58.

5. Davidson, A.R., Prise, G., and French, C., "Successful High-Temperature/High-Pressure Well Testing from a Semisubmersible Drilling Rig," SPE Drilling & Completion, March 1993, pp. 7-13.

6. Ramey, H.J., "Wellbore Heat Transmission' " JPT, April 1962, pp. 427-35.

7. Curtis, M.R., and Witterholt, E.J. "Use of the Temperature Log for Determining Flow Rates in Producing Wells," SPE Paper No. 4637, 48th Annual Fall Meeting, Las Vegas, Nev., Sept. 30-Oct. 3 1973.

8. Willhite, G.P., "Over-all Heat Transfer Coefficients in Steam and Hot Water Injection Wells," JPT, May 1967, pp. 607-15.

9. Shiu, K.C., and Beggs, H. D. "Predicting Temperatures in Flowing Oil Wells," in Beggs, H.D., Gas Production Operations, OGCI, Tulsa, 1984, pp. 126.

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