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A method for calculating tubing behavior in HPHT wells
Journal of Petroleum Science and Engineering 41 (2004) 183 – 188 www.elsevier.com/locate/petrol
A method for calculating tubing behavior in HPHT wells De-Li Gao *, Bao-Kui Gao School of Petroleum Engineering, University of Petroleum, Changping, Beijing 102249, PR China Received 6 November 2002; received in revised form 5 April 2003
Abstract During well testing and production, the helical buckling often occurs near the lower end of tubing and friction appears. When loading continues or temperature increases, the tubing deformation will develop under post buckling condition, and it is difficult to describe the axial force with analytic method, that is why numerical method is needed. In this paper, plastic incremental theory in plastic mechanics is used to calculate the axial force distribution of buckled tubing strings, according to which the final axial force and deformation are determined by both the initial state and the loading process. It is found that the axial force distribution of tubing with friction is changed significantly and is strongly dependent upon the operation steps. The friction can prevent the large loads caused by high pressure and/or high temperature from spreading so that only a small part of tubing deforms severely. This calculation method was verified by experiments and has been used in oil fields. D 2003 Elsevier B.V. All rights reserved. Keywords: Petroleum engineering; High pressure – high temperature; Oil and gas well; Tubular mechanics; Well testing safety
1. Introduction Tubular string for testing or production is commonly very long. Its top end hangs from wellhead and its lower end has a packer and many other accessories. The packer is firmly fixed with casing near the bottom-hole, as shown in Fig. 1. The axial force of tubing varies with hole depth, which is tension near the wellhead and compression near the bottom-hole so that the tubing’s upper section stays straight and its lower part may be helically buckled. The axial force may distribute linearly if there is no buckling. When buckling occurs, the
* Corresponding author. E-mail address: [email protected] (D.-L. Gao). 0920-4105/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0920-4105(03)00152-9
buckled segment will be in contact with casing and thus friction will appear. The friction can make the axial force distribution change significantly. Temperature, pressure, liquid density and fluid velocity within tubing may change with hole depth, time and operations, so that the axial force changes constantly. It is very important to predict the axial forces for the safety evaluation of high-pressure – high-temperature (HPHT) wells. A large compression load at low end can induce the tubing plastic deformation and make the packer damage. A large tension load at the top end may unpack the packer or cause the tubing to break. If the tubing failed, the whole borehole can hardly maintain its integrity and safety (Gao and Gao, 2002). In this paper, the plastic incremental theory in plastic mechanics is used to calculate the axial force
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D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
Multiply the contact force by friction factor to get the friction force. h ¼ lN
ð4Þ
When axial load, temperature, and liquid pressures inside and outside the tubing are set originally, the tubing behavior can be described as follows:
The length of helically buckled segment. The contact force and friction between casing and tubing. Distribution of axial force along the tubing axis.
2.2. Engineering problems Fig. 1. Conditions of tubular string in HPHT testing.
distribution of buckled tubular strings, and some examples are presented.
2. Basic formulas and problems 2.1. Basic formulas Set the coordinate as shown in Fig. 1. Let the position of wellhead be the origin of x coordinate axis, and assume the axial force is positive when tension. Define the virtual axial force of tubing as follows (Mitchell, 1996): Ff ðxÞ ¼ Fa ðxÞ þ po ðxÞAo pi ðxÞAi
2.2.1. Damage of testing string below wellhead Fig. 2 shows an offshore case. When anchors move, the platform floats away from its normal position, and the sub-sea safe system will cut the tubing and seal the sub-sea wellhead. Thus, annular will have high top pressure and well control liquid with high density that may destroy the casing and the packer. 2.2.2. Testing string leaking near the packer Piercing or wrong actions of down hole tools make some channels between well control liquid and formation fluid, then HPHT oil and gas go upward annular and replace well control liquid, as shown in Fig. 3. This accident can result when well
ð1Þ
According to formula (1), we can judge if the tubing is buckled. If Ff (x) < 0, then buckling will occur. The axial shortage of the tubing caused by helical buckling is dðDxÞhel ¼
Ff ðxÞr2 Dx 4EI
ð2Þ
The contact force between helically buckled tubing and casing within a axial unit length can be expressed as follows (Chen, 1987):
N¼
rFf2 ðxÞ 4EI
ð3Þ
Fig. 2. Top damage.
D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
185
3. Draw ideas of plastic mechanics into calculation of tubing behavior
Fig. 3. Bottom leaking.
control liquid infiltrates into and pollutes the formation, and a little annulus pressure drop leads to a high top pressure under wellhead and threaten the casing. It is reported that Keshen-1# was destroyed in this way. 2.3. Calculation problems If loading is simple, for example, only loading without unloading, it is easy to calculate the tubing deformation. Actually, with the change of liquid pressure, temperature and flow rate, loading and unloading take place repeatedly, and the following problems are caused:
The axial force of tubing increases and decreases alternatively so that the value and direction of friction often change, which leads to the axial force being not in monogamy with strain. The temperature may rise and the axial length of tubing may increase when fluid flows upward in tubing. However, the distance between the wellhead and the packer is fixed so that the surplus length must be balanced by increasing the axial force and the length of buckled tubing. Temperature and fluid pressure change constantly during well testing. The axial force and deformation of tubing cannot resume completely when temperature and pressure resume their original values.
According to the stress – strain relationship of plastic mechanics (Xu, 1988), values between stresses and strains are not always in monogamy when plastic deformation of tubing occurs. The strains are related not only to stresses, but also to loading history. If the deformation process is not clear, the strain cannot be identified only by the stress. So in plastic mechanics, the incremental theory has been used to study stress – strain relationship. Comparing with plastic deformation, the testing tubing deformation has many similar points as shown in Table 1. In order to calculate the testing string deformation more accurately, plastic incremental theory is adopted in the study, but some changes have been made to suit the particular case.
4. Applications of the method Let us consider an original state under which testing string is balanced and helical buckling occurs at lower segment. Assume that the following load is caused only by temperature increase. This makes the analysis simple. It is assumed that the upper end and lower end of string do not move. Analysis may carry as follows.
Table 1 Comparison of similar points Testing tubing deformation
Incremental theory of plastic mechanics
When load increase monotonically, axial compression has limit. When buckling occurs, axial force distributes nonlinearly. Deformation relates to both original state and operation procedure. Final deformation is multiform function of axial force. Maximum axial length remains unchanged. Mechanical friction affects the relation between axial force and deformation.
Stresses have yield surface. Relationship between stresses and strains are nonlinear. Strains relate to both stress states and loading history. Final strains are multiform function of stresses. Strains satisfy equations of compatibility. Material internal friction affects stress – strain relations.
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D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
4.1. Using incremental method deal with loading To the string, increase of temperature means loading. If deformation is free, axial strain can be expressed as: eT ¼ aT
ð5Þ
If the tubing is bent originally, the friction can affect the deformation, and the axial force cannot be expressed analytically so that numerical method is needed. In order to make the result more accurate, the temperature increment T is divided into some increments such as that dT = T/n, where n is an integer. The tubing deformation under each dT is calculated until T. 4.2. Axial force Consider an infinitesimal element of tubing, Dx, where x = x0, and let two ends of the element fix. The length prolonged by thermal increment (dT) can be expressed as: dðDxÞ ¼ eT Dx ¼ adT Dx
ð6Þ
d(Dx) causes additional axial compressive force ( FT), and in return, FT compresses Dx leading to a axial shortage of tubing: dðDxÞa ¼
FT Dx EAc
where if dT is the same along the whole string. Thus, we have the conclusion that, for the original balanced string, thermal incremental dT will not lead to any axial movement, and the original friction does not change. The axial force increases FT after a thermal increment is produced. 4.3. Examples Packer is sealed at depth 4000 m. The outer diameter of the tubing is 8.89 cm and the inner diameter is 69.85 cm. Friction factor is 0.2, thermal expansion coefficient is 1.2 10 5 1/jC, T is assumed the same along the whole string. 4.3.1. Example 1 Assume a 4001-m-long tubing is run into a well with the inner diameter of casing of 15.25 cm, and consider the tubing’s axial force. In the following five cases, it is calculated to show the effects of friction: l = 0, T = 0 jC, l = 0, T = 30 jC, l = 0.2, T = 0 jC, l = 0.2, Ta = 30 jC (temperature increase 30 jC after the tubing has been set), (5) l = 0.2, Tb = 30 jC (temperature increase 30 jC before the tubing is set). (1) (2) (3) (4)
ð7Þ
Another effect of FT is to make the tubing buckle more severely (when Ff (x)>0), and more axial shortage: dðDxÞb ¼
FT r 2 Dx 4EI
ð8Þ
Therefore, for the element, we have dðDxÞ dðDxÞa dðDxÞb ¼ 0 or FT ¼ adT
1 r2 þ EAc 4EI
ð9Þ
Formula (9) shows that the additional axial compressive force ( FT) will take the same value every-
Fig. 4. Axial force distribution of tubing under different cases of loading.
D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188 Table 2 Key values under different loading case Loading case
(1)
(2)
187
Table 3 Verification data (3)
(4)
(5)
Top force (kN) 293 140 240 82 166 Bottom force (kN) 490 644 305 447 309 Helical buckling length (m) 1509 2408 1820 2748 4001 100 kN spreading length (m) 4001 4001 635 420 586
The axial force distributions are shown in Fig. 4 and key values are listed in Table 2. It is clear that (1) The axial forces change linearly if l = 0, and nonlinearly if l >0. (2) Case (4) and case (5) show that the loading order affect the axial force distribution of tubing. (3) The axial forces at lower end in case (3) and case (5) tend to a limit value with the upward friction, while the axial force at lower end in case (4) breaks the limit value with the downward friction. (4) Length of the tubing in helical buckling changes greatly, shown in Table 2. The last row of Table 2 gives the distance that a compression force of 100 kN exerting at lower end can spread upward the tubing. In case (4), the spreading distance is the shortest with a local higher compression. 4.3.2. Example 2 Casing has section change at 400 m above the packer. The upper clearance is 7.5 cm, and the lower clearance is 5 cm. Initially, the tubing segment of 1000 m above the packer is buckled.
Well number
Recorded ueal (kN)
Recorded leal (kN)
Calculated leal (kN)
Relevant errors (%)
1 2 3 4 5
64 102 136 188 190
147 127 128 168 132
145.26 123.78 125.67 170.77 130.30
1.18 2.52 1.82 1.65 1.30
Six curves are obtained in the order of temperature going up in this example. If temperature drops or goes up and down crossly, the pattern of curves is different. At the point where casing diameter becomes large, the axial forces change sharply, and the higher the increase in temperature, the greater the interruption will be. Fig. 5 shows the axial force distribution under different thermal increment (T).
5. Applications Simulating tests in laboratory are conducted to verify the correctness of the calculation method. Data from experiments prove that the calculation values are precise. The calculation method is used in several wells that have normal formation pressure and temperature. In order to compare the results, two subs are fixed at each end of testing string. These two subs record the axial forces during operation. Based on recorded axial force at upper end (UAF), lower end axial force (LAF) is calculated. The relevant errors can be obtained by comparing the calculated values with the recorded values. See Table 3. After being proven by both the laboratory experiment and the field’s testing, the method is used in high-pressure – high-temperature well testing. Three HPHT wells are tested successfully.
6. Conclusions
Fig. 5. Axial force distributions along well depth under different increments of temperature.
The incremental theory in plastic mechanics is introduced to calculate the axial force distribution of buckled tubing. It is found that the axial force distribution of tubing, with friction considered, can be changed significantly and has strong dependence
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D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
upon operation steps. The friction can prevent large loads caused by high pressure and/or high temperature from spreading, and as a result, a small part of tubing deforms severely. Both the laboratory experiments and the field’s testing data have proved that the method is feasible, and successful applications in testing of three HPHT wells in China verify its practicality. Nomenclature Ai Ao Ac E I EI Fa(x) pi(x) po(x) r T
cross-sectional area computed using the inside diameter of tubing cross-sectional area computed using the outside diameter of tubing net cross-sectional area of tubing, Ac = Ao Ai Young’s modulus or elastic modulus of tubular materials moment of inertia of tubing bending stiffness of tubing axial force of tubular string fluid pressure inside the tubing fluid pressure outside the tubing radial clearance between casing and tubing increment of temperature
l a
frictional factor between tubular string and borehole coefficient of thermal expansion of tubing
Acknowledgements The financial support received from the National Natural Science Foundation of China (Grant No. 59825115 and 50234030) is gratefully acknowledged.
References Chen, Y., 1987. Post Buckling Behavior of a Circular Rod Constrained Within an Inclined Hole. MS Thesis, Rice U, Houston, 1987. Gao, D.-L., Gao, B.-K., 2002. Discussion on Safety of HPHT Oil and Gas Wells, Presentation at the Third International Symposium on Safety Science and Technology (2002 ISSST) held in Tai-An, China, Oct. 10 – 13. Mitchell, R.F., 1996. Comprehensive Analysis of Buckling With Friction, SPE DE, USA, pp. 178 – 184. September. Xu, B.Y., 1988. Plastic Mechanics. Higher Educational Publishing House, Beijing, China, pp. 54 – 92.
A method for calculating tubing behavior in HPHT wells De-Li Gao *, Bao-Kui Gao School of Petroleum Engineering, University of Petroleum, Changping, Beijing 102249, PR China Received 6 November 2002; received in revised form 5 April 2003
Abstract During well testing and production, the helical buckling often occurs near the lower end of tubing and friction appears. When loading continues or temperature increases, the tubing deformation will develop under post buckling condition, and it is difficult to describe the axial force with analytic method, that is why numerical method is needed. In this paper, plastic incremental theory in plastic mechanics is used to calculate the axial force distribution of buckled tubing strings, according to which the final axial force and deformation are determined by both the initial state and the loading process. It is found that the axial force distribution of tubing with friction is changed significantly and is strongly dependent upon the operation steps. The friction can prevent the large loads caused by high pressure and/or high temperature from spreading so that only a small part of tubing deforms severely. This calculation method was verified by experiments and has been used in oil fields. D 2003 Elsevier B.V. All rights reserved. Keywords: Petroleum engineering; High pressure – high temperature; Oil and gas well; Tubular mechanics; Well testing safety
1. Introduction Tubular string for testing or production is commonly very long. Its top end hangs from wellhead and its lower end has a packer and many other accessories. The packer is firmly fixed with casing near the bottom-hole, as shown in Fig. 1. The axial force of tubing varies with hole depth, which is tension near the wellhead and compression near the bottom-hole so that the tubing’s upper section stays straight and its lower part may be helically buckled. The axial force may distribute linearly if there is no buckling. When buckling occurs, the
* Corresponding author. E-mail address: [email protected] (D.-L. Gao). 0920-4105/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0920-4105(03)00152-9
buckled segment will be in contact with casing and thus friction will appear. The friction can make the axial force distribution change significantly. Temperature, pressure, liquid density and fluid velocity within tubing may change with hole depth, time and operations, so that the axial force changes constantly. It is very important to predict the axial forces for the safety evaluation of high-pressure – high-temperature (HPHT) wells. A large compression load at low end can induce the tubing plastic deformation and make the packer damage. A large tension load at the top end may unpack the packer or cause the tubing to break. If the tubing failed, the whole borehole can hardly maintain its integrity and safety (Gao and Gao, 2002). In this paper, the plastic incremental theory in plastic mechanics is used to calculate the axial force
184
D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
Multiply the contact force by friction factor to get the friction force. h ¼ lN
ð4Þ
When axial load, temperature, and liquid pressures inside and outside the tubing are set originally, the tubing behavior can be described as follows:
The length of helically buckled segment. The contact force and friction between casing and tubing. Distribution of axial force along the tubing axis.
2.2. Engineering problems Fig. 1. Conditions of tubular string in HPHT testing.
distribution of buckled tubular strings, and some examples are presented.
2. Basic formulas and problems 2.1. Basic formulas Set the coordinate as shown in Fig. 1. Let the position of wellhead be the origin of x coordinate axis, and assume the axial force is positive when tension. Define the virtual axial force of tubing as follows (Mitchell, 1996): Ff ðxÞ ¼ Fa ðxÞ þ po ðxÞAo pi ðxÞAi
2.2.1. Damage of testing string below wellhead Fig. 2 shows an offshore case. When anchors move, the platform floats away from its normal position, and the sub-sea safe system will cut the tubing and seal the sub-sea wellhead. Thus, annular will have high top pressure and well control liquid with high density that may destroy the casing and the packer. 2.2.2. Testing string leaking near the packer Piercing or wrong actions of down hole tools make some channels between well control liquid and formation fluid, then HPHT oil and gas go upward annular and replace well control liquid, as shown in Fig. 3. This accident can result when well
ð1Þ
According to formula (1), we can judge if the tubing is buckled. If Ff (x) < 0, then buckling will occur. The axial shortage of the tubing caused by helical buckling is dðDxÞhel ¼
Ff ðxÞr2 Dx 4EI
ð2Þ
The contact force between helically buckled tubing and casing within a axial unit length can be expressed as follows (Chen, 1987):
N¼
rFf2 ðxÞ 4EI
ð3Þ
Fig. 2. Top damage.
D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
185
3. Draw ideas of plastic mechanics into calculation of tubing behavior
Fig. 3. Bottom leaking.
control liquid infiltrates into and pollutes the formation, and a little annulus pressure drop leads to a high top pressure under wellhead and threaten the casing. It is reported that Keshen-1# was destroyed in this way. 2.3. Calculation problems If loading is simple, for example, only loading without unloading, it is easy to calculate the tubing deformation. Actually, with the change of liquid pressure, temperature and flow rate, loading and unloading take place repeatedly, and the following problems are caused:
The axial force of tubing increases and decreases alternatively so that the value and direction of friction often change, which leads to the axial force being not in monogamy with strain. The temperature may rise and the axial length of tubing may increase when fluid flows upward in tubing. However, the distance between the wellhead and the packer is fixed so that the surplus length must be balanced by increasing the axial force and the length of buckled tubing. Temperature and fluid pressure change constantly during well testing. The axial force and deformation of tubing cannot resume completely when temperature and pressure resume their original values.
According to the stress – strain relationship of plastic mechanics (Xu, 1988), values between stresses and strains are not always in monogamy when plastic deformation of tubing occurs. The strains are related not only to stresses, but also to loading history. If the deformation process is not clear, the strain cannot be identified only by the stress. So in plastic mechanics, the incremental theory has been used to study stress – strain relationship. Comparing with plastic deformation, the testing tubing deformation has many similar points as shown in Table 1. In order to calculate the testing string deformation more accurately, plastic incremental theory is adopted in the study, but some changes have been made to suit the particular case.
4. Applications of the method Let us consider an original state under which testing string is balanced and helical buckling occurs at lower segment. Assume that the following load is caused only by temperature increase. This makes the analysis simple. It is assumed that the upper end and lower end of string do not move. Analysis may carry as follows.
Table 1 Comparison of similar points Testing tubing deformation
Incremental theory of plastic mechanics
When load increase monotonically, axial compression has limit. When buckling occurs, axial force distributes nonlinearly. Deformation relates to both original state and operation procedure. Final deformation is multiform function of axial force. Maximum axial length remains unchanged. Mechanical friction affects the relation between axial force and deformation.
Stresses have yield surface. Relationship between stresses and strains are nonlinear. Strains relate to both stress states and loading history. Final strains are multiform function of stresses. Strains satisfy equations of compatibility. Material internal friction affects stress – strain relations.
186
D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
4.1. Using incremental method deal with loading To the string, increase of temperature means loading. If deformation is free, axial strain can be expressed as: eT ¼ aT
ð5Þ
If the tubing is bent originally, the friction can affect the deformation, and the axial force cannot be expressed analytically so that numerical method is needed. In order to make the result more accurate, the temperature increment T is divided into some increments such as that dT = T/n, where n is an integer. The tubing deformation under each dT is calculated until T. 4.2. Axial force Consider an infinitesimal element of tubing, Dx, where x = x0, and let two ends of the element fix. The length prolonged by thermal increment (dT) can be expressed as: dðDxÞ ¼ eT Dx ¼ adT Dx
ð6Þ
d(Dx) causes additional axial compressive force ( FT), and in return, FT compresses Dx leading to a axial shortage of tubing: dðDxÞa ¼
FT Dx EAc
where if dT is the same along the whole string. Thus, we have the conclusion that, for the original balanced string, thermal incremental dT will not lead to any axial movement, and the original friction does not change. The axial force increases FT after a thermal increment is produced. 4.3. Examples Packer is sealed at depth 4000 m. The outer diameter of the tubing is 8.89 cm and the inner diameter is 69.85 cm. Friction factor is 0.2, thermal expansion coefficient is 1.2 10 5 1/jC, T is assumed the same along the whole string. 4.3.1. Example 1 Assume a 4001-m-long tubing is run into a well with the inner diameter of casing of 15.25 cm, and consider the tubing’s axial force. In the following five cases, it is calculated to show the effects of friction: l = 0, T = 0 jC, l = 0, T = 30 jC, l = 0.2, T = 0 jC, l = 0.2, Ta = 30 jC (temperature increase 30 jC after the tubing has been set), (5) l = 0.2, Tb = 30 jC (temperature increase 30 jC before the tubing is set). (1) (2) (3) (4)
ð7Þ
Another effect of FT is to make the tubing buckle more severely (when Ff (x)>0), and more axial shortage: dðDxÞb ¼
FT r 2 Dx 4EI
ð8Þ
Therefore, for the element, we have dðDxÞ dðDxÞa dðDxÞb ¼ 0 or FT ¼ adT
1 r2 þ EAc 4EI
ð9Þ
Formula (9) shows that the additional axial compressive force ( FT) will take the same value every-
Fig. 4. Axial force distribution of tubing under different cases of loading.
D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188 Table 2 Key values under different loading case Loading case
(1)
(2)
187
Table 3 Verification data (3)
(4)
(5)
Top force (kN) 293 140 240 82 166 Bottom force (kN) 490 644 305 447 309 Helical buckling length (m) 1509 2408 1820 2748 4001 100 kN spreading length (m) 4001 4001 635 420 586
The axial force distributions are shown in Fig. 4 and key values are listed in Table 2. It is clear that (1) The axial forces change linearly if l = 0, and nonlinearly if l >0. (2) Case (4) and case (5) show that the loading order affect the axial force distribution of tubing. (3) The axial forces at lower end in case (3) and case (5) tend to a limit value with the upward friction, while the axial force at lower end in case (4) breaks the limit value with the downward friction. (4) Length of the tubing in helical buckling changes greatly, shown in Table 2. The last row of Table 2 gives the distance that a compression force of 100 kN exerting at lower end can spread upward the tubing. In case (4), the spreading distance is the shortest with a local higher compression. 4.3.2. Example 2 Casing has section change at 400 m above the packer. The upper clearance is 7.5 cm, and the lower clearance is 5 cm. Initially, the tubing segment of 1000 m above the packer is buckled.
Well number
Recorded ueal (kN)
Recorded leal (kN)
Calculated leal (kN)
Relevant errors (%)
1 2 3 4 5
64 102 136 188 190
147 127 128 168 132
145.26 123.78 125.67 170.77 130.30
1.18 2.52 1.82 1.65 1.30
Six curves are obtained in the order of temperature going up in this example. If temperature drops or goes up and down crossly, the pattern of curves is different. At the point where casing diameter becomes large, the axial forces change sharply, and the higher the increase in temperature, the greater the interruption will be. Fig. 5 shows the axial force distribution under different thermal increment (T).
5. Applications Simulating tests in laboratory are conducted to verify the correctness of the calculation method. Data from experiments prove that the calculation values are precise. The calculation method is used in several wells that have normal formation pressure and temperature. In order to compare the results, two subs are fixed at each end of testing string. These two subs record the axial forces during operation. Based on recorded axial force at upper end (UAF), lower end axial force (LAF) is calculated. The relevant errors can be obtained by comparing the calculated values with the recorded values. See Table 3. After being proven by both the laboratory experiment and the field’s testing, the method is used in high-pressure – high-temperature well testing. Three HPHT wells are tested successfully.
6. Conclusions
Fig. 5. Axial force distributions along well depth under different increments of temperature.
The incremental theory in plastic mechanics is introduced to calculate the axial force distribution of buckled tubing. It is found that the axial force distribution of tubing, with friction considered, can be changed significantly and has strong dependence
188
D.-L. Gao, B.-K. Gao / Journal of Petroleum Science and Engineering 41 (2004) 183–188
upon operation steps. The friction can prevent large loads caused by high pressure and/or high temperature from spreading, and as a result, a small part of tubing deforms severely. Both the laboratory experiments and the field’s testing data have proved that the method is feasible, and successful applications in testing of three HPHT wells in China verify its practicality. Nomenclature Ai Ao Ac E I EI Fa(x) pi(x) po(x) r T
cross-sectional area computed using the inside diameter of tubing cross-sectional area computed using the outside diameter of tubing net cross-sectional area of tubing, Ac = Ao Ai Young’s modulus or elastic modulus of tubular materials moment of inertia of tubing bending stiffness of tubing axial force of tubular string fluid pressure inside the tubing fluid pressure outside the tubing radial clearance between casing and tubing increment of temperature
l a
frictional factor between tubular string and borehole coefficient of thermal expansion of tubing
Acknowledgements The financial support received from the National Natural Science Foundation of China (Grant No. 59825115 and 50234030) is gratefully acknowledged.
References Chen, Y., 1987. Post Buckling Behavior of a Circular Rod Constrained Within an Inclined Hole. MS Thesis, Rice U, Houston, 1987. Gao, D.-L., Gao, B.-K., 2002. Discussion on Safety of HPHT Oil and Gas Wells, Presentation at the Third International Symposium on Safety Science and Technology (2002 ISSST) held in Tai-An, China, Oct. 10 – 13. Mitchell, R.F., 1996. Comprehensive Analysis of Buckling With Friction, SPE DE, USA, pp. 178 – 184. September. Xu, B.Y., 1988. Plastic Mechanics. Higher Educational Publishing House, Beijing, China, pp. 54 – 92.