US 4,382,381 AGrant
Determining Stresses and Length Changes in Well Production Tubing
Issue Date:1983-05-10
•10 Claims
•6 Drawing Sheets
Abstract
In the production or stimulation of a well, the length change of a string of tubing caused by temperature and pressure is determined for an inclined well. The weight of each section of the tubing is resolved into the axial component applied to the next successive section. For each of the successive sections the buckling force is determined from the actual force and the axial component of weight. This buckling force is compared to a threshold to determine if buckling occurs. The length change of the tubing between the initial condition and the condition of fluid flow in the tubing caused by the pressure and temperature of the fluid and caused by buckling if it is present is determined. An output indicates the change in length of the tubing and the stress applied to the tubing.
Metadata
Assignee
- Mobil Oil Corporation
Inventor
- Edy Soeiinah
Application Information
Application Number:US 2974525
Filing Date:1981-08-28
Priority Date:1981-08-28
Art Unit:244
Classifications
IPC:
E21B 47024
Field of Search:
731753336415145303;304;313422
Patent Drawings (6 sheets)
Description
Background of the Invention
This invention relates to the production and stimulation of oil wells and more particularly, to a method of determining the length change of a string of tubing in an inclined well.
Gas wells and flowing oil wells are usually completed and treated through a string of tubing and a packer. Changes in temperature and pressure during stimulation and production of a well usually result in changes in tubing length, tubing stress, and packer load. These changes in tubing length and stress are quite substantial especially in deep high temperature, high pressure wells. Costly failure occurs if the stresses exceed the tubing mechanical strength, or if the seal length is inadequate to compensate for the length change. If the fluid pressure inside the tubing is much greater than that outside, the tubing may buckle helically, even if there is packer-to-tubing tension.
The forces acting on a tubing string which undergoes changes in temperature and in pressure, and a study of helical buckling is contained in "Helical Buckling of Tubing Sealed in Packers," A. Lubinski, W. S. Althouse and J. L. Logan, Petroleum Transactions June 1962, pp. 655-670. This study is extended to combination completions having varying tubing and/or casing sizes in "Movement, Forces and Stresses Associated With Combination Tubing Strings Sealed in Packers," D. J. Hammerlindl, February, 1977, J. of Pet. Tech., pp. 195-208. "Tubing Movement, Forces, and Stresses in Dual Flow Assembly Installations," Kenneth S. Durham, SPE 9265, Paper presented at the 55th Annual Fall Technical Conference of the Society of Petroleum Engineers of AIME, Dallas, Texas, Sept. 21-24, 1980, extends the study to situations involving dual flow assembly installations.
The present invention is an improvement on the techniques discussed in the foregoing prior art. More particularly, the present invention is an improvement which can be used in sharply inclined wells where buckling may or may not occur, depending on the forces which are applied to the tubing string. The presence or absence of buckling is an important component of length change. The present invention provides an improvement in the accuracy in the determination of length change because it determines whether or not buckling has occurred.
Related Applications
"Preventing Buckling In Drill String", Dellinger, Gravley and Walraven, Ser. No. 292,061, filed Aug. 11, 1981, discloses the determination of the buckling of a drill string, 20 and more particularly, discloses the criteria for developing the threshold for buckling. This is incorporated herein by reference.
Summary of the Invention
In accordance with the present invention, the length change of a string of tubing in a well caused by fluid flow through tubing during production or stimulation of the well is determined by using the inclination of successive sections of the tubing string to resolve the weight of each section into the axial component applied to the next successive segment. This axial component is combined with the actual force applied to each of the tubing segments from fluid pressure acting upon the cross-sectional area of the tubing. For each of the successive sections, the buckling force is determined from the actual force and from the axial component of weight. This buckling force is compared to a threshold to determine if there is buckling of the tubing string. The length change of the tubing between the initial condition and the condition of fluid flow in the tubing caused by pressure and temperature of the fluid and caused by buckling if it is present, is determined. An output indicating the change in the length of the tubing and the stress applied to the tubing is produced.
In accordance with another aspect of the present invention, an improvement in the determination of length change of the tubing due to radial pressure forces over that shown in the aforementioned Hammerlindl reference, is obtained by separately determining the length change caused by the ballooning effect and the length change caused by the fluid frictional drag due to flow.
The foregoing and other objects, features and advantages of the invention will be better understood from the following more detailed description and appended claims.
Short Description of the Drawings
FIG. 1 shows an inclined well with a tubing string to which the present invention is applicable;
FIGS. 2A and 2B together show a flow sheet of the present invention;
FIGS. 3A and 3B show the force and resolved weight acting on one segment of a tubing string in a vertical and an inclined well respectively;
FIG. 4 shows a well which was used in an example of the performance of the invention; and
FIG. 5 shows more details of the seal unit and receptacle of the well of FIG. 4.
Description of the Preferred Embodiment
FIG. 1 shows an inclined well having a casing 11 and a string of tubing 12 which extends through the annulus 13 at the surface of the well. A packer 14 and a seal 15 on the casing separate the formation pressure P.sub.1 from the casing pressure P.sub.0. Normally, the casing outside of the tubing is filled with casing fluid, the pressure of which at any depth is directly related to the hydrostatic head. The formation pressure P.sub.1 is known from surveys. In accordance with the present invention, it is assumed that the string of tubing is made up of a number of sections, each having an inclination .theta..sub.1, .theta..sub.2, and .theta..sub.3 and so on.
During normal production, fluids or hot gas under formation pressure enter the bottom of the string of tubing 12. During stimulation, the flow is in the opposite direction with high pressure steam, or relatively cold acid entering the string of tubing at the surface. Changes in temperature and pressure during stimulation or production of a well result in changes in tubing length, tubing stress and load on the packer 14. Changes may be substantial and may result in failure of the system. For example, if the change of length of a tubing string is greater than the length of the seal 15, the pressure seal will be lost. If the stress on the tubing string is greater than its capability to withstand stress, fracturing of the tubing will occur. In accordance with the present invention, computer 16 produces an output .DELTA.L indicating change in the length of the tubing and outputs S.sub.0 and S.sub.i representing the combined stresses on the tubing. By monitoring these outputs, failure of an operating system can be prevented. Alternatively, the present invention can be used to simulate an operating well to provide the engineer with design criteria.
Change in length of the string of tubing is caused by several factors. The formation pressure acting on the cross-sectional area of the tubing exerts a compressive force in accordance with Hooke's law. A temperature change causes a change in length of the tubing dependent upon the thermal coefficient of expansion of the tubing material. Fluid flow through the tubing causes a length change due to the frictional drag of the fluid on the walls of the tubing. It has been found that difference in pressure also induces a length change caused by ballooning (or contraction) of the diameter of the tubing. That is, high pressure inside the tubing will cause ballooning of the tubing which shortens the length; conversely, high pressure outside the tubing contracts its diameter and lengthens the tubing. Finally, a very significant change in length occurs depending upon whether or not there is buckling of the string of tubing. This is of particular concern in inclined wells to which the present invention is directed because sometimes the string of tubing buckles, and at other times it does not. The present invention determines length change of a string of tubing in an inclined well.
The invention is depicted in the flow chart of FIGS. 2A and 2B. The following nomenclature will be used in describing the invention.
A.sub.i --Area corresponding to tubing ID
A.sub.o --Area corresponding to tubing OD
A.sub.p --Area corresponding to packer-bore ID
A.sub.s --Cross-sectional area of the tubing wall
D--OD of the tubing
E--Young's modulus (for steel, E=30.times.10.sup.6 psi)
F--Force (positive if a compression)
F.sub.a --Resultant actual force at the lower end of tubing, resulting from pressures and packer restraint
F.sub.f --Resultant fictious force in presence of packer restraint
F.sub.p --Packer-to-tubing force
F.sub.fr --Fluid friction drag
I--Moment of inertia of tubing cross-section with respect to its diameter: I=.pi./64 (D.sup.4 -d.sup.4), where D is OD and d is ID
L--Length of tubing, L.sub.1 =length of Section 1, L.sub.2 =length of Section 2, etc.
.DELTA.L.sub.1 =Length change of the tubing due to Hooke's law
.DELTA.L.sub.2 --Length change of the tubing due to helical buckling
.DELTA.L.sub.3 --Length change of the tubing due to radial pressure forces
.DELTA.L.sub.4 --Length change of the tubing due to temperature change
.DELTA.L.sub.5 --Length change of the tubing due to fluid flow through the tubing
P.sub.i --Pressure inside the tubing
P.sub.o --Pressure outside the tubing
.DELTA.P.sub.o --Change in pressure outside the tubing
.DELTA.P.sub.i --Change in pressure inside the tubing
r--Tubing-to-casing radial clearance
R--Ratio OD/ID of the tubing
W--Weight per unit length, in air, same as W.sub.s ; in liquid, W is given by the equation for Wi herein.
.beta.--Coefficient of thermal expansion of the tubing material (for steel,=6.9.times.10.sup.6 /1.degree. F.)
.delta.--Pressure drop in the tubing due to flow per unit length, psi/1000 ft.
.DELTA.t--Change in average tubing temperature
.rho..sub.i --Density of liquid in the tubing
.rho..sub.o --Density of liquid in the annulus
.DELTA..rho..sub.i --Change in density of liquid in the tubing
.DELTA..rho..sub.o --Change in density of liquid in the annulus
.mu.--Poisson's ratio of the material (for steel, .mu.=0.3)
.sigma..sub.a --Normal axial stress (i.e., F/A.sub.s)
.sigma..sub.b --Bending stress at the outer fiber
S.sub.i --Combined stress at inner wall of tubing
S.sub.o --Combined stress at outer wall of tubing
.theta.--Angle of inclination
Referring now to FIGS. 2A and 2B the pressure inside the tubing P.sub.i and the pressure outside the tubing P.sub.0 form inputs as indicated by the step 20. These pressures are determined from the measured formation pressure P.sub.1, known from a survey for example, and from the measured fluid pressure beneath the annulus and the hydrostatic head of the casing fluid. As indicated at 21, the inclination of the sections of the tubing string, .theta..sub.1, .theta..sub.2, .theta..sub.3, are determined from a well survey. As indicated at 22, the weight W.sub.i of each section of tubing in the mud is determined from the weight of the tubing section in air, W.sub.s, and from the mud density under the initial condition and under the final condition, .rho..sub.0 and .rho..sub.i, respectively and from the inside and outside cross-sectional areas of the tubing, A.sub.i, A.sub.o. The weight of each section is determined in accordance with:
As indicated at 23, the actual force on the bottom of the drill string due to pressure is determined in accordance with
The actual force at the bottom of the string is equal to the inside pressure multiplied by the difference in packer bore area and the inside cross section area, minus the outside pressure multiplied by the difference in packer bore areas and the outside cross section area. To this is added the weight supported by the packer, F.sub.p, which is commonly referred to as the slack-off weight.
In order to determine the actual force applied to successive sections of the tubing string, the weight on each section must be resolved into the component acting axially along the tubing string. This step is indicated at 24. This can best be explained with reference to FIGS. 3A and 3B. Assume first that the tubing string is vertical as shown in FIG. 3A and that the section has a weight LW. (W is weight per unit length, e.g. lb per foot therefore the weight of the string is WL). The actual force applied to the bottom of the section is F.sub.a1. The force applied to the next successive section is:
On the other hand, when the tubing string is inclined as shown in FIG. 3B, the force applied to the next succeeding section will be:
After the weight of each section has been resolved into its axial components, the buckling force F.sub.f.sbsb.i for each successive section can be determined as indicated at 25. The force on each section in the presence of a restraint by the packer, has been referred to in the literature as the "fictious force". This force is
Whether or not there is buckling of each section is determined by comparing this buckling, or fictious, force to a threshold as indicated by the step 26. The threshold is a critical force F.sub.cr which is given by: ##EQU1## The manner in which this threshold is developed is more fully explained in the aforementioned Dellinger, Gravley and Walraven application.
In accordance with step 27, if the buckling force applied to a section is greater than a threshold, determination of length change due to buckling is made. This step is indicated at 28. Where buckling is present, the resultant length change in the tubing is: ##EQU2##
The length changes due to temperature and pressure are determined as indicated at 29. These length changes are: ##EQU3##
Referring now to FIG. 2B, the determination of length change due to radial pressure is divided into two steps. First, as indicated by step 30, the component caused by ballooning is determined in accordance with: ##EQU4## In the step indicated at 31, the component of length change caused by fluid frictional drag is determined from: ##EQU5##
Next, the combined stresses on the string of tubing are determined as indicated by the step 32. These stresses are based on maximum-distortion-energy theory as follows: ##EQU6##
An example of a computer program for carrying out the invention on a Control Data Corporation Computer, Model No. 750 is included in the appendix. This is but one example of programming which can be used to carry out the invention.
The operation of the invention will be better understood from its application to an actual example. The example is a dual completion well shown in FIG. 4. During the short string completion test, the well developed communication between the long string and the short string completions. When the failure occurred, the long string was full of 14.0 lb/gal CaBr.sub.2 fluid and the short string was producing 8 MMSCFD of gas with an estimated flowing bottom hole pressure at the seal of 3700 psig. The present invention was used to analyze the failure. The following inputs were provided.
1. Packer type number is 2; packers permitting limited motion. Packer bore ID is 2.812". Assume a slack off weight of 5,000 lb.
2. Assume a vertical hole. Assume the surface is at the dual hydraulic packer. The packer depth is therefore 10862-10491=371'.
3. Tubing sizes: ID-1.995", OD-2.375", Weight=4.7 #/ft., MD=371'.
4. Casing ID: Use 47 #/ft. with an ID of 8.681" for the 95/8" casing and 4" ID for the screen assembly.
a. ID=8.681", MD=10584-10491=93'
b. ID=4.00", MD=371'
5. Fluids
a. Initial condition
Casing=14 ppg
Tubing=14 ppg
b. Present condition
Casing=1.5 ppg (0.7 gravity gas @ 3700 psig and 210.degree. F.)
Tubing=14 ppg
6. Surface Pressure
a. Initial completion condition
Surface pressure for both tubing and casing (@ dual hydraulilc packer)=14.times.10491.times.0.052=7637 psig
b. Present condition
Tubing surface pressure=7637 psig
Casing surface pressure=3700-371.times.1.5.times.0.052=3671 psig
7. Temperature
a. Initial condition: 210.degree. F.
b. Present condition: 210.degree. F.
8. Fluid frictional pressure loss: assume zero. The output is shown below.
(The input and output print out are shown on the following page.) ##SPC1##
The following conclusions can be drawn from the program output:
1. The tubing only shortened by 4.4 inches. The seal unit length is 2.57', therefore the communication between the short and long string was not caused by the seal movement.
2. The section of the tubing inside the 41/2" screen assembly between 10584' and 10862' measured depths had combined stresses well below 80% of the minimum yield. No tubing failure would occur in this section. The minimum yield for N-80 tubing is 80,000 psi.
3. The combined stresses for the section of the tubing between 10491' and 10584' measured depths were well above 80% of the minimum yield. The whole section would be permanently corkscrewed, though not necessarily ruptured. Since there was a communication between the short string and long string and the communication was not caused by seal movement, this section of tubing was concluded to be ruptured or parted at its weakest point somewhere between 10491' and 10584'. The weakest point is not necessarily, though likely, at the point where the calculated combined stress is highest. Remember that the combined stress is calculated based on uniform wall thickness. The actual wall thickness might be thicker or thinner and the actual yield strength might also be higher than the minimum yield at that particular point.
When the production assembly was pulled, it was found that the 23/8" tubing was badly corkscrewed between the top of the short string GP packer and the dual hydraulic packer, and the joint of tubing directly below the dual hydraulic packer was ruptured and had parted. This agreed with the conclusions based on the program output.
The following alternatives could be used to avoid the failure:
1. Limit the pressure differential across the seal to 3,000 psi by limiting the drawdown during the completion test.
2. Upgrade the 23/8" N-80 tubing to P-110.
3. Use a string of 23/8", N-80 blast joints or a string of 23/8", N-80, 5.95 lb/ft. tubing between the dual hydraulic packer and the GP packer.
While a particular embodiment of the invention has been shown and described, various modification are within the true spirit and scope of the invention. The appended claims are, therefore, intended to cover all such modifications.
Claims
What is claimed is:
1. The method of determining the length change of a string of tubing in a vertical or deviated well caused by fluid flow through said tubing during production or stimulation of the well comprising: measuring the fluid pressure where it enters said tubing; for successive sections of said tubing, determining the actual force applied to said tubing from said fluid pressure acting upon the cross-sectional area of said tubing; measuring the inclination of said sections of said tubing; determining the weight of each section; resolving the weight of each section into the axial component applied to the next successive segment, said axial component being related to the measured inclination of the sections; for each of said successive sections, determining the buckling force from said actual force and said axial component of weight; comparing said buckling force to a threshold to determine if there is buckling of said tubing; determining the length change of said tubing between the initial condition and the condition of fluid flow in said tubing caused by the pressure and temperature of said fluid and caused by buckling if it is present as determined from the preceding step; and producing an output indicating the change in length of said tubing.
2. The method recited in claim 1 further comprising: determining the length changes of the tubing due to radial pressure forces by separately determining the length change caused by ballooning or compression of said tubing due to pressure and determining the length change caused by frictional drag.
3. The method recited in claim 1 wherein said tubing string is supported in a packer having a seal, said method further comprising: measuring the hydrostatic pressure outside of the tubing above said packer, and wherein the step of determining the actual force applied to said tubing includes determining the differential in said fluid pressure where it enters said tubing and said fluid pressure outside of said tubing.
4. The method recited in claim 3 wherein said length change is determined during production of said well, wherein said fluid pressure where it enters said tubing is the formation pressure at the bottom of said tubing string, wherein said pressure outside of said tubing string is the hydrostatic pressure of the casing fluid just above said packer, and wherein the step of determining the actual force applied to said tubing is carried out for successive sections of said tubing starting at the bottom thereof.
5. The method recited in claim 3 wherein said length change is determined during stimulation of said well, wherein said fluid pressure where it enters said tubing is the pressure of the stimulation fluid at the top of said tubing string, wherein said pressure outside of said tubing string is the hydrostatic pressure of the casing fluid just below the annulus, and wherein the step of determining the actual force applied to said tubing is carried out for successive sections of said tubing starting at the top thereof.
6. The method recited in claim 1 further comprising: determining the stress applied to each section of said tubing; and producing an output indicating said stress.
7. The method recited in claim 1 wherein the step of determining the weight of each section includes determining the buoyed weight of each section from the weight of the section in air, the density of the fluid in which the section is immersed, and the cross-sectional area of the section.
8. The method of producing an output useful in the analysis of a well, in which fluid flows through a string of tubing during production or stimulation of the well, from inputs representing the fluid pressure where it enters said tubing, the inclination of sections of said tubing and the physical parameters of said tubing comprising: for successive sections of said tubing, determining the actual force applied to said tubing from said fluid pressure acting upon the cross-sectional area of said tubing; determining the weight of each section; resolving the weight of each section into the axial component applied to the next successive segment, said axial component being related to the inclination of the sections; for each of said successive sections, determining the buckling force from said actual force and said axial component of weight; comparing said buckling force to a threshold to determine if there is buckling of said tubing; determining the length change of said tubing between the initial condition and the condition of fluid flow in said tubing caused by the pressure and temperature of said fluid and caused by buckling if it is present as determined from the preceding step; determining the stress applied to each section of said tubing; and producing an output indicating the change in length of said tubing and the stress applied to each section of said tubing.
9. The method recited in claim 8 further comprising: determining the length changes of the tubing due to radial pressure forces by separately determining the length change caused by ballooning or compression of said tubing due to pressure and determining the length change caused by frictional drag.
10. The method recited in claim 8 wherein the step of determining the weight of each section includes determining the buoyed weight of each section from the weight of the section in air, the density of the fluid in which the section is immersed, and the cross-sectional area of the section.
Non-Patent Literature (3)
- "Helical Buckling of Tubing Sealed in Packers," A. Lubinski, W. S. Althouse and J. L. Logan, Petroleum Transactions, Jun. 1962, pp. 655-670.
- "Movement, Forces, and Stresses Associated With Combination Tubing Strings Sealed in Packers," D. J. Hammerlindl, Feb. 1977, J. of Pet. Tech., pp. 195-208.
- "Tubing Movement, Forces, and Stresses in Dual Flow Assembly Installations," Kenneth S. Durham, SPE 9265, Paper presented at the 55th Annual Fall Technical Conference of the Society of Petroleum Engineers of AIME, Dallas, Texas, Sep. 21-24, 1980.