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United States Patent Application 20180252056
Kind Code A1
Pabon; Jahir ;   et al. September 6, 2018

GRADUALLY TAPERED COILED TUBING IN EXTENDED REACH WELLBORES

Abstract

Maximum penetration of coiled tubing in extended reach horizontal wellbores is significantly increased by using a continuously tapered portion of coiled tubing. In particular, the tubing wall thickness is continuously tapered such that the well thickness gradually decreases toward the far end of the coiled tubing. The exterior dimension remains constant while the internal diameter gradually increases. The non-linear thickness profile of the tubing is optimized for extended reach in horizontal wellbores. In some cases, the thickness profile is optimized for providing a uniform axial strain on the tubing in the horizontal section of the well.


Inventors: Pabon; Jahir; (Newton, MA) ; Su; Tianxiang; (Somerville, MA)
Applicant:
Name City State Country Type

SCHLUMBERGER TECHNOLOGY CORPORATION

Sugar Land

TX

US
Family ID: 1000002508526
Appl. No.: 15/447629
Filed: March 2, 2017


Current U.S. Class: 1/1
Current CPC Class: E21B 19/22 20130101; E21B 17/20 20130101
International Class: E21B 17/20 20060101 E21B017/20; E21B 19/22 20060101 E21B019/22

Claims



1. A coiled tubing configured for extended reach in a horizontal portion of a well, the tubing comprising a tapered portion having a near end and a far end, and configured for deployment from a surface location into the horizontal portion of the well such that the far end is further from the surface location, the tapered portion having a tubing wall thickness that continuously and non-linearly decreases from the near end to the far end, thereby facilitating further reach into the horizontal portion of the well by the far end without buckling of the tubing when compared to non-tapered tubing, discontinuously tapered tubing and linearly tapered tubing.

2. The coiled tubing according to claim 1, wherein the tubing wall thickness of the tapered portion decreases at an exponential rate from the near end to the far end.

3. The coiled tubing according to claim 1, wherein the tubing wall thickness of the tapered portion has a thickness profile that is optimized for reach.

4. The coiled tubing according to claim 1 wherein the tubing wall thickness of the tapered portion has a thickness profile that is optimized for uniform axial strain along the tapered portion.

5. The coiled tubing according to claim 1, wherein the far end of the tapered portion is fitted to a bottom hole assembly.

6. The coiled tubing according to claim 1, further comprising a non-tapered portion joined to the near end.

7. The coiled tubing according to claim 1, wherein the tapered portion is free of welds that join tubing having different wall thicknesses at a location of a joint.

8. A method of deploying coiled tubing into a horizontal portion of a wellbore from a surface location comprising: lowering a tapered portion of the coiled tubing into the wellbore from the surface location, the tapered portion having near end and a far end, the far end being lowered into the wellbore first, the tapered portion having a tubing wall thickness that continuously and non-linearly decreases from the near end to the far end; and pushing the tapered portion into the horizontal portion of the wellbore such that the far end reaches further into the horizontal portion without buckling of the tubing when compared to non-tapered tubing, discontinuously tapered tubing and linearly tapered tubing.

9. The method according to claim 8, wherein the tubing wall thickness of the tapered portion decreases at an exponential rate from the near end to the far end.

10. The method according to claim 8, wherein the wall thickness of the tapered portion has a thickness profile that is optimized for reach.

11. The method according to claim 8, wherein the wall thickness of the tapered portion has a thickness profile that is optimized for uniform axial strain along the tapered portion.

12. The method according to claim 8, wherein the wellbore includes a vertical portion above the horizontal portion, and said pushing is performed at least in part by lowering a non-tapered portion of the coiled tubing into the vertical portion, the non-tapered portion being joined to the near end of the tapered portion.

13. The method according to claim 8, wherein the horizontal portion traverses a hydrocarbon bearing subterranean rock formation and the method further comprises performing an intervention operation on the wellbore at the location of the far end using said coiled tubing.
Description



FIELD

[0001] The subject disclosure generally relates to the field of coiled tubing in wellbores. More particularly, the subject disclosure relates to techniques for deploying coiled tubing in extended reach wellbores.

BACKGROUND

[0002] Coiled tubing has been used in many extended reach wells. Due to its inherent characteristics, coiled tubing has rather limited extended reach capability. Many wells that can be successfully drilled by the drillers cannot be properly serviced by conventional coiled tubing deployment techniques. Buckling and lockup are problems for coiled tubing operations in horizontal wells. The problems originate from the friction exerted by the wellbore/casing on the coiled tubing during run-in-hole (RIH) operations. The friction compresses the tubing, causing axial strain to develop along its length. When enough tubing is injected and the compressive strain reaches a critical level, the tubing will buckle, undergo large deformation, and eventually lockup inside the wellbore. When lockup happens, most of the pushing force applied on the surface will not be transmitted to the downhole end, and consequently the coiled tubing cannot be pushed further into the hole.

[0003] Known techniques for extending the reach of coiled tubing include pumping friction reducer into the tubing-wellbore/casing annulus, using agitator and/or tractor to shake/pull the tubing at the bottom end. Tapered coiled tubing has been designed, but it is mainly used for reducing the hanging weight of tubing in vertical wells. When tapered tubing is used for extended reach purpose, people optimize the length of different sections of the tubing with different diameters.

[0004] I. McCourt and J Kubie, "Limits on the penetration of coiled tubing in horizontal oil wells: effect of the pipe geometry," Proc. IMechE Vol. 219 Part C: J. Mechanical Engineering Science, pp 1191-1197 (2005) (hereinafter "McCourt et al.") discusses increasing the penetration of coiled tubing in horizontal oil wells using coiled tubing made by welding together several non-tapered tubing sections. Each successive tubing section has a uniform, but decreased wall thickness. The tubing is thus made up of several "steps" where non-tapered tubing sections having different wall thickness are welded together. This discontinuous tapering profile is shown to have some increased penetration properties over non-tapered coiled tubing.

[0005] K. Stanley et al, "Continuously Taped Coiled Tubing," Soc. of Petroleum Engineers inc. Paper No. 68881 (2001) (hereinafter "Stanley et al.") discusses designing and manufacturing coiled tubing having a continuously linear profile over part of the tubing length. That is, the tapered section of the tubing changes thickness in a linear fashion.

SUMMARY

[0006] This summary is provided to introduce a selection of concepts that are further described below in the detailed description. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to be used as an aid in limiting the scope of the claimed subject matter.

[0007] According to some embodiments, coiled tubing is configured for extended reach in a horizontal portion of a well. The tubing includes a tapered portion with a tubing wall thickness that continuously and non-linearly decreases from the near end to the far end of the tapered portion. The tapered portion of tubing facilitates further reach into the horizontal portion of the well without buckling of the tubing when compared to non-tapered tubing, discontinuously tapered tubing and linearly tapered tubing.

[0008] According to some embodiments, the tubing wall thickness of the tapered portion decreases at an exponential rate from the near end to the far end. The thickness profile can be optimized for reach, or in some cases for uniform axial strain. According to some embodiments, the tapered portion is free of welds that join tubing having different wall thicknesses at the location of the joint.

[0009] According to some embodiments, a method is described for deploying coiled tubing into a horizontal portion of a wellbore. The method includes lowering a tapered portion of the coiled tubing into the wellbore from a surface location. The tapered portion of the tubing has a tubing wall thickness that continuously and non-linearly decreases from the near end to the far end. The method also includes pushing the tapered portion of the tubing into the horizontal portion of the wellbore, such that the far end reaches further into the horizontal portion without buckling of the tubing when compared to non-tapered tubing, discontinuously tapered tubing and linearly tapered tubing.

[0010] As used herein the term "coiled tubing" refers to a type of tubing that is typically supplied spooled on a large reel on the surface. The term "coiled tubing" does not mean that the tubing is in a coiled form when deployed in a wellbore.

[0011] Further features and advantages of the subject disclosure will become more readily apparent from the following detailed description when taken in conjunction with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0012] The subject disclosure is further described in the detailed description which follows, in reference to the noted plurality of drawings by way of non-limiting examples of the subject disclosure, in which like reference numerals represent similar parts throughout the several views of the drawings, and wherein:

[0013] FIG. 1 is a diagram of continuously tapered coiled tubing being deployed in an extended reach portion of a wellbore, according to some embodiments;

[0014] FIG. 2 is a plot of tubing thickness profiles for coiled tubing optimized for extended reach in horizontal wells, according to some embodiments;

[0015] FIG. 3 is a plot of tubing thickness profiles for coiled tubing optimized for uniform axial stress in horizontal wells, according to some embodiments;

[0016] FIGS. 4A and 4B are diagrams showing results of two dynamic simulations in which non-tapered tubing and tubing tapered to optimize reach, according to some embodiments, are injected into a horizontal channel;

[0017] FIG. 5 is a graph plotting simulated axial displacement (movement) of the downhole end of the non-tapered and the tubing tapered according to some embodiments;

[0018] FIGS. 6A and 6B are diagrams showing results of two further dynamic simulations in which non-tapered tubing and tubing tapered to optimize reach, according to some embodiments, are injected into a horizontal channel;

[0019] FIG. 7 is a graph plotting axial displacement (movement) of the downhole end of the non-tapered and the tapered tubing with 3000 meters of tubing placed initially in the well; and

[0020] FIG. 8 is a plot comparing a linearly tapered tubing profile with an optimally tapered tubing profile, according to some embodiments.

DETAILED DESCRIPTION

[0021] The particulars shown herein are by way of example and for purposes of illustrative discussion of the examples of the subject disclosure only, and are presented in the cause of providing what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the subject disclosure. In this regard, no attempt is made to show structural details in more detail than is necessary, the description taken with the drawings making apparent to those skilled in the art how the several forms of the subject disclosure may be embodied in practice. Furthermore, like reference numbers and designations in the various drawings indicate like elements. As used herein, the terms and phrases "deviated section or portion of the well", "deviated section or portion", "horizontal section or portion of the well", and "horizontal section or portion" are used interchangeably to indicate the section of the well that departs from the vertical wellbore.

[0022] As mentioned, buckling and lockup are problems for coiled tubing operations in horizontal wells. Friction exerted by the wellbore/casing on the coiled tubing during run-in-hole (RIH) operations compresses the tubing, causing axial strain to develop along its length. It has been found that typically the compressive strain is small near the downhole end, but it increases gradually towards the surface end. When enough tubing is injected and the compressive strain reaches a critical level, the tubing will buckle, undergo large deformation, and eventually lockup inside the wellbore. When lockup happens, most of the pushing force applied on the surface will not be transmitted to the downhole end, and consequently the coiled tubing cannot be pushed further into the hole. According to some embodiments, techniques are described that (1) reduce the friction, and (2) increase the tubing's capability to resist buckling. According to some embodiments, uniform axial strain along the tubing can be achieved. According to some embodiments, the tubing thickness profile along its length is maximized to achieve maximum reach in a horizontal wellbore. According to some embodiments, tubing thickness is optimized to keep its deformation in the reversible elastic, instead of the irreversible plastic, regime.

[0023] FIG. 1 is a diagram of continuously tapered coiled tubing being deployed in an extended reach portion of a wellbore, according to some embodiments. Wellbore 130 is shown penetrating the earth from wellsite 120 at the surface through earth 100 and into subterranean rock formation 110. The wellbore 130 includes an extended horizontal portion 132 within formation 110, which according to some embodiments is a hydrocarbon-bearing rock formation. Coiled tubing 140 is being used to perform an intervention on the well. In this case the intervention is performed in the horizontal portion 132 of wellbore 130 at locations that include the far end of the horizontal section, near the well bottom 134, using bottom hole assembly (BHA) 152 that is attached to the far end of coiled tubing 140. According to some embodiments, a tapered portion of coiled tubing 150 is used within the horizontal portion 132 of wellbore 130. The tapered portion 150 has a continuously tapered tubing wall profile such that the tubing wall becomes gradually thinner and thinner near the far end (where the BHA is located). In FIG. 1, the horizontal reach distance d is shown.

[0024] According to some embodiments, the tapered portion 150 of coiled tubing 140 is designed having a thickness profile along its length that maximizes the reach of the tubing in horizontal wellbores. Reducing the thickness of the tubing while keeping its outer diameter fixed can reduce the tubing weight (per unit length) and therefore reduce dry friction. At the same time, however, the tubing's capability to resist buckling is compromised with small wall thickness. Therefore, to achieve maximum reach, the tubing thickness should be reduced to a level such that the tubing can still resist buckling everywhere along its length: F(x)=F.sub.cr. Here, F(x) is the internal compressive load along the tubing, which has been parameterized by the arc length x, with x=0 denoting the downhole end and x increases towards the surface end. Also, in the above equation, the critical buckling force F.sub.cr is given by F.sub.cr=2 {square root over (EI.rho.gA/r)}, where E, I, .rho., A, g, r are respectively the Young's modulus, second area moment of inertia, volumetric density, cross section area of the tubing, gravity constant and clearance between the tubing and the wellbore/casing. Moreover, noticing that the compression load F(x) should balance the friction along the tubing, i.e., F(x)=.intg..sub.0.sup.x .mu..rho.gA(s)ds in a quasi-static equilibrium situation, we obtain the following integral equation:

2 EI ( x ) .rho. gA ( x ) r = .intg. 0 x .mu. .rho. gA ( s ) ds ( 1 ) ##EQU00001##

[0025] If tubing thickness t is much smaller than the outer radius r.sub.0, we can Taylor expand the above equation, keeping only the terms that is on the linear order of tubing thickness, and obtain the following solution for tubing thickness profile:

t ( x ) = t 0 exp ( x - x 0 .lamda. 1 ) ( 2 ) ##EQU00002##

[0026] In the above equation, t.sub.0 is the thickness of the tubing at a reference location x=x.sub.0. The solution suggests that, for a thin wall tubing (t<<r.sub.0), the optimal thickness increases from the downhole end towards the surface end in an exponential manner, with a characteristic length constant .lamda..sub.1 given by .lamda..sub.1= {square root over (2Er.sub.0.sup.2/.mu..sup.2.rho.g.DELTA.r)}. Plugging in the typical values for coiled tubing operations, this length scale is about 15,000 feet.

[0027] The above equation for optimal thickness profile holds under the assumption that the tubing wall is thin. When the thickness increases to a level that is not negligible with respect to the tubing outer radius, this equation will be invalid. The exact solution to Eq.(1), without assuming small thickness, is:

x - x 0 .lamda. 1 / 2 = 3 t _ 2 ? 2 t _ + 2 2 2 a tanh ( t _ 2 - 2 t _ + 2 2 ) 3 t _ 0 2 ? 2 t _ 0 + 2 + 2 2 a tanh ( t _ 0 2 - 2 t _ 0 + 2 2 ) ? indicates text missing or illegible when filed ( 3 ) ##EQU00003##

Here t=t/r.sub.0 is the dimensionless tubing thickness.

[0028] FIG. 2 is a plot of tubing thickness profiles for coiled tubing optimized for extended reach in horizontal wells, according to some embodiments. In particular, the two curves show thickness profiles that have been optimized for extended reach. Curve 210 is a sample approximation valid for small thicknesses (using Eq. 2), while curve 212 is an exact solution for the optimized thickness profile (using Eq. 3). On the vertical axis, thickness t is normalized by the outer radius of the tubing r.sub.0. When this normalized thickness value is 1, the tubing is solid. On the horizontal axis, the tubing length x is normalized by the length scale .lamda..sub.1=2Er.sub.0.sup.2/.mu..sup.2.rho.g.DELTA.r. The typical value of this length scale is about 15,000 feet. As can be seen, the approximate solution matches well with the exact solution when the wall thickness is small.

[0029] In the discussion supra, the optimization of tubing thickness was calculated for extended reach purposes. According to to some embodiments, the tubing thickness can be optimized such that the entire tubing is under a uniform axial stress. For example, for a uniform stress .sigma..sub.y we can solve:

.sigma..sub.y A(x)=.intg..sub.0.sup.x .mu..rho.gA(s)ds (4)

The exact solution is an exponential relation:

2 t _ - t _ 2 2 t _ 0 - t _ 0 2 = exp ( x - x 0 .lamda. 2 ) ( 5 ) ##EQU00004##

where .lamda..sub.2=.sigma..sub.y/.mu..rho.g is the characteristic length scale. The typical value for .lamda..sub.2 is about 52,000 feet. For thin wall tubing with t<<r0, the above full solution simplifies to:

t = t 0 = exp ( x - x 0 .lamda. 2 ) ( 6 ) ##EQU00005##

[0030] FIG. 3 is a plot of tubing thickness profiles for coiled tubing optimized for uniform axial stress in horizontal wells, according to some embodiments. Curve 310 is the exact solution (using Eq. 5), while the curve 312 is the approximate solution for thin wall tubing (using Eq. 6). On the vertical axis, thickness t is normalized by the outer radius of the tubing r.sub.0. On the horizontal axis, the tubing length x is normalized by the length scale .lamda..sub.2=.sigma..sub.y/.mu..rho.g. The typical value of this length scale is about 52,000 feet. As expected, the approximate solution matches well with the exact solution when the wall thickness is small.

[0031] According to some embodiments, thickness profiles are calculated based on both extended reach and also on uniform stress purposes using the yielding stress. The profile that calls for a thicker wall can be selected. In general, the solution calling for the thicker tubing wall will be the one optimizing the reach because .lamda..sub.1<.lamda..sub.2 (the wall thickness increases faster from the downhole end to surface end for the extended reach profile).

[0032] Four dynamic simulations were performed to test if tapered tubing has better buckling resistance than non-tapered tubing. In these simulations, we inject coiled tubing into a horizontal wellbore and solve Newton's motion law to obtain the configurations of the tubing as a function of time. In particular, the tubing is modeled as Kirchhoff rod with linear elasticity properties and the channel is modeled as a frictional viscoelastic contact.

[0033] FIGS. 4A and 4B are diagrams showing results of two dynamic simulations in which non-tapered tubing and tubing tapered to optimize reach, according to some embodiments, are injected into a horizontal channel. In the first simulation, we put coiled tubing of uniform thickness (i.e. non-tapered) into the wellbore of inner diameter 6.125 inches. The outer and inner diameter of the tubing are respectively 2.375 and 1.995 inches. Therefore, thickness of the tubing is uniformly t=0.19 inches (16% of the outer radius) and clearance between the tubing and the wellbore is r=1.875 inches. Young's modulus and density of the tubing are set as E=207 GPa and .rho.=7900 kg/m.sup.3, respectively. Moreover, the friction coefficient between the tubing and the wellbore is set as .mu.=0.25. Using these parameters, we can estimate the length of the tubing that can be injected into a horizontal wellbore before the first sinusoidal buckling wave forms is L.sub.sin=2 {square root over (EI/(.rho.gA.DELTA.r))}/.mu.=1180 m=3871 feet. Also, the length of the tubing that can be injected before first buckled helix forms is L.sub.hei=2 {square root over (2EI/(.rho.gA.DELTA.r))}/.mu.=1669 m=5476 feet. In this simulation, we initially (time=0 sec) put a length of 1800 m tubing inside the wellbore and then begin to push and inject more tubing at time>0 sec. To minimize the transient effect, we perform injection by controlling the force at the injection end to target an injection speed of v.sub.target=40 feet/min. In particular, for the first 5 seconds, we linearly ramp up the pushing force from zero to exactly balance the friction force correspondingly to 1800 m tubing. After 5 seconds (time>5 sec), the pushing force is set as:

F = mg { 1 + 1 2 ( 1 ? v v target ) [ 1 ? exp ( ? time ? 5 10 ) ] } ##EQU00006## ? indicates text missing or illegible when filed ##EQU00006.2##

Here mg is the weight of the tubing currently inside the wellbore (it is increasing as more and more tubing is injected), v is the current tubing velocity at the injection end in the wellbore axial direction. The exponential function is set to avoid a sudden jump in the pushing force at time=5 seconds. As expected (since L.sub.hei<1800 m), in this simulation, the coiled tubing buckles into helices. FIG. 4A illustrates the first 200 m of the un-tapered tubing near the injection end where buckling occurs.

[0034] In a second simulation, we keep all parameters the same as the one discussed above, except the thickness profile of the tubing is set to be non-uniform. In particular, for a section of tubing L=519 m near the downhole end, we design a uniform tubing thickness of 0.06 inches (inner tubing diameter ID=2.2563 inch, outer diameter OD=2.375 inch). We expect this length of the tubing near the downhole end is short enough to withstand friction without buckling during injection. Behind this section towards the surface end, the thickness of the tubing gradually increases accordingly to Equation (3), from 5% of the tubing outer radius to 16%. In this simulation, we again initially put 1800 m of the tubing inside the wellbore and then apply a linear ramp-up pushing force to target an injection speed of V.sub.target=40 feet/min. Not like the first simulation, in this case, the tubing remains straight and does not buckle. FIG. 4B shows the 200 m of the tapered tubing near the injection end. The simulation results demonstrate that tapered tubing has a better resistance to buckling and can be injected further without lockup.

[0035] FIG. 5 is a graph plotting simulated axial displacement (movement) of the downhole end of the non-tapered and the tubing tapered, according to some embodiments. Curve 510 shows the movement of the downhole end of the uniform thickness tubing as a function of time. Curve 512 shows the movement of the end of a tapered tubing. As can be seen under the same pushing condition, the uniform tubing moves slower than the non-uniform one due to buckling. Note that in these first two simulations, the downhole ends of both tubing's are moving at all times.

[0036] In the third and fourth simulations, we put tubing with longer initial length inside the wellbore. In particular, the initial length of the tubing inside the wellbore is 3000 m instead of 1800 m. These two simulations again compare a tubing with uniform thickness with a tubing tapered to optimize reach. The thickness of the uniform tubing is the same as in the first simulation, while the thickness of the tapered tubing gradually increases from the downhole end accordingly to Equation (3), from 1% of the tubing outer radius to 16%. All other parameters are the same as the first two simulations. FIGS. 6A and 6B are diagrams showing results of third and fourth simulations. FIG. 6A shows simulation results of a non-tapered coiled tubing injected into a horizontal channel at a constant velocity of 40 feet/min with 3000 meters of tubing initially in the channel. As can be seen, the tubing buckles into helices. FIG. 6B shows simulation results of a tubing with the optimized thickness profile (using Eq. 3) injected in the channel. The tapered tubing can be injected much further than the first case without buckling. FIG. 7 is a graph plotting axial displacement (movement) of the downhole end of the non-tapered and the tubing tapered according to some embodiments in the third and fourth simulations (i.e. with 3000 meters of tubing placed initially). Curve 710 shows the movement of the downhole end of a uniform thickness tubing as a function of time. Curve 712 shows the movement of the end of a tapered tubing. The two tubing's are under the same pushing condition. As can be seen, the tapered tubing moves into the wellbore, while the uniform thickness tubing locks up and cannot be pushed forward. The simulation results demonstrate that tapered tubing has a better resistance to buckling and can be injected further without lockup.

[0037] Note that in the discussion and simulations, supra, optimized tubing profiles have been compared to non-tapered tubing. In the following discussion, optimized tubing profiles are compared to linearly tapered tubing.

[0038] We estimate the reach of a linearly tapered tubing, whose outer radius is r.sub.0 and inner radius increases from r.sub.i1 to r.sub.i2 linearly. We equate the critical buckling load with the friction-induced tubing compressive load:

2 EI .rho. gA r = .intg. 0 s .mu. .rho. gAds + F 0 . ##EQU00007##

Here F.sub.0 is the applied load at the downhole end of the tubing. For example, due to a BHA attached to the bottom end of the tubing. The above equation leads to the estimation of reach of the tubing as:

L = 1 .mu. ( r o 2 ? r i 2 2 ) ( r o 2 + r i 2 2 ) E .rho. g r r o 2 ? ( r i 2 3 ? r i 1 3 ) 3 ( r i 2 ? r i 1 ) . ? indicates text missing or illegible when filed ##EQU00008##

[0039] With this formula, we now compare the reach of a linear tapered tubing and one with optimal thickness profile. For a value of F.sub.0=6728.8N, the needed thickness of the coiled tubing at the bottom is 5% of r.sub.0. If we limit the maximum thickness to 16% of r.sub.0 at the upper end of the tubing, then the reach with the optimal thickness profile (optimized for extended reach, using eq. e) is 1281 meters. The reach with a linear thickness profile is 1141 meters. For a value of F.sub.0=1401.1 N, the needed thickness of the coiled tubing at the bottom is 1% of r.sub.0. If we again limit the maximal thickness to 16% of r.sub.0 at the upper end of the tubing, then the reach with the optimal thickness profile is 3261.7 meters. The reach with a linear thickness profile in this case is 2001.8 meters. Note that in these calculations, we have replaced the portion of constant thickness tube on the downhole end and with the effective force F.sub.0. In practice the BHA size and weight will determine the force F.sub.0 for any particular application. For many applications, such as pumping acid, the BHA can be quite minimal, and therefore even the smaller of force F.sub.0 values that were simulated, 1401.1 N would be more than adequate. As can be seen from the simulation, the lighter the BHA load, the greater the advantage of exponential, optimally tapered tubing profile has over a linearly tapered tubing profile. However, the simulations demonstrate that even for relatively heavy BHAs (e.g. coiled tubing drilling applications), the optimal tubing still will have a longer reach than the linear tapered tubing.

[0040] FIG. 8 is a graph comparing a linearly tapered tubing profile with an optimally tapered tubing profile, according to some embodiments. The graph plots tubing wall thickness against tubing length. The wellbore inner diameter was 6.125 inch. In both cases the tubing outer diameter was 2.375 inch (outer radius is 1.1875 inch), the maximum tubing thickness was 16% of outer radium and the minimum tubing thickness was 1% of outer thickness. The Young's modulus was 207 GPa and density 7900 kg/m.sup.3, which reflect values for steel tubing. As can be seen, the reach-optimized tubing profile shown in curve 812 has a distinctive exponential shape compared to the linear tubing profile shown in curve 810. As was calculated in the simulations, discussed supra, the exponentially shaped, reach-optimized tubing profile is capable of much further horizontal reach without buckling than the linear profile tubing.

[0041] The non-linear continuously tapered tubing profile techniques as described herein according to many embodiments also have advantages over a step-wise discontinuous taper profiles such as discussed in McCourt et al. For example, the continuously tapered tubing profile does not rely on welding together mismatched thicknesses. The continuously tapered profile can potentially be made with much fewer numbers of welds than the step-wise profile, since each "step" relies on an additional weld. Furthermore, an optimized continuously tapered portion of coiled tubing can offer better performance than a step-wise profile due to being closer to optimal thickness profile. Note that although the continuously tapered tubing profile, according to some embodiments, does not rely on welding together mismatched thicknesses, there may be times when it is useful to nevertheless weld sections of the tapered tubing together. In such cases, however, the thickness vs. length relationship over the entire tapered portion should still be satisfied and as a consequence the ends of tubing being welded will have the same cross-section (rather than being of mismatched thicknesses).

[0042] Although only a few examples have been described in detail above, those skilled in the art will readily appreciate that many modifications are possible in the examples without materially departing from this subject disclosure. Accordingly, all such modifications are intended to be included within the scope of this disclosure as defined in the following claims. In the claims, means-plus-function clauses are intended to cover the structures described herein as performing the recited function and not only structural equivalents, but also equivalent structures. Thus, although a nail and a screw may not be structural equivalents in that a nail employs a cylindrical surface to secure wooden parts together, whereas a screw employs a helical surface, in the environment of fastening wooden parts, a nail and a screw may be equivalent structures. It is the express intention of the applicant not to invoke 35 U.S.C. .sctn. 112, paragraph 6 for any limitations of any of the claims herein, except for those in which the claim expressly uses the words `means for` together with an associated function.

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