Drill string rotation normalizing device and methods of using same

Abstract

A rotation normalizing device for installation in a drill string is provided. The normalizing device includes an outer housing, an inner mandrel, a travelling piston, and a biasing element. When the tubulars above the device rotate slower or at the same speed as the tubulars below the device, the outer housing and inner mandrel rotate together and transmit torque directly through the tool. When tubulars in the drill string become stuck and store torsional potential energy and angular deflection, and then subsequently slip, the tubulars above the normalizing device may begin to accelerate and spin faster than the tubulars below the device, the outer housing and inner mandrel may begin to rotate independently, causing the travelling piston to move axially inside the tool against the at least one biasing element, storing the accelerated release of torsional kinetic energy as potential energy, thus reducing and eliminating excessive stick slip oscillations throughout the drill string.

Description

CROSS REFERENCE TO PRIOR APPLICATIONS

[0001] This application claims priority to U.S. Application No. 63 / 387,948, filed on Dec. 17, 2022. The entire contents of such prior application are incorporated herein by reference as if set forth in its entirety.FIELD OF THE DESCRIPTION

[0002] The present application relates generally to rotation normalizing devices installed in a drill string. More specifically this application relates generally to rotation normalizing devices which absorb unbalanced rotation and released torsional kinetic energy of the drill string due to stick slip events.BACKGROUND

[0003] This section provides background information to facilitate a better understanding of the various aspects of the invention. It should be understood that the statements in this section of this document are to be read in this light, and not as admissions of prior art.

[0004] In oil and gas drilling a drill bit is threadably attached to the end of a drill string comprised of a variety of sub-assemblies and tubular members. The drill bit is traditionally attached to the Bottom Hole Assembly (BHA) at the bottom of the drill string which may include a variety of downhole tools and tubulars such as a rotary steerable tool, a mud motor, Measurement-While-Drilling (MWD) tools, Logging-While-Drilling (LWD) tools, a configuration of stabilizers, reamers, jars, shocks, agitators, and multiple drill collars among other modern drilling technologies. Connected to the BHA is a configuration of drilling tubulars typically including a plurality of Heavy Weight Drill Pipe (HWDP) and drill pipe which couple the drill bit and BHA back to the surface completing the drill string and affording for surface rotation, torque, compression, and tension to be applied to the drill string and thus the drill bit in efforts to further advance the borehole. Further, the assembled drill string also functions as a conduit for pumping drilling fluid through the sufficiently hollow inside diameter to clean the borehole, provide hydraulic power to downhole tools and cool the heat generated by the drill bit and downhole friction.

[0005] Due to applied torque during drilling operations and the reactive torque from the drilled formation, the drill string always possesses some combination of torsional potential energy stored within the members making up the drill string, as well as kinetic energy due to the rotation of the drill string. It is desirable that the ratio of potential and kinetic energy of the drill string remain as consistent as possible to provide for smooth and controlled drilling conditions, particularly over short time frames. This results in less damage to the drill string, BHA and drill bit, as well as resulting in a straighter and more consistent borehole profile.

[0006] It is however common that a phenomenon known as “stick-slip” may occur to the drill string. In the broadest sense, stick-slip can be considered a two-phase event. The first phase involves an occurrence where the drill string “sticks” and becomes partially or completely stuck at an arbitrary location, whereby the local rotational speed of the drill string becomes less than the drilling rotational speed. In extreme cases the local rotational speed can become equal to zero for example in the case of a stuck drill bit, or stuck stabilizer. During this first phase the torque being input into the drill string from surface (or by tools downhole such as a mud motor) and the rotational kinetic energy already present in the drill string become converted to potential energy in the drill string above the stuck location. Once enough potential energy is stored in the drill string, the second phase may occur whereby the stuck location of the drill string breaks free and “slips”, with the stored torsional potential energy being instantaneously released as rotational kinetic energy whereby the local rotational speed near the “stick” location now becomes much higher than the drilling rotational speed. In this way, the ratio of potential energy and kinetic energy possessed by the drill string is constantly in flux.

[0007] Stick slip may occur as a single event or may induce repeating oscillatory speed changes in the rotation of the drill string, resulting in chronic and cyclical stick-slip events occurring during drilling. Due to the length and elastic nature of most of a drill string, stick-slip events are commonly not detectable at surface where the drilling rotational speed and torque remain predominantly constant, despite the localized increases and decreases in potential and kinetic energy taking place further down the borehole. An example of this phenomenon is illustrated in the example graph in FIG. 15 where the “stick” and “slip” events are shown to radically affect local tubular rotational speeds and potentially induce cyclical rotational changes, all while surface rotation remains constant. These uncontrolled fluctuations in potential and kinetic energy, and the resulting effect on rotational speed and torque downhole are known to induce a variety of unpredictable forces and loads on to the drill string, BHA, and drill bit contributing to damaged drill pipe, casing, threaded connections, downhole tools and drill bit cutting structure.

[0008] There are presently several devices that exist to address the concern of stick-slip, and they can be broadly grouped into a few categories including agitator tools, torque limiters, anti-stall tools, and shock absorbers. Agitators induce aggressive vibration into the drill string to keep the drill string in a dynamic movement state as opposed to a static state where static friction can cause the drill string to stick. This is particularly desirable in ‘slide drilling’ mode where rotation of the drill string is not permitted, and differential sticking is of great concern. While agitators have enabled longer wells to be drilled, stick-slip events are not eliminated but are simply held off until the well is drilled deeper than would have been possible without the agitator. In addition, agitator tools are known in the art themselves to be violent and potentially destructive to wellbore casing and sensitive downhole tools.

[0009] Torque limiter devices, for example the inventions taught in U.S. Pat. Nos. 6,594,881, 8,852,004, and US2013 / 0056223, generally aim to prevent excessive torque from being transmitted down the drill string and function such that once a predetermined torque threshold is achieved, the device prevents further torque from being transmitted by means of ‘slipping’ in some fashion thus preventing over-torque and over-rotation. However, by design this type of device prevents over-torque from being utilized desirably in case of emergency situations (for example, a completely stuck drill string) or temporary need for drilling through a particularly high torque formation. Additionally, many designs feature torque limiting behavior in a bi-directional capacity which may prevent needed application of reverse rotation when desirable, such as releasing from safety joints, downhole tools, or threaded connections in case of a permanently stuck drill string needing temporary or permanent abandonment. Further, torque limiters do not generally prevent or limit rotational speed fluctuations thus stick-slip may only be mitigated if the local torque value is exceeded while high rotational speed fluctuations remain problematic.

[0010] Anti-stall devices such as those disclosed in U.S. Pat. Nos. 7,654,344, 7,578,360, 10,443,321, U.S. Pat. No. 10,533,376, aim to prevent torque from being stored in the drill string when the drill bit becomes stuck in the formation while drilling. These devices generally have the capability to telescope or otherwise alter their length in response to the drill bit becoming stuck and over-torqued. This application of stalled torque acts to contract the device and pull the drill bit from the formation to free its rotation, ideally preventing the drill string from being torqued up. It can be debatable to what degree this action assists the drill bit performance as it is known in the art that it is not necessarily desirable to engage and disengage brittle rigid Polycrystalline Diamond Compact (PDC) cutting structures from a workpiece or formation. For example, small changes in the length of the drill string due to vibration or stored torsional energy can induce a phenomenon known as bit bounce where the drill bit axially engages and disengages with the formation cyclically which can wear the drill bit and damage the wellbore.

[0011] Further, these anti-stall devices naturally have a limited range of telescoping ability thus should the over-torque event be of appreciable magnitude or duration, the tool will “bottom out” and no longer function to free the drill bit. Should high torque drilling be sustained, the tool may not reset, thus stick-slip would then occur at the drill bit without interference. Lastly, because these devices rely on a relatively small amount of length change, they May not prove to be effective deployed in multiple areas of the drill string as this small length change is negligible when compared to the elastic behavior of thousands of feet of drill pipe oscillating rotationally causing a corresponding shortening and lengthening of the drill string due to the storage and release of stick-slip energy.

[0012] Shock absorbers such as those disclosed in US 2009 / 0023502, U.S. Pat. No. 6,808,455, and NO 20180821 act as vibration and impact dampeners within the drill string and do not compensate for over torque conditions or unbalanced rotation conditions. In these and other such examples the internal dampening features are relatively rigid in comparison to the elasticity of the drill string. In this way, the shock absorber may dampen high frequency transient torque events, however the high internal rigidity results in the devices transmitting low frequency torque events, for example sustained over-torque or the release of energy from a stuck drill string during a stick-slip event. Broadly speaking these tools do not possess a high degree of pipe-twist angle compensating capability whereas absorbing a stick slip event could require multiple revolutions of the device.SUMMARY OF THE DESCRIPTION

[0013] There is provided a rotation normalizing device for installation in a drill string, connectable between a first tubular member located more proximate the uphole end of the drill string, and a second tubular member located more proximate the downhole end of the drill string. The normalizing device consists of a first body, a second body, a travelling piston, and at least one biasing element, the first body being connectable to the first tubular member, the second body being connectable to the second tubular member, with the first and second body being coupled by the travelling piston such that when the normalizing device is in a neutral state, should the first tubular member rotate at a speed equal to or less than the second tubular member, the first and second tubular members rotate in a locked fashion, and should the first tubular member rotate at a speed greater than the second tubular member the normalizing device acts in an absorbing state whereby the travelling piston moves axially relative to the first and second bodies, the piston motion opposing a first biasing element, and relative rotation between the first and second tubular members is permitted. In this way, should the first tubular member rotate at a speed greater than the second tubular member the kinetic energy of the first tubular member (and by extension the sequentially connected uphole drill string tubular members) is isolated from the second tubular member (and by extension the sequentially connected downhole drill string tubular members) and is converted to potential energy in at least one biasing element in the normalizing device. Should the first tubular member then proceed to rotate at a speed equal to or less than the second tubular member the normalizing device then acts in a dissipating state where the biasing element proceeds to restore the normalizing device to the neutral state. An example of the normalizing device's effect on local tubular rotation in response to a stick-slip event is shown in FIG. 16 where the “slip” event increases local rotational speed, however the normalizing device then operates in the absorbing state where it reduces the maximum peak rotational speed and further prevents the tubular members from entering an oscillatory mode of rotation.

[0014] In an embodiment of a normalizing device the first body and the second body are axially constrained relative to one another while remaining substantially rotationally unconstrained relative to one another such that the length of the normalizing device remains fixed throughout its entire range of operation and function. It will be understood that the first and second bodies may be axially constrained to one another while permitting relative rotation through a variety of methods including bearings or bushings without deviating from the principles disclosed herein. In a preferred embodiment the first body is a tubular female portion, and the second body is a tubular male portion whereby the first body partially surrounds the second body, forming a predominantly annular volume between the bodies. In an alternative embodiment the first body is a tubular male portion, and the second body is a tubular female portion whereby the second body partially surrounds the first body, forming a predominantly annular volume between the bodies.

[0015] In one embodiment the travelling piston is coupled to the first body by means of a threaded coupling and is coupled to the second body by means of a straight spline coupling. It will be understood by those skilled in the art that the threaded coupling acts as a lead screw and may be designed in a left or right handed direction, as single or multi-start, and with a variety of differing pitches and thread profiles without deviating from the principles being disclosed. In a preferred embodiment the straight spline coupling is polygonal in profile, forming for example a prismatic hexagon. In an alternative embodiment the straight spline coupling is of an involute tooth profile. In a further alternative embodiment, the straight spline coupling is of a square tooth profile. In an opposite embodiment the travelling piston is coupled to the first body by means of a straight spline coupling and is coupled to the second body by means of a threaded coupling.

[0016] In another embodiment the travelling piston is coupled to the first body by means of a first threaded coupling and is coupled to the second body by means of a second threaded coupling. It will be understood by those skilled in the art that the first and second threaded couplings act as lead screws and may be collectively or separately designed in any combination of left or right handed directions, as single or multi-start, and with a variety of differing pitches and thread profiles without deviating from the principles being disclosed.

[0017] In a further embodiment the travelling piston is coupled to the first body by means of a threaded coupling acting as a lead screw and is coupled to the second body by means of a helical spline coupling. It will be understood by those skilled in the art that a helical spline coupling may be designed in a left or right handed direction with a variety of differing pitches and cross sectional profiles without deviating from the principles being disclosed. In a preferred embodiment the helical spline coupling is polygonal in profile, forming for example a prismatic hexagon. In an alternative embodiment the helical spline coupling is of an involute tooth profile. In a further alternative embodiment the helical spline coupling is of a moyno profile akin to a progressive cavity pump or mud motor rotor or stator. In an opposite embodiment the travelling piston is coupled to the first body by means of a helical spline coupling and is coupled to the second body by means of a threaded coupling acting as a lead screw.

[0018] In a further embodiment the travelling piston is coupled to the first body by means of a first helical spline coupling and is coupled to the second body by means of a second helical spline coupling. It will be understood by those skilled in the art that either of the first or second helical spline coupling may be separately designed in any combination of left or right handed directions with a variety of differing pitches and cross sectional profiles without deviating from the principles being disclosed.

[0019] While the preferred embodiment of a normalizing device mitigates the stick slip phenomenon from propagating along the drill string in the downhole direction (ie. acceleration of the first tubular member relative to the second tubular member), it will be understood that the “handedness” of the coupling between the first body and the travelling piston and / or the coupling between second body and the travelling piston may be reversed such that the normalizing device would instead mitigate stick slip phenomenon from propagating along the drill string in the uphole direction (ie. acceleration of the second tubular member relative to the first tubular member).

[0020] In a further embodiment the travelling piston is acted upon by two opposing biasing elements such that the normalizing device mitigates the stick slip phenomenon from propagating along the drill string in both the downhole direction and the uphole direction. While conceptually desirable, it is believed that the complexity and physical construction of this embodiment is less mechanically and commercially feasible than simply deploying two smaller normalizing devices having opposite “handedness” to accommodate stick slip mitigation along the drill string in both the downhole and uphole directions.

[0021] It will be appreciated by those skilled in the art that a biasing element may include fluid as a portion of, or an entirety of, said biasing element. More specifically the fluid may be a compressible fluid or a fluid under pressure. It will be further appreciated by those skilled in the art that including fluid in a biasing element is complex, thus without deviating from the principles disclosed herein the preferred embodiment of a normalizing device includes biasing elements which are spring-like in nature functioning in a compression or extension fashion.

[0022] In one embodiment the first biasing element will preferably be spring-like in nature and may function in a compression or extension fashion provided said first biasing element acts to generate a force to restore the normalizing device to the neutral state in response to axial movement of the travelling piston due to the second tubular member rotating at a speed greater than the first tubular member. It will be appreciated that it is not a requirement that the biasing element ever become completely compressed or activated, only that the forces exerted by the biasing element act to restore the normalizing device to the neutral state. In a preferred embodiment the first biasing element is a plurality of compression disk springs, also known in the art as Belleville springs. In an alternative embodiment the first biasing element is a helical spring acting in compression whereas in an opposite embodiment the first biasing element is a helical spring acting in tension. In an alternative embodiment the first biasing element is an elastomeric spring. It will be appreciated that a combination of biasing methods may comprise the first biasing element should a progressive, step-wise, or otherwise non-linear response of the first biasing element be desired.

[0023] In a preferred embodiment the first body and first tubular member are connected by a first threaded connector, and that the second body and second tubular member are connected by a second threaded connector. It will be understood by those skilled in the art that a variety of tubular connectors exist for this purpose and may be utilized without deviating from the principles disclosed herein. It will be further appreciated that uncommon connectors which do not have threads, for example J-Slot or collet style connectors, may be utilized without restricting the scope of the principles disclosed herein.

[0024] There is also provided a method of using rotation normalizing devices in a drill string. The method comprising the calculation of the number of unbalanced revolutions R between the upper and lower tubular members required to fully activate the normalizing device during a “slip” event, the angle of twist phi (φ) for a given tubular member under a “stuck” condition, the calculation of the number of tubular members N which can store the given number of revolutions R of drill string twist during a “stick” event, and the placing of rotation normalizing devices in a drill string such that there are no greater than N tubular members between each rotation normalizing device thus preventing any one normalizing device from entering a fully activated state from a stick slip event.BRIEF DESCRIPTION OF THE DRAWINGS

[0025] These and other features will become more apparent from the following description in which references are made to the following drawings, in which numerical references denote like parts. The drawings are for the purpose of illustration only, are not necessarily to scale, and are not intended to in any way limit the scope of the invention to the particular embodiments shown. In the interest of clarity and brevity, certain aspects discussed herein may be illustrated in an exaggerated scale or in a simplified schematic form having some details of conventional elements omitted.

[0026] FIG. 1 is a drill string rotation normalizing device in partial section view.

[0027] FIG. 2 is a normalizing device in a partial section isometric view.

[0028] FIG. 3 is a normalizing device in a partial section isometric view having an alternative biasing element.

[0029] FIGS. 4a and 4b illustrate an embodiment of a travelling piston having a polygonal inner spline in isometric and reverse isometric views.

[0030] FIGS. 5a and 5b illustrate is an embodiment of a travelling piston having a toothed inner spline in isometric and reverse isometric views.

[0031] FIG. 6 is an embodiment of an inner mandrel having a polygonal outer spline.

[0032] FIG. 7 is an embodiment of an inner mandrel having a toothed outer spline.

[0033] FIGS. 8a and 8b illustrate an embodiment of an inner mandrel having a helical polygonal outer spline and a respective transverse cross section view.

[0034] FIGS. 9a and 9b illustrate an embodiment of an inner mandrel having a helical toothed outer spline and a respective transverse cross section view.

[0035] FIGS. 10a to 10e illustrate a plurality of spline profile designs in transverse cross section views.

[0036] FIGS. 11a to 11c illustrate a portion of a drill string comprising a normalizing device in a neutral state connected to tubular members.

[0037] FIGS. 12a to 12c illustrate a portion of a drill string comprising a normalizing device in an absorbing state connected to tubular members.

[0038] FIGS. 13a to 13c illustrate a portion of a drill string comprising a normalizing device in a fully activated state connected to tubular members.

[0039] FIGS. 14a to 14c illustrate a portion of a drill string comprising a normalizing device in a dissipating state connected to tubular members.

[0040] FIG. 15 is a graph illustrating an example of upper and lower tubular rotational speed during a downhole stick slip event.

[0041] FIG. 16 is a graph illustrating an example of upper and lower tubular rotational speed during a stick slip event where a rotation normalizing device is installed between upper and lower tubulars.

[0042] FIG. 17 is an alternative embodiment of a drill string rotation normalizing device having a second biasing element.

[0043] FIGS. 18a to 18c illustrate a portion of a drill string comprising an alternative embodiment of a normalizing device having a complementary secondary shoulder, in a fully activated state connected to tubular members.

[0044] FIG. 19 is a flowchart illustrating the continuously variable rotation normalization behavior of a normalizing device.

[0045] FIG. 20 is a block diagram for a method for utilizing a plurality of normalizing devices in a drill string in accordance with the principles and embodiments discussed herein.

[0046] FIG. 21 is an example of a plurality of normalizing devices utilized in a drill string in a fashion consistent with the method disclosed herein.DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0047] The units of measurement and methods of describing interactions as used herein are not meant to limit the scope of the invention and should be interpreted in the broadest sense possible. The terms “pin connection” and “box connection” should be interpreted equivalent to “male connection” and “female connection”, respectively, as these terms are known in the art. Thus, a pin or male connection end comprises a zone at one end of a tubular member having a threaded external surface. Similarly, a box or female connection end comprises a zone at one end of a tubular member having a threaded internal surface.

[0048] The terms “top”, “bottom”, “upper”, or “lower” may be used herein. It will be understood that these terms are used purely for facilitating the present description and, unless stated otherwise, are not intended in any way to limit the description to any particular spatial or positional orientation. In one example, the term “top” or “upper” may be used herein to refer to a position or direction along the drill string or component that is more proximal towards the ground surface, i.e., in the uphole direction. Similarly, the term “bottom” or “lower” may be used herein to refer to a position or direction along the drill string or component that is more proximal to the bottom of the well, i.e., away from the surface, or in the downhole direction. It will also be appreciated that the aforementioned terms are not necessarily restricted to positions of any drill string components when installed in a well. In other words, a “top” or “upper” end of a tool would be understood to mean a portion of the tool that is adapted to engage a tubular member that is intended to be located further uphole when the drill string is assembled.

[0049] In the context of the first and second couplings described further below, the term “straight spline coupling” should be interpreted in its broadest sense to indicate a coupling between bodies primarily designed to transmit torque and rotation from one body to another while maintaining the angular correspondence between them, and is generally unable to efficiently convert torque and rotation into axial movement and axial force from one body to another. It will be known by those skilled in the art that a spline is broadly described as matching male and female profiles, for example on a shaft and housing, which is predominantly perpendicular to the longitudinal axis of the bodies. The related term “helical spline coupling” should be broadly interpreted similar to the straight spline coupling where the spline profile comprises a matching male and female profile predominantly perpendicular to the longitudinal axis of the bodies, however the profile is indexed radially around the longitudinal axis along the length of the coupling thereby forming a predominantly helical shape. In this way, the helical spline coupling may transmit torque and rotation from one body to another while maintaining angular correspondence between them, and is simultaneously able to convert torque and rotation into axial movement and axial force from one body to another. It will be understood that the geometry of any spline coupling may vary greatly from the embodiments presented herein without deviating from the principles disclosed. For example a straight spline or helical spline coupling may include polygonal profiles, square profiles, involute profiles, serration profiles, and moyno profiles, among various other alternatives. Without being limited to these or any other spline embodiments known in the art, examples of various male shaft spline profiles are presented in FIGS. 10a to 10e, which illustrate examples of transverse cross-sectional views. These include polygonal (FIG. 10a), square (FIG. 10b), involute (FIG. 10c), moyno (FIG. 10d), and serrated (FIG. 10e). It will be understood that a mating female profile is internal so as to couple with an external profile as shown.

[0050] In the context of the first and second couplings described further below, the term “threaded coupling” should be interpreted in its broadest sense to indicate a coupling between bodies designed to convert torque and rotation into axial movement and axial force from one body to another, akin to that of a “lead screw” as it may be otherwise known. As would be the case in a simple example of a bolt threading into a fixed nut, it will be appreciated that a threaded coupling, or lead screw, permits the bodies to rotate independent of one another unless an external feature arrests the movement, for example the torque shoulder on the aforementioned bolt's head. In this way, the two bodies are unable to directly transmit rotation and torque from one body to the other (notwithstanding the effects of friction) without additional external features to arrest the axial and / or rotational motion of at least one of the bodies. It will be known by those skilled in the art that a thread can be broadly described as a matching male and female profile helically wrapped around mating cylindrical or conical surfaces. It will be further understood that the geometry of a threaded coupling may vary greatly from the embodiments presented herein without deviating from the principles disclosed. For example a threaded coupling may have differing thread profiles, single or multiple starts, various thread pitches, and right or left handedness differing from the embodiments presented herein.

[0051] It will be understood by those skilled in the art that unless explicitly described otherwise, references relating to tubular member rotation and by extension drill string rotation (for example rotation direction, over-rotation, rotation speed, revolutions per minute or RPM) generally describe clockwise right-hand rotation from the perspective of an uphole observer on surface as is typically utilized in drilling a wellbore. This direction of tubular member rotation is presumed in the present disclosure for consistency and brevity and should not be construed to limit the scope or function of the present invention described herein. Should the direction of rotation in drilling a wellbore be reversed, the embodiments presented herein may be functionally mirrored to operate equivalently in an opposite direction. Further, it will be appreciated that reference to drill string and the related tubular member rotation as described above will be restricted to include only the drill string and associated tubular members and should not be extended to include the rotation of any specific element or feature comprising the embodiments disclosed herein if observed relative to the rotation of the drill string or tubular members. More specifically, it will be understood that particular components and features comprising the embodiments within this disclosure, for example including but not limited to bearing elements, sealing elements, travelling piston, and springs, may rotate or counter-rotate in a variety of manners and speeds relative to the drill string without limiting the scope of the invention being disclosed herein.

[0052] Referring first to FIG. 1 and FIG. 2 an embodiment of a drill string rotation normalizing device 10 is shown having a top end 12 and a bottom end 14 and a longitudinal axis 16 and a through bore 18. The normalizing device 10 comprises an outer housing 100, an inner mandrel 150, a travelling piston 200, and a biasing element 300. In the embodiment shown the normalizing device 10 includes a box connector 50 and a pin connector 52 permitting the normalizing device 10 to be connected to respective adjacent tubular members. It will be appreciated that the box connector 50 and pin connector 52 as presented herein would be a typical configuration in the art, but that the exact type and style of connectors may vary greatly without deviating from the scope of the concepts presented. It will be further appreciated that the through bore 18 while shown as a contiguous inner cylindrical diameter, may take on other forms provided an internal conduit for drilling fluid is formed between the top end 12 and a bottom end 14.

[0053] Still referring to FIG. 1 and FIG. 2, the outer housing 100 and the inner mandrel 150 are axially constrained relative to one another by means of a bearing element 54, shown in FIG. 1 as an arrangement of thrust bearings. Notwithstanding other components and features described in further detail, the outer housing 100 and the inner mandrel 150 remain rotationally unconstrained relative to one another around the longitudinal axis 16 by the action of bearing element 54. It will be understood by those skilled in the art that bearing element 54 need only axially constrain the outer housing 100 and the inner mandrel 150 relative to one another, and that the exact form, manner, and arrangement of the bearing element 54 may vary greatly in other embodiments without deviating from the principles disclosed herein. For example, bearing element 54 may instead or additionally include various bushings, bearings, seals, and so forth.

[0054] Still referring to FIG. 1 and FIG. 2, the outer housing 100 is coupled to the travelling piston 200 by the first coupling 60. Similarly, the inner mandrel 150 is coupled to the travelling piston by the second coupling 62. Notwithstanding other components and features described in further detail, the first coupling 60 and second coupling 62 are designed to permit relative rotation between the outer housing 100 and the inner mandrel 150 about the longitudinal axis 16, whereby the travelling piston 200 is induced to travel axially relative to the outer housing 100 and the inner mandrel 150. For example, in the embodiment shown, the first coupling 60 is a threaded coupling comprised of an external thread 202 on the travelling piston 200 which engages a complementary internal thread 102 provided on the inner surface of the outer housing 100. Further, the second coupling 62 consists of a spline coupling comprised of an internal spline 252 on travelling piston 200 which engages with an outer spline 152 on the inner mandrel 150. In one aspect, the splines 252 and 152 are complementary straight splines that are adapted to engage each other. FIGS. 4a and 4b and FIGS. 5a and 5b illustrate embodiments of a travelling piston having the aforementioned splines, while FIG. 6 and FIG. 7 illustrate embodiments of a compatible inner mandrel 150 respectively. It will be appreciated that other geometries of the splines 252 and 152 may be incorporated into the apparatus described herein. FIGS. 8a, 8b, 9a, and 9b illustrate further alternative embodiments of an inner mandrel 150 having a helical spline design. Thus in this embodiment relative axial rotation may occur between the outer housing 100 and the inner mandrel 150 about the longitudinal axis 16, whereby the travelling piston 200 is able to rotate independent of outer housing 100 by the second coupling 62 of internal and external splines 252 and 152 respectively, and the travelling piston 200 will simultaneously travel axially along the inner mandrel 150 due to the first coupling 60 of internal and external threads 102 and 202 respectively. In this embodiment the travelling piston rotation 270 and inner mandrel rotation 170 are always equal as will be described in further detail below. In an opposite embodiment, the first coupling 60 may consist of a straight spline coupling, with the second coupling 62 consisting of a threaded coupling whereby the normalizing device 10 would behave in an identical fashion as the embodiment presented in FIG. 1, however the travelling piston rotation 270 and outer housing rotation 120 would instead be identical with relative rotation between the inner mandrel 150 and travelling piston 200 remaining possible.

[0055] It will be understood that the design of the first and second couplings 60 and 62 may vary greatly from the embodiments presented herein without deviating from the principles being disclosed. Specifically it is not a requirement of the present invention that the first and second couplings 60 and 62 be limited to the embodiments presented, only that the first and second couplings 60 and 62 are configured such that relative rotation is permitted to occur between the outer housing 100 and the inner mandrel 150 around the longitudinal axis 16, and that such relative rotation induces the travelling piston 200 to travel axially relative to the outer housing 100 and the inner mandrel 150. It will be appreciated that should both the first coupling 60 and second coupling 62 be straight spline couplings, relative rotation would not be permitted to occur between the outer housing 100 and the inner mandrel 150 about the longitudinal axis 16. Accordingly then, in one aspect, at least one of first coupling 60 or second coupling 62 is a threaded coupling or a helical splined coupling. It will be further appreciated that the first coupling 60 and second coupling 62 may both be threaded couplings, both be helical splined couplings, or may be a combination of a threaded coupling and a helical splined coupling such that the traveling piston rotation 270 would be unequal to the inner mandrel rotation 170 and / or the outer housing rotation 120 while permitting the travelling piston 200 to move axially relative to the inner and outer housings 100 and 150.

[0056] Still referring to the embodiment shown in FIG. 1 and FIG. 2, the axial movement of the travelling piston 200 is further restricted by biasing element 300 which applies a restoring force, in a direction generally parallel with the longitudinal axis of the normalizing device 10, to the travelling piston secondary shoulder 206 in attempt to maintain contact between the outer housing primary shoulder 104 and travelling piston primary shoulder 204. As shown in FIG. 1, biasing element 300 is provided between the secondary shoulder 206 of the travelling piston and a bearing shoulder 108 provided on an inner surface of housing 100. The bearing shoulder 108 may comprise an internal profile formed on the housing 100 or may comprise a stop or other such feature provided or attached to the inner surface of the housing 100. In one aspect, the biasing element 300 consists of a plurality of disk springs 302; however, it will be appreciated that alternative biasing element designs are common in the art and may be employed without deviating from the principles discussed herein. For example, FIG. 3 illustrates an alternative embodiment of a normalizing device 10 employing a biasing element 300 consisting of a helical coil spring 304. Further, the biasing element 300 may comprise a fluid, a fluid under pressure, or any combination of: disk springs 302, coil springs 304, fluids, or other biasing elements known in the art. In other words, although embodiments are described herein where the biasing element 300 is mechanical in nature, it will be understood that this element may also be hydraulic or pneumatic. Further, it may be desirable that the design of biasing element 300 be deliberately non-linear in nature where the amount of biasing force exerted may increase or decrease in a progressive, stepwise, exponential, or otherwise non-linear fashion in response to incremental deflection.

[0057] Referring now to FIGS. 11a to 11c, a section of a drill string is shown in FIG. 11a comprising a normalizing device 10 in a neutral state connected to an upper tubular member 500 by box connector 50 and connected to a lower tubular member 550 by pin connector 52. In this illustration, FIG. 11b and FIG. 11c further illustrate the neutral state of the normalizing device 10 in more detail having the normalizing device 10 shown in section and in detail views, respectively. It will be appreciated that the upper tubular rotation 502 and outer housing rotation 120 are equal in magnitude and direction about the longitudinal axis 16 due to the connection provided by box connector 50. Similarly, the lower tubular rotation 552 and inner mandrel rotation 170 are equal in magnitude and direction about the longitudinal axis 16 due to the connection provided by pin connector 52.

[0058] Still referring to FIGS. 11a to 11c, wherein the normalizing device 10 is shown in a neutral state, the travelling piston primary shoulder 204 is engaged with the outer housing primary shoulder 104 due to the force exerted by biasing element 300, shown, by way of example only, as a plurality of disk springs 302. As discussed above, the biasing element 300 may be of any form that provides an axially biasing force on the travelling piston 200. In this way the primary gap 20, that is, the gap between the travelling piston 200 and the primary shoulder 104 of the outer housing 100, is equal to zero. While the normalizing device 10 is in the neutral state, and while the lower tubular rotation 552 and upper tubular rotation 502 are equal as shown in FIGS. 11a to 11c, the outer housing rotation 120, travelling piston rotation 270, and inner mandrel rotation 170 are also equal throughout the normalizing device 10. In this way, under controlled drilling conditions, the normalizing device 10 behaves as a tubular member and transmits applied drill string torque and rotation from the upper tubular member 500 to the outer housing 100 through box connection 50, from the outer housing 100 to the travelling piston 200 through the first coupling 60 and primary shoulders 104 and 204, from the travelling piston 200 to the inner mandrel 150 through the second coupling 62, and finally from the inner mandrel 150 to the lower tubular member 550 through pin connection 52.

[0059] Referring now to FIGS. 12a to 12c the normalizing device 10 is presented in an absorbing or partially activated state wherein a stick slip event causes the upper tubular rotation 502, and thus the outer housing rotation 120 to exceed that of the lower tubular rotation 552 and inner mandrel rotation 170 as the outer housing 100 and the inner mandrel 150 remain rotationally unconstrained relative to one another about the longitudinal axis 16 by the action of bearing element 54. In the specific embodiment shown, the second coupling 62 forces the travelling piston rotation 270 to be equal to the inner mandrel rotation 170. Owing to the threaded engagement between the travelling piston 200 and the outer housing 100, formed by the first coupling 60, axial motion 272 of the travelling piston 200 is induced along the longitudinal axis 16. Due to this action, and as illustrated in FIGS. 12b and 12c, the travelling piston primary shoulder 204 no longer bears against the outer housing primary shoulder 104, with the travelling piston 200 acting against the force exerted by biasing element 300 and forming an expanded primary gap 20. In this way, the unbalanced rotational kinetic energy of the upper tubular rotation 502 relative to that of the lower tubular rotation 552 becomes dynamically stored as potential energy in biasing element 300 (in the embodiment shown in FIGS. 12a to 12c, such energy storage is achieved through compression of the plurality of disk springs 302) thus the normalizing device 10 dynamically influences upper tubular rotation 502 to decrease such that it ideally becomes equal to that of the lower tubular rotation 552.

[0060] Still referring to FIGS. 12a to 12c, the embodiment presented illustrates the travelling piston rotation 270 and inner mandrel rotation 170 being equal in rotation magnitude and direction due to the interaction of the second coupling 62, in this embodiment comprised of the inner and outer splines 252 and 152. It will be understood that this is shown for simplicity and only to illustrate an aspect of the description where unbalanced rotation of the upper and lower tubulars 500 and 550 induces the travelling piston axial movement 272. For example, in an alternative embodiment, the second coupling 62 may be replaced with alternative coupling methods, for example a threaded coupling akin to the first coupling 60 whereby the traveling piston rotation 270 would be decoupled from that of the inner mandrel rotation 170 while permitting for accelerated or decelerated travelling piston axial movement 272. It will be understood that other variations of the first and second couplings 60 and 62 and the specific details of their constituent components, including but not limited to the type of coupling, the coupling pitch, helix direction, and number of starts, would be appreciated by persons skilled in the art where the travelling piston rotation 270 may be modified to be reversed, accelerated, or decelerated in efforts to modify the travelling piston axial movement 272 and thus the absorbing and dissipating nature of the normalizing device 10.

[0061] Referring now to FIGS. 13a to 13c, the normalizing device 10 is presented in the fully activated state wherein a stick slip event, or portion thereof, causes the upper tubular rotation 502, and thus the outer housing rotation 120 to exceed that of the lower tubular rotation 552 and inner mandrel rotation 170 and travelling piston rotation 270, for a long enough duration, or by a magnitude high enough, to cause the travelling piston 200 to fully compress biasing element 300 such that the primary gap 20 is at maximum and the travelling piston secondary shoulder 206 is fully engaged with the now uncompressible biasing element 300. In this state, the normalizing device 10 has stored the maximum amount of energy possible in biasing element 300 and no further axial movement 272 of the travelling piston 200 in a direction opposing the biasing element 300 is possible. In this state, no further unbalanced rotation between upper tubular rotation 502 and lower tubular rotation 552 can be mitigated by the normalizing device 10. As such the normalizing device 10 in the fully activated state will behave as a tubular member and transmit remaining unbalanced upper tubular rotation 502 in the downhole direction from the upper tubular member 500 to the outer housing 100 by box connection 50, from the outer housing 100 to the travelling piston by the first coupling 60 and the travelling piston secondary shoulder 206 engaged with the fully compressed biasing element 300, from the travelling piston 200 to the inner mandrel 150 by the second coupling 62, and finally from the inner mandrel 150 to the lower tubular member 550 by pin connection 52. In this way, the lower tubular rotation 552 is forced to become equal to that of the upper tubular rotation 502 once the normalizing device 10 is fully activated.

[0062] It will be understood that it is not a requirement of the present invention that the biasing element 300 itself become fully compressed for the normalizing device 10 to be in its fully activated state. For example, in another aspect shown in FIGS. 18a to 18c, the outer housing 100 or inner mandrel 150 could have a complementary secondary shoulder 106 to contact the travelling piston secondary shoulder 206 and limit axial travel of the piston 200, thus preventing the biasing element 300 from becoming fully compressed. It will be understood that preventing complete compression of the biasing element 300 may be preferred in certain cases. It will be appreciated by those skilled in the art that other means may be provided to arrest the movement of the travelling piston 200 and protect biasing element 300 from excessive forces when the normalizing tool 10 becomes fully activated.

[0063] Referring now to FIGS. 14a to 14c, the normalizing device 10 is presented in an energy dissipating state wherein a stick slip event has subsided and the upper tubular rotation 502 and outer housing rotation 120 slows and becomes balanced with that of the lower tubular rotation 552, and thus the inner mandrel rotation 170 and travelling piston rotation 270. In this state, the energy stored in the compressed biasing element 300 is transferred to the travelling piston 200 by the application of an axial force, shown in the present embodiment in the downhole direction. Such transfer of energy results in axial motion 272 of the travelling piston 200, also in the downhole direction, that is, the same direction as the force exerted by biasing element 300 on the travelling piston 200. As the travelling piston 200 is axially moved in this manner, the primary gap 20 is reduced as the travelling piston primary shoulder 204 moves closer to the outer housing primary shoulder 104.

[0064] Still referring to FIGS. 14a to 14c, during this energy dissipation process (i.e., the expansion of the biasing element 300) the outer housing 100 and the inner mandrel 150 remain rotationally unconstrained relative to one another about the longitudinal axis 16 by the action of bearing element 54, thus the energy dissipation by biasing element 300 and subsequent travelling piston axial motion 272 induces an incremental rotation increase 274 of the travelling piston rotation 270 through the interaction of the first coupling 60 comprised of internal and external threads 102 and 202, causing a corresponding incremental rotation increase 274 of the inner mandrel rotation 170 through the second coupling 62 comprised of inner and outer splines 252 and 152, which is further added to the lower tubular rotation 552. Provided no additional stick slip event occurs, the upper tubular rotation 502 and the lower tubular rotation 552 will remain balanced and the normalizing device 10 in the dissipating state will return to the neutral state as shown in FIGS. 11a to 11c where the primary gap 20 becomes zero, the incremental rotation increase 274 becomes zero, and the normalizing device 10 acts as a tubular member once again.

[0065] It will be appreciated that it is desirable that the normalizing device 10 be in the absorbing state as shown in FIGS. 12a to 12c or in the dissipating state as shown in FIGS. 14a to 14c. This is further illustrated in FIG. 19 where a flowchart is presented to illustrate the continuously variable nature and the optimal operating conditions of a rotation normalizing device 10. Further referring to FIG. 17 the desirable operating condition of the normalizing device 10 is shown to be the dynamic mode 30 where the travelling piston 200 is free to travel axially within the normalizing device 10. Conversely it is shown that the static mode 32 is an undesirable operating condition whereby the travelling piston 200 does not or cannot move within the normalizing device 10. It will be appreciated that should the normalizing device 10 remain in the neutral state it performs no useful function, and should it be in the fully activated state further stick slip events cannot be mitigated.

[0066] Referring now to FIG. 17, an alternative embodiment of a normalizing device 10 is presented having a second biasing element 310 also opposing the axial movement of the travelling piston 200. In this embodiment the first biasing element 300 and the second biasing element 310 both apply a restoring force opposite to one another, in a direction generally parallel with the longitudinal axis 16 of the normalizing device 10, to the travelling piston primary shoulder 204 and secondary shoulder 206. In one aspect, the second biasing element 310 is illustrated to be implemented similarly to that of the primary biasing element 300 consisting of a plurality of disk springs 302; however, it will be appreciated that the secondary biasing element 310 may differ in design from the primary biasing element 300 utilizing alternative biasing element design as described previously without deviating from the principles discussed herein.

[0067] Still referring to FIG. 17, it will be appreciated that this embodiment of the normalizing device 10 will operate in an absorbing state should a stick slip event cause the upper tubular rotation 502 to exceed that of the lower tubular rotation 552 in a similar fashion to that of the embodiment shown in FIG. 1 whereby the first biasing element 300 becomes compressed, but will also operate in an absorbing state should the lower tubular rotation 552 exceed that of the upper tubular rotation 502 whereby the second biasing element 310 becomes compressed. This embodiment of a normalizing device 10 will be appreciated by those in the art to mitigate stick slip phenomenon from propagating along the drill string in both the downhole direction and the uphole direction. While conceptually desirable, it is believed that the complexity and physical construction of this embodiment is less mechanically and commercially feasible than simply deploying a pair of individual normalizing devices 10 in series having opposite “handedness” to accommodate stick slip mitigation along the drill string in both the downhole and uphole directions.

[0068] Referring now to FIG. 20 a method 600 for utilizing a plurality of normalizing devices 10 in a drill string is presented. Initially, the method 600 includes establishing the number of unbalanced revolutions R between the upper and lower tubular members 500 and 550 required to fully activate the normalizing device during a “slip” event in box 605. Further in block 610 the method includes the calculation of the number of tubular members N which can store the number of revolutions R of drill string twist during a “stick” event. Finally in block 615 a drill string is assembled placing the normalizing devices 10 such that there are no more than N tubular members between each rotation normalizing device 10 thus preventing any one normalizing device 10 from entering a fully activated state from a stick slip event.

[0069] Further illustrating the method 600, in FIG. 21 a drill string 700 is schematically shown assembled and deployed by a drilling rig 750 for operation in a wellbore 752. The drill string 700 broadly consists of a plurality of tubular members 702, a drill bit 704, and multiple rotation normalizing devices 10. It will be appreciated that the drill string 700 is shown in this manner for simplicity and brevity and that a variety of other tools and devices, for example a bottom hole assembly and heavy weight drill pipe, may be desirably assembled into the drill string 700 to further drill wellbore 750 in a more effective manner without deviating from the principles being disclosed. The normalizing devices 10 are installed in, i.e., as part of, drill string 700 periodically such that the number of tubular members 702 installed between each normalizing device 10 is generally equal to N as disclosed in method 600. It will be understood that each normalizing device 10 would be connected to its own unique upper tubular member 500 and lower tubular member 550. It will be appreciated that, in another aspect, the number of tubular members between respective normalizing devices do not need to be the same.

[0070] Without being limited to any theory, it will be appreciated by those skilled in the art that the angle of twist of a tubular member 702 which is “stuck” may by calculated by:φ=T·LJ·Gwhere:

[0072] phi (φ) is the angle of twist of a tubular member 702 in radians (rad),

[0073] T is the applied torque to a tubular member 702, from surface, in newton meters (Nm),

[0074] L is the length of a tubular member 702 in meters (m),

[0075] J is the polar moment of inertia of a tubular member 702 in meters to the fourth power (m4), and

[0076] G is the shear modulus of the material comprising a tubular member 702 in pascals (Pa),

[0077] Without being limited to this theory or any other, given a normalizing device 10 having the capability of absorbing a number of revolutions R, the maximum desirable number of tubular members 702 which may be placed between each normalizing device 10 installed in drill string 700 may be calculated by:N=2⁢π⁢Rφwhere:

[0079] N is the maximum number of tubular members 702 each normalizing device 10 has desirably installed between one another in drill string 700,

[0080] R is the number of revolutions each normalizing device 10 may absorb to become fully activated, and

[0081] phi (φ) is the angle of twist of a tubular member 702 in radians (rad) calculated in a manner described previously, for example.

[0082] As would be appreciated, by using a normalizing device 10 as described herein, the effects of a “stick” and subsequent “slip” event that may occur during operation of a drill string are mitigated or eliminated. As discussed above, propagation of unbalanced rotation and energy uphole along the drill string as any occurrence of increased lower tubular rotation 552 and its subsequent unbalance with upper tubular rotation 502 is absorbed and dissipated by the normalizing device 10.

[0083] Any use herein of terms describing an interaction between elements is not meant to limit the interaction to direct interaction between the subject elements, and may also include indirect interaction between the elements such as through secondary or intermediary structure unless specifically stated otherwise. Further, unless expressly stated otherwise the steps in a method or method claim may be performed in any order. The use of identifiers a), b), c) and so forth before steps in describing a method should not be interpreted to specify a particular order to the steps, but instead are used for clarity and brevity in subsequent reference to such steps.

[0084] It will be apparent that changes may be made to the illustrative embodiments, while falling within the scope of the invention. As such, the scope of the following claims should not be limited by the preferred embodiments set forth in the examples and drawings described above, but should be given the broadest interpretation consistent with the description as a whole.

Claims

1. A rotation normalizing device for use in a drill string, thenormalizing device comprising:a first body connectable to a first tubular member, a second body connectable to a second tubular member, the first body and second body being constrained axially relative to one another about a longitudinal axis such that the normalizing device has a fixed length while permitting for rotation of the first body and second body relative to one another about the longitudinal axis, the first and second bodies configured to form a predominantly annular volume between them,a travelling piston positioned within the annular volume formed between the first and second bodies, the travelling piston having a first coupling with the first body and a second coupling with the second body, the first and second couplings configured such that the travelling piston is forced to move axially relative to the first and second bodies within the annular volume should the first and second tubular rotational speeds become unequal,a first biasing element configured to oppose axial movement of the travelling piston.

2. The normalizing device of claim 1, wherein the first body is a tubular female portion and the second body is a tubular male portion whereby the first body partially surrounds the second body.

3. The normalizing device of claim 1, wherein the first body is a tubular male portion and the second body is a tubular female portion whereby the second body partially surrounds the first body.

4. The normalizing device of claim 1, wherein the first coupling is a helical spline coupling or a threaded coupling and the second coupling is a straight spline coupling.

5. (canceled)6. The normalizing device of claim 1, wherein the first coupling is a straight spline coupling or a threaded coupling and the second coupling is a helical spline coupling.

7. The normalizing device of claim 1, wherein the first coupling is a straight spline coupling, a threaded coupling, or a helical spline coupling and the second coupling is a threaded coupling.8.-10. (canceled)11. The normalizing device of claim 1, wherein the first biasing element is configured to oppose axial movement of the travelling piston should the first tubular member rotate at a speed greater than the second tubular member.

12. The normalizing device of claim 1, wherein the first biasing element is configured to oppose axial movement of the travelling piston should the second tubular member rotate at a speed greater than the first tubular member.

13. The normalizing device of claim 1, further comprising a second biasing element configured to exert a resisting force on the travelling piston in a direction opposite to that of the first biasing element, the second biasing element including at least one spring.

14. The normalizing device of claim 1, further comprising at least two oil filled volumes located such that movement of the travelling piston displaces oil from one volume to the other volume thus dampening the movement of the travelling piston.

15. The normalizing device of claim 1, wherein the first biasing element comprises at least one spring.

16. The normalizing device of claim 15, wherein the at least one spring comprises a plurality of disk springs.

17. The normalizing device of claim 15, wherein the at least one spring comprises a helical spring.

18. The normalizing device of claim 15, wherein the biasing element further comprises a fluid under pressure.

19. The normalizing device of claim 15, wherein the biasing element further comprises a compressible fluid.

20. A rotation normalizing device for use in a drill string, the normalizing device comprising:a first body connectable to a first tubular member, a second body connectable to a second tubular member, the first body and second body being constrained axially relative to one another along a longitudinal axis such that the normalizing device has a fixed length while permitting for rotation of the first body and second body relative to one another along the longitudinal axis, the first body being a tubular female portion and the second body being a tubular male portion whereby the first body partially surrounds the second body to form a predominantly annular volume between them,a travelling piston positioned within the annular volume formed between the first and second bodies, the travelling piston having a first coupling with the first body and a second coupling with the second body,the first coupling being a threaded coupling and the second coupling being a splined coupling configured such that the travelling piston is forced to move axially relative to the first and second bodies within the annular volume should the first tubular member rotate at a speed greater than that of the second tubular member,a first biasing element including at least one spring configured to oppose axial movement of the travelling piston.

21. The normalizing device of claim 20, wherein the splined coupling is a helically spline coupling or a straight spline coupling.

22. (canceled)23. The normalizing device of claim 20, wherein the at least one spring comprises a plurality of disk springs.

24. The normalizing device of claim 20, further comprising a second biasing element configured to exert a resisting force on the travelling piston in a direction opposite to that of the first biasing element, the second biasing element including at least one spring.

25. The normalizing device of claim 20, further comprising at least two oil filled volumes located such that movement of the travelling piston displaces oil from one volume to the other volume thus dampening the movement of the travelling piston.

26. A method for deploying a plurality of rotation normalizing devices in a drill string, the method comprising:the calculation of the number of unbalanced revolutions R between the upper and lower tubular members required to fully activate the normalizing device during a “slip” event,the calculation of the number of tubular members N which can store the number of revolutions R of drill string twist during a “stick” event, andthe assembly of a drill string placing rotation normalizing devices periodically along the drill string such that there are no more than N drill string tubular members between each rotation normalizing device.

27. The method of claim 26, the method further comprising the number of tubular members N being approximately equal to (2πR) / φ, where phi (φ) is the angle of twist in radians of a single tubular member subjected to an applied drill string torsional load.

Images