FOR DIRECTIONAL DRILLING

TECHNICAL FIELD

[0001] The present technology generally pertains to drilling in earth formations, and more specifically, to feedback controls for path tracking in directional drilling.

BACKGROUND

[0002] Directional drilling or controlled steering is commonly used to guide drilling tools in the oil, water, and gas industries to reach resources that are not located directly below a wellhead. Directional drilling particularly provides access to reservoirs where vertical access is difficult if not impossible. In general, directional drilling refers to steering a drilling tool according to a predefined well plan, having target coordinates and drilling constraints, created by a multidisciplinary team (e.g., reservoir engineers; drilling engineers; geo-steerers; geologists, etc.) to optimize resource collection/discovery.

[0003] As the future of directional drilling moves toward exploiting complex reservoirs and difficult to reach resources, it becomes increasingly important for the drilling tool to follow these predefined well plans as closely as possible. Deviations from such pre-defined well plans may result in a waste of resources, damage the drilling tools, or even undermine the stability of earth formations surrounding a reservoir. Path tracking along the predefined well plans often presents new challenges due, in part, physical and operational constraints of the drilling tools, characteristics of rock formations, complex well geometries, and the like.

BRIEF DESCRIPTION OF THE DRAWINGS

[0004] The embodiments herein may be better understood by referring to the following description in conjunction with the accompanying drawings in which like reference numerals indicate analogous, identical, or functionally similar elements. Understanding that these drawings depict only exemplary embodiments of the disclosure and are not therefore to be considered to be limiting of its scope, the principles herein are described and explained with additional specificity and detail through the use of the accompanying drawings in which:

[0005] FIG. 1 is a schematic diagram of a directional drilling environment, showing measurement while drilling (MWD) operations;

[0006] FIG. 2 is a schematic diagram of a directional drilling device;

[0007] FIG. 3 is a schematic diagram of a three-dimensional (3D) wellbore environment, showing a drilling tool following a well path defined by a collection of waypoints;

[0008] FIG. 4A is a graph showing two-dimensional (2D) wellbore path divergences for directional drilling using attitude azimuth correction;

[0009] FIG. 4B is a graph showing 2D wellbore path divergences for directional drilling using attitude position correction;

[0010] FIG. 5 is a graph showing wellbore path convergence for directional drilling using curvature-based feedback control operations, according to one embodiment of this disclosure;

[0011] FIG. 6 is an exemplary simplified procedure for directional drilling based on the curvature-based feedback control operations, particularly from the perspective of a drilling tool.

[0012] FIGS. 7-9 are graphs showing simulated curved wellbore paths and corresponding curvatures value calculations for a directional drilling tool that employs curvature-based feedback controls;

[0013] FIGS. 10-11 are graphs showing simulated curved wellbore paths and corresponding curvatures value calculations when a drift value is constant; and

[0014] FIGS. 12-14 are graphs showing simulated curved wellbore paths and corresponding curvatures value calculations when a proportional uncertainty is added to curvature values.

DETAILED DESCRIPTION

[0015] Various embodiments of the disclosure are discussed in detail below. While specific implementations are discussed, it should be understood that this is done for illustration purposes only. A person skilled in the relevant art will recognize that other components and configurations may be used without parting from the spirit and scope of the disclosure. Additional features and advantages of the disclosure will be set forth in the description which follows, and in part will be obvious from the description, or can be learned by practice of the herein disclosed principles. The features and advantages of the disclosure can be realized and obtained by means of the instruments and combinations particularly pointed out in the appended claims. These and other features of the disclosure will become more fully apparent from the following description and appended claims, or can be learned by the practice of the principles set forth herein.

[0016] As used herein, the term "coupled" is defined as connected, whether directly or indirectly through intervening components, and is not necessarily limited to physical connections. The term "substantially" is defined to be essentially conforming to the particular dimension, shape or other word that substantially modifies, such that the component need not be exact. For example, substantially rectangular means that the object in question resembles a rectangle, but can have one or more deviations from a true rectangle. The "position" of an object can refer to a placement of the object, location of the object, plane of the object, direction of the object, distance of the object, azimuth of the object, axis of the object, inclination of the object, horizontal position of the object, vertical position of the object, and so forth. Moreover, the "position" of an object can refer to the absolute or exact position of the object, the measured or estimated position of the object, and/or the relative position of the object to another object

[0017] This disclosure generally relates to directional drilling and steering a drilling tool along a planned well path, and more specifically, provides curvature-based feedback control techniques suitable for directional drilling systems having steering assemblies that direct a drill bit as it creates a borehole along a desired path (i.e., trajectory). The steering assemblies may include, for example, rotary steerable systems ("RSS") that can change a direction of the drill string while in operation. However, it is also appreciated these techniques may be employed by other known directional drilling tools.

[0018] These curvature-based feedback control techniques address the challenges present in path tracking along a predefined well plan, and particularly, provide feedback control laws, processes, methods, systems, devices, and the like, to continuously adjust/correct wellbore paths based on curvature values. These techniques provide simultaneous path convergence between a predefined well plan and a current wellbore path, both in terms of position and attitude (e.g., inclination and azimuth). These techniques may be used to track a current position and a current attitude of a steerable drilling tool, determine a curvature value for a curved path that intersects the current position (tangent to the current attitude), and a curvilinear position on a target wellbore path (tangent to a target attitude), instruct the steerable drilling tool to generate a wellbore path based on the curvature value, and update the current position or the current attitude based on at least one of a change in the current position or a change in the current attitude. These and other features of the subject curvature-based feedback control techniques are described in greater detail with reference to the figures.

[0019] FIG. 1 is a schematic diagram of a directional drilling environment, particularly showing a measurement-while-drilling (MWD) system 100, in which the presently disclosed techniques may be deployed. As depicted, MWD system 100 includes a drilling platform 102 having a derrick 104 and a hoist 106 to raise and lower a drill string 108. Hoist 106 suspends a top drive 110 suitable for rotating drill string 108 and lowering drill string 108 through a well head 112. Notably, drill string 108 may include sensors or other instrumentation for detecting and logging nearby characteristics and conditions of the wellbore and surrounding formation.

[0020] In operation, a top drive 1 10 supports and rotates drill string 108 as it is lowered through well head 112. In this fashion, drill string 108 (and/or a downhole motor) rotate a drill bit 114 coupled to a lower end of drill string 108 to create a borehole 116 through various subsurface formations. A pump 120 circulates drilling fluid through a supply pipe 122 to top drive 110, down through an interior of drill string 108, through orifices in drill bit 114, back to the surface via an annulus around drill string 108, and into a retention pit 124. The drilling fluid transports cuttings from wellbore 116 into retention pit 124 and helps maintain wellbore integrity. Various materials can be used for drilling fluid, including oil-based fluids and water-based fluids.

[0021] As shown, drill bit 114 forms part of a bottom hole assembly 150, which further includes drill collars (e.g., thick-walled steel pipe) that provide weight and rigidity to aid drilling processes. Detection tools 126 and a telemetry sub 128 are coupled to or integrated with one or more drilling collars.

[0022] Detection tools 126 may gather MWD survey data or other data and may include various types of electronic sensors, transmitters, receivers, hardware, software, and/or additional interface circuitry for generating, transmitting, and detecting signals (e.g., sonic waves, etc.), storing information (e.g., log data), communicating with additional equipment (e.g., surface equipment, processors, memory, clocks, input/output circuitry, etc.), and the like. In particular, detection tools 126 can measure data such as position, orientation, weight-on-bit, strains, movements, borehole diameter, resistivity, drilling tool orientation, which may be specified in terms of a tool face angle (rotational orientation), an inclination angle (the slope), and compass direction, each of which can be derived from measurements by sensors (e.g., magnetometers, inclinometers, and/or accelerometers, though other sensor types such as gyroscopes, etc.).

[0023] Telemetry sub 128 communicates with detection tools 126 and transmits telemetry data to surface equipment (e.g., via mud pulse telemetry). For example, telemetry sub 128 can include a transmitter to modulate resistance of drilling fluid flow thereby generating pressure pulses that propagate along the fluid stream at the speed of sound to the surface. One or more pressure transducers 132 operatively convert the pressure pulses into electrical signal(s) for a signal digitizer 134. It is appreciated other forms of telemetry such as acoustic, electromagnetic, telemetry via wired drill pipe, and the like may also be used to communicate signals between downhole drilling tools and signal digitizer 134. Further, it is appreciated telemetry sub 128 can store detected and logged data for later retrieval at the surface when bottom hole assembly 150 is recovered.

[0024] Digitizer 134 converts the pressure pulses into a digital signal and sends the digital signal over a communication link to a computing system 137 or some other form of a data processing device. In at least some embodiments, computer system 137 includes processing units to analyze collected data and/or perform other operations by executing software or instructions obtained from a local or remote non-transitory computer-readable medium. As shown, computer system 137 includes input device(s) (e.g., a keyboard, mouse, touchpad, etc.) as well as output device(s) (e.g., monitors, printers, etc.). These input/output devices provide a user interface that enables an operator to interact and communicate with the borehole assembly ISO, surface/downhole directional drilling components, and/or software executed by computer system 137.

[0025] For example, computer system 137 enables an operator to select or program directional drilling options, review or adjust types of data collected, modify values derived from the collected data (e.g., measured bit position, estimated bit position, bit force, bit force disturbance, rock mechanics, etc.), adjust borehole assembly dynamics model parameters, generate drilling status charts, waypoints, a desired borehole path, an estimated borehole path, and/or to perform other tasks. In at least some embodiments, the directional drilling performed by borehole assembly 150 is based on a surface and/or downhole feedback loops, as discussed in greater detail below.

[0026] MWD system 100 also includes a controller 152 that instructs or steers bottom hole assembly 150 as drill bit 114 extends wellbore 116 along a desired path 119 (e.g., within one or more boundaries 140). Controller 1S2 includes processors, sensors, and other hardware/software such as a rotary steerable system (RSS). In operation, controller 152 applies a force to flex or bend a drilling shaft coupled to bottom hole assembly 150 thereby imparting an angular deviation to a current the direction traversed by drill bit 114. Notably, controller 152 can communicate real-time data with one or more components of bottom hole assembly 150 and/or surface equipment. In this fashion, controller 152 can analyze real-time data and generate steering signals according to, for example, the feedback control techniques discussed herein. While controller 152 represents one type of directional steering system, it is appreciated other steering mechanisms such as steering vanes, a bent sub, and the like, may also be employed.

[0027] It is further appreciated, the environment shown in FIG. 1 is provided for purposes of discussion only, not for purposes of limitation. The detection tools, drilling devices, and curvature-based feedback control techniques discussed herein may be suitable in any number of drilling environments.

[0028] FIG. 2 is a block diagram of an exemplary device 200, which can include controller 152 (or components thereof)- Device 200 is configured to perform the curvature-based feedback control techniques discussed herein and communicates signals that steer or direct the drilling tool along a well path. In operation, device 200 communicates with one or more of the above- discussed borehole assembly 150 components and may also be configured to communication with remote devices/systems such as computer system 137.

[0029] As shown, device 200 includes hardware and software components such as network interfaces 210, at least one processor 220, sensors 260 and a memory 240 interconnected by a system bus 250. Network interface(2) 210 include mechanical, electrical, and signaling circuitry for communicating data over communication links, which may include wired or wireless communication links. Network interfaces 210 are configured to transmit and/or receive data using a variety of different communication protocols, as will be understood by those skilled in the art. [0030] Processor 220 represents a digital signal processor (e.g., a microprocessor, a microcontroller, or a fixed-logic processor, etc.) configured to execute instructions or logic to perform tasks in a wellbore environment. Processor 220 may include a general purpose processor, special-purpose processor (where software instructions are incorporated into the processor), a state machine, application specific integrated circuit (ASIC), a programmable gate array (PGA) including a field PGA, an individual component, a distributed group of processors, and the like. Processor 220 typically operates in conjunction with shared or dedicated hardware, including but not limited to, hardware capable of executing software and hardware. For example, processor 220 may include elements or logic adapted to execute software programs and manipulate data structures 245, which may reside in memory 240.

[0031] Sensors 260 typically operate in conjunction with processor 220 to perform wellbore measurements, and can include special-purpose processors, detectors, transmitters, receivers, and the like. In this fashion, sensors 260 may include hardware/software for generating, transmitting, receiving, detection, logging, and/or sampling magnetic fields, seismic activity, and/or acoustic waves.

[0032] Memory 240 comprises a plurality of storage locations that are addressable by processor 220 for storing software programs and data structures 245 associated with the embodiments described herein. An operating system 242, portions of which are typically resident in memory 240 and executed by processor 220, functionally organizes the device by, inter alia, invoking operations in support of software processes and/or services executing on device 200. These software processes and/or services may comprise an illustrative "curvature-based feedback control" process/service 244, as described herein. Note that while process/service 244 is shown in centralized memory 240, some embodiments provide for these processes/services to be operated in a distributed computing network.

[0033] It will be apparent to those skilled in the art that other processor and memory types, including various computer-readable media, may be used to store and execute program instructions pertaining to the borehole evaluation techniques described herein. Also, while the description illustrates various processes, it is expressly contemplated that various processes may be embodied as modules having portions of the curvature-based feedback control process 244 encoded thereon. In this fashion, the program modules may be encoded in one or more tangible computer readable storage media for execution, such as with fixed logic or programmable logic (e.g., software/computer instructions executed by a processor, and any processor may be a programmable processor, programmable digital logic such as field programmable gate arrays or an ASIC that comprises fixed digital logic. In general, any process logic may be embodied in processor 220 or computer readable medium encoded with instructions for execution by processor 220 that, when executed by the processor, are operable to cause the processor to perform the functions described herein.

[0034] As mentioned, the curvature-based feedback control techniques address challenges in accurately controlling directional drilling and steering a drilling tool along a planned well path for a predefined well plan. Well plans can be described by a 3D path in an earth formation and defined by a collection of waypoints. Generally, each waypoint typically corresponds to a position in the 3D space, and possibly, higher order information about the path at the specified location. For example, in this context, a 3D waypoint may take the form of:

represent first derivatives of the well

plan with respect to a path length coordinate associated with the well plan, and

represent second derivatives of the well plan with respect to the path length coordinate associated with the well plan. Notably, attitude information, which can include inclination and azimuth, is typically defined as part of the well plan, or it may also be inferred based on known interpolation schemes for smoothly interpolating multiple waypoints. In addition, may be optionally included as part of the definition of a waypoint.

[0035] FIG. 3 is a schematic diagram of a 3D wellbore environment 300, showing a drilling tool 305 creating a wellbore path that substantially follows a well path 310, which is defined by a collection of waypoints, labeled as Notably, each waypoint may include higher order information (e.g., derivatives) such as a steering angle or attitude angle φ (e.g., labeled as " Φ ₁ " through " Φ ₆ "). Wellbore environment 300 represents an ideal environment where drilling tool 305 creates a stable wellbore path that accurately tracks well path 310. In real-world environments, however, the wellbore path may be subject to various instabilities, disturbances, noise, faults, and the like, which may require path correction or adjustment in order to minimize path divergence or deviation.

[0036] Various control techniques may be employed to adjust and conform a current wellbore path of a drilling tool to a planned well path. One example of these control techniques includes an attitude control, which attempts to control a drilling tool's attitude (inclination and azimuth) to minimize wellbore path divergence from the predetermined well plan. However, when a well plan is described by a tool attitude (including inclination and azimuth), and only attitude control is applied for and path correction/convergence on tool attitude relative to the well plan, the actual drilled wellbore path can deviate considerably from the planed well path.

[0037] FIGS. 4 A and 4B provide graphs 401 and 402, respectively, showing wellbore path divergences caused by attitude azimuth correction (graph 401) and attitude inclination or position correction (graph 402). Here, graph 401 illustrates an intended or target well path 405a (dashed line), defined by "target" waypoints actual wellbore

path 405b (solid line) created or traversed by the drilling tool, defined by actual waypoints [X ₁ , y ₁ ], [Χ2, Y2], and [x ₃ ,y ₃ ]. In operation, the drilling tool may include a controller (e.g., controller 152) that performs path tracking and steers the drilling tool through waypoints as it creates the wellbore path. As shown, the controller applies attitude azimuth correction or attitude hold that matches a current attitude for a position on actual wellbore path 405b to a target attitude (inclination) for a corresponding position on the intended well path 405a. Put differently, the controller employs an attitude hold that directs the drill tool to actual positions/actual waypoints so that the drilling tool has the same attitude (inclination) as the corresponding target waypoint (e.g., the inclination of drilling tool at waypoint is the same as the target inclination at waypoint Although attitude hold controls ensure attitude convergence between the actual wellbore path and the intended well path, deviations may be present or even increase depending on distances traversed and a complexity of the well plan.

[0038] In FIG. 4B, graph 402 illustrates an intended well path 410a (dashed line) and an actual wellbore path 410b (solid line) when the controller applies position hold controls. Here, both well path 410a and wellbore path 410b are defined by the same waypoints [X ₁ , y ₁ ], [Χ2, Y2], and [x ₃ ,y ₃ ]. In operation, the controller steers the drilling tool along the same waypoints of both paths and matches the target position for each target waypoint. As shown, actual wellbore path 410b represents a position hold control, which directs the drill tool to traverse the target waypoints. While such position hold controls ensure wellbore path 410b substantially traverses each target waypoint, such position hold controls may create oscillating behavior and divergences between intended well path 410a and wellbore path 410b. This oscillation may be caused, in part, by differences between an actual steering angles (labeled as " Φ ₁ " through " Φ ₃ ") of the drill tool and target steering angles (labeled as " Φ _lt " through " φ _6t ") at each waypoint.

[0039] Accordingly, the curvature-based feedback control techniques disclosed herein mitigate and minimize the undesired path divergences shown by graphs 401 and 402, and provide simultaneous convergence for position and attitude with respect to a target well path. In some cases, such simultaneous convergence may incorporate higher order derivatives to provide smooth convergence corrections or adjustments.

[0040] More specifically, a 3D well path can be projected into two perpendicular planes, and represented by a unique curve in each plane. Therefore, without loss of generality, curvature- based feedback control techniques may control the evolution of wellbore in a 2D plane and establish a desired convergence in the 2D plane. Convergence in the 3D space logically follows. For example, the following kinematic equation can represent an arbitrary evolution of wellbore in a 2D plane with Cartesian coordinates {x and y), where s is a path length coordinate (e.g., a curvilinear coordinate defined along the wellbore path), φ is a steering angle, and x is the curvature. When x and y define a vertical plane, φ may be interpreted as inclination when φ e [Ο,ττ]. Notably, in the following equations, φ e (-∞,∞), and the equations can generate arbitrary path with continuous first derivatives in the x-y plane.

Equation 1: x'(s) = cos(0(s))

Equation 2: y'(s) = sin(0(s))

Equation 3: φ'(β) = K(S)

[0041] The above equations uniquely determine a curvature K(S) for curved wellbore path as a function of a current position and attitude. Preferably, the controller (e.g., an RSS force/bending controller, etc.) may continuously compute desired curvature values in a state feedback control law and steers the drilling tool as it generates a wellbore curvature close to the desired curvature values (e.g., by applying the appropriate amount of RSS force and bending, etc.). For example, the state feedback control law may have the following forms:

Equation 4: K(S) = SFB(x(s),y(s) _f x'(s\y'(s),x _d ,y _d ,x _dl y _d )

Equation 5: K(S) = SFB{x{s),y{s^,x'(s),y'{s),x _d ,y _d ,x _dl y _d ' _t x _d '\y _d '\ ...)

[0042] In the alternative (or in addition to the above), the curvature-based feedback control law may compute desired curvature values as a function of sensor outputs/inputs. [0043] Generally, the curvature values represent a curvature of a deflection beam that satisfies position constraint and slope constraints between a current location and a target waypoint.

[0044] For example, FIG. 5 illustrates a graph 500 that shows path convergence using the curvature-based feedback control techniques. Graph 500 provides a deflection beam 505 that represents a curved convergence path for directing drill tool 510 from its current position [x ₀ , y ₀ ] to a desired position and also providing simultaneous attitude (e.g., derivatives of position) convergence such that drill tool 510 traverses desired position [xa.ya] at a desired orientation or attitude φ _α . A current orientation of drill tool 510 at current position [χ ₀ ,γ ₀ ] is represented φ, and derivatives of x and y positions are specified according to:

Equation 6: x' (0) = cos φ , y ' (0) = sin φ

Where a desired location and attitude (waypoint) are represented by

[0045] Deflection beam 505 provides a curved convergence path that intersects current position [x ₀ ,y ₀ ] tangent to current attitude φ and desired position at desired orientation φ _ά . For purposes of illustration and discussion herein, assume the tangent direction of the path at is parallel to the x axis (i.e., φα = 0). However, for non-parallel tangents, another set of x— y coordinates may be determined by rotating the original x-y system to ensure a parallel relation. The coordinate transform may be performed from x-y to x— y to establish equivalent boundary conditions at current and target positions in the x— y domain, as is appreciated by those skilled in the art.

[0046] Applying small angle assumptions for φ, deflection beam 505 may be determined using the Euler-Bernoulli beam equation. For example, the deflection and slope of deflection beam 505 may be given by:

Equation 7: y = ax ³ + bx ² + cx + d

Equation 8: y' = 3ax ² + 2bx + c

Where a, b,c,d are constants in the Euler-Bernoulli beam equation.

[0047] Boundary conditions for x and x _d are given by:

[0048] Constants a, b, c, d may be calculated from:

[0049] Accordingly, a curvature of deflection beam SOS, at current position [x ₀ ,yo] is given

[0050] The above calculations represent a non-linear state feedback law, which is singular when L = 0, for which κ is not bounded. Accordingly, in certain instances, when x is very close proximity or distance to x _d and y and y' has not converged to the desired value yet, a large or steep curvature value is needed for path convergence with respect to both position and attitude. Preferably, however, when x is sufficiently close to x _d (e.g., x is within a threshold distance from x _d ) the current target waypoint may be assigned to a "next" target waypoint on the planned path. For example, the next or subsequent waypoint on the planned path may be selected when x (a current position) is within a threshold distance of x _d and/or a curvature value for the drilling tool to pass proximate (or through) x _d is above/below a threshold tolerance, and the like. Alternatively (or in addition), the "next" target waypoint may continuously move along the planned path as the drill tool moves forward to avoid any steep curvatures and minimize potential oscillations.

[0051] With respect to three dimension (3D) coordinates, the waypoint can be selected based on:

Where X _c = (x _c , yc _> ^z c) ^{ls the} current position, and [Xp(.s),y _p (s),z _p (sy] defines the planned path, s is depth, and s _c denotes the depth at which the position of the well plan is closest to the current position.

[0052] Equation 13 identifies a target position [x _p , γp,, Ζρ], and derivatives of the target position correspond to a target attitude. If a curvature value for a curved path from the current position to the target position is larger than a threshold, τ is increased. Equations 12 and 13 may be iteratively calculated as the drilling tool moves forward. [0053] The above curvature calculations describe curvature values based on the Euler-Bernoulli beam equation, however such curvature calculations are provided for purposes of example, not limitation, and it is appreciated any suitable curved path equation may be used.

[0054] As discussed above, the curvature-based feedback control techniques employed by a controller that performs path tracking for a drilling tool such as drilling tool 510. In operation, the controller for drilling tool 510 tracks a current position and a current attitude (ø) and determines a curvature value (K(S)) for a curved path (deflection beam 505) that intersects the current position, tangent the current attitude, and a curvilinear or target position (

or substantially proximate to a target wellbore path, tangent to a target attitude (φ _α ). Here, φ _α at the curvilinear position is parallel to the x axis (i.e., ø _d = 0). The controller may further provide the curvature value to force/bending hardware to generate a curved wellbore path for drilling tool 510.

[0055] Drilling tool 510 further updates its current position/attitude and re-calculates the curvature values to adjust the curved wellbore path (and correct for disturbances, noise, etc.). In this fashion, drilling tool 510 may employ a feedback control loop to continuously and iteratively compute new curvature values to ensure substantial path convergence, which minimizes deviations from faults, noise, seismic activity, or other disturbances.

[0056] In addition, drilling tool 510 may also update curvilinear position ([½,)¾]) on the target well path to avoid oscillating behavior. For example, drilling tool 510 may update the curvilinear position based on a threshold distance or threshold proximity between drilling tool 510 and the curvilinear position in order to avoid steep curvatures that violate drilling constraints (e.g., dogleg severity constraints (DLS), etc.). Further, the target curvilinear position may also be continuously updated and assigned to a new position on or substantially proximate to the well path (e.g., when drilling tool 510 updates its current position). This new position may include a "next" waypoint position and/or it may include any number of other positions on the well path.

[0057] Moreover, while FIG. 5 illustrates a curved path, shown by deflection beam 505, that converges or intersects the target well path at target curvilinear position, having a target

attitude <p _d = 0, it is appreciated that such view is shown for purposes of discussion, not limitation. Specifically, it is appreciated that convergence or intersection between the curved path and the target well path may not be possible (or even desired) in certain instances. In such instances, the curved path may represent a "best" path having positions that are substantially close or proximate to one or more positions that define the target well path and at a target attitude substantially similar a well path attitude for corresponding positions.

[0058] In other embodiments of this disclosure, x and y coordinates may be described by functions of an independent variable t as follows:

and

[0059] Boundary conditions are related to coefficients of the cubic function as:

[0060] Derivative boundary conditions are given by:

[0061] Coefficients a _x , b _x , c _x , d _x and d _y , b _y , c _y , d _y can be computed from above two equations, and are functions of L and boundary conditions at t = 0 and t = L, and L may be given by:

[0062] Second derivatives of x and y can be calculated based on the coefficients and the curvature value for a curved convergence path may be determined by:

[0063] Notably, higher order polynomials of an independent variable may also be used for generating curvature feedback as is appreciated by those skilled in the art.

[0064] For three-dimensional (3D) coordinates, a curvature value may be directly calculated based on state/output feedback. Such curvature value is further used to determine a magnitude and an orientation of the drilling tool actuation. For example, using 3D Cartesian coordinates x, y and z, where the current position of the tool is defined by γ = (x, y, z), the curvature κ may be given by:

Where x', y", y', x", z', z" are calculated based on the current position and attitude of the tool and the desired waypoint (position, attitude, and/or higher order derivatives of the path).

[0065] Generally, a normal direction in 3D space is used for determining a direction for generating the curvature value. For example, the normal direction for applying the curvature is given by the following vector: E

[0066] Notably, both the normal direction and the curvature value are used to instruct drilling tool actuation, as is appreciated by those skilled in the art.

[0067] FIG. 6 is an exemplary simplified procedure 600 for directional drilling based on the curvature-based feedback techniques discussed above. Procedure 600 begins at step 605 and continues on to step 610 where, as discussed, a target wellbore path is defined by a plurality of target waypoints. A drilling tool drills executes the curvature-based feedback techniques to drill a wellbore path that substantially conforms to the target wellbore path. In particular, procedure 600 continues to step 615 where the drilling tool tracks its current position and its current attitude. For example, the drilling tool may be configured with one or more sensors, modules, and/or other hardware/software that communicate with each other to determine and track its position and attitude.

[0068] Drilling tool further determines, in step 620, a curvature value for a curved path that intersects the current position, tangent the current attitude, and a curvilinear position (e.g., a target waypoint, etc.) on the target wellbore path, tangent to a target attitude (which tangent may be parallel to an axis of the target wellbore path).

[0069] The drilling tool may be instructed, at step 62S, to generate a wellbore path according to the curvature value, which can cause its controller (e.g., RSS force/bending controller) to actuate or bend a drilling shaft shaft thereby steering the drilling tool along the curved path. Notably, the curvature value may be constrained by thresholds to prevent violations of predefined drilling constraints (e.g., dogleg constraints). In such embodiments, a new position may be assigned as the target position (e.g., a position on the wellbore path further from the drilling tool) and a new curvature value may be calculated and compared against the threshold(s). This process may continue until the new curvature value falls within certain thresholds to provide smooth path convergence. Alternatively (or in addition), derivatives of the current/target positions may be used to also facilitate smooth path convergence.

[0070] Procedure 600 also provides a feedback loop, shown at step 635where the drilling tool updates its current position/attitude based on movement or change in the current position/attitude. From step 635, procedure 600 iteratively repeats steps 620 through step 635 while the drilling tool generates its wellbore path. In this fashion, the drilling tool continuously receives position/attitude information, calculates appropriate curvature values for curved path convergence, and adjusts its drilling direction/movement. Procedure 600 may subsequently end at step 640, but as discussed, it may begin again at step 615 where, as discussed above, the drilling tool tracks its current position/attitude.

[0071] It should be noted that certain steps within procedure 600 may be optional, and further, the steps shown in FIG. 6 are merely examples for illustration - certain other steps may be included or excluded as desired. Further, while a particular order of the steps is shown, this ordering is merely illustrative, and any suitable arrangement of the steps may be utilized without departing from the scope of the embodiments herein.

[0072] Collectively, FIGS. 7, 8, and 9 each provide a pair of graphs showing simulated wellbore paths (e.g., 100 simulations) and curvature values for a drilling tool as it moves from an initial position at an initial attitude to a target position having a target attitude. Notably, the simulated wellbore paths demonstrate robust features of the curvature-based feedback techniques discussed herein and show continuous and iterative changes to the curvature values for each estimated path as the drilling tool is subjected to various disturbances (e.g., formation changes, etc.). [0073] FIG. 7 illustrates graphs 701 and 702, FIG. 8 illustrates graphs 801 and 802, and FIG. 9 illustrates graphs 901 and 902. For each of these graphs, the drilling tool begins at the same initial position and the same initial attitude and ends at the same final position, but with different target attitudes. In particular, in graphs 701 and 702, Φ(0)=0; 0 _d =-pi/2; Δκ e [-0.5,0.5]; in graphs 801 and 802, 0(O)=O; φ _ά = -pi/3; and in graphs 901 and 902, Φ(0)=0; φ _α = -pi/3.

[0074] Graphs 701, 801, and 901 each illustrate simulated curvature values for estaimated curved wellbore paths produced or traversed by the drilling tool from the initial position to the final position, while corresponding graphs 702, 802, and 902 each illustrate a continuously computed curvature value K(S) using the curvature-feedback calculations discussed above (e.g., based on the drilling tool's current position, current altitude, as well as the target position and target attitude, etc.). The drilling tool is subjected to various disturbances as it traverses the wellbore path and it continuously adjusts the curvature values, shown by graphs 702, 802, and 902, in order to provide the smoothly curved wellbore path shown in graphs 701, 801, and 901. Notably, the actual curvatures for the curved wellbore paths may be represented by K(S) + Δκ, where Δκ is a random number or uncertainty that may result from various sources (e.g., measurement noise or inaccuracies, faulty tool responses, estimation errors for χ,γ,φ, and the like).

[0075] These curved wellbore paths shown in graphs 701, 801, and 901 provide a curved convergence path for the drilling tool as it moves toward the final position/attitude. Notably, this final position/attitude may represent a target waypoint, or some other curvilinear position on a planned wellbore path. Moreover, these curved convergence paths demonstrate substantially simultaneous convergence between the actual wellbore path and the planned wellbore path, both in terms of position and attitude (e.g., inclination and azimuth), which demonstrates a robustness and an accuracy of the curvature-based feedback operations despite the various disturbances applied to the drilling tool.

[0076] FIGS. 10 and 11 illustrate a pair of graph 1001/1002 and 1101/1102, respectively, showing path convergence when Δκ is assigned a constant value (e.g., a constant drift), rather than the random uncertainty (e.g., ref. FIGS. 7-9). In particular, Δκ = 0.1 for graphs 1001 and 1002 and Δκ = 0.25 for graphs 1101 and 1102. Graphs 1001 and 1101 particularly illustrate actual curvatures of a curved wellbore path produced by the drilling tool as it travels from the initial position to the final/target position, and corresponding graphs 1002 and 1102 illustrate the continuously computed curvature values K(S) as the drilling tool accounts for disturbances/uncertainty along the curved wellbore path.

[0077] FIGS. 12, 13, and 14 illustrate a pair of graphs -here, 1201 and 1202, 1301 and 1302, and 1401 and 1402, respectively- showing position and attitude convergence when a proportional uncertainty is added to the curvature value. Graphs 1201, 1301, and 1401 illustrate actual curvatures of a curved wellbore path produced by the drilling tool as it travels from the initial position to the final/target position, and corresponding graphs 1202, 1302, and 1402 illustrate the continuously computed curvature values K(S) as the drilling tool accounts for disturbances/uncertainty along the curved wellbore path. Regarding the proportional uncertainty the drilling tool produces an actually curvature (l+q)K(s), where q£(-l ,∞), instead of producing a desired curvature K(S). Notably, for graphs 1201 and 1202, q=-0.5; graphs 1301 and 1302, q=-2; and graphs 1401 and 1402, q=10.

[0078] While there have been shown and described illustrative embodiments for curvature-based feedback controls that provide simultaneous convergence for positions and attitudes between an actual wellbore path and a planned well path, , it is to be understood that various other adaptations and modifications may be made within the spirit and scope of the embodiments herein. For example, the embodiments have been shown and described herein with curvature values determined by specific deflection beam equations. However, the embodiments in their broader sense are not as limited, and may, in fact, be used with any curved path equations. In addition, the embodiments are shown with certain devices/modules performing certain operations, however, it is appreciated that various other sensors/devices may be readily modified to perform operations without departing from the sprit and scope of this disclosure.

[0079] The foregoing description has been directed to specific embodiments. It will be apparent, however, that other variations and modifications may be made to the described embodiments, with the attainment of some or all of their advantages. For instance, it is expressly contemplated that the components and/or elements described herein can be implemented as software being stored on a tangible (non-transitory) computer-readable medium, devices, and memories (e.g., disks/CDs/RAM/EEPROM/etc.) having program instructions executing on a computer, hardware, firmware, or a combination thereof. Further, methods describing the various functions and techniques described herein can be implemented using computer-executable instructions that are stored or otherwise available from computer readable media. Such instructions can comprise, for example, instructions and data which cause or otherwise configure a general purpose computer, special purpose computer, or special purpose processing device to perform a certain function or group of functions. Portions of computer resources used can be accessible over a network. The computer executable instructions may be, for example, binaries, intermediate format instructions such as assembly language, firmware, or source code. Examples of computer-readable media that may be used to store instructions, information used, and/or information created during methods according to described examples include magnetic or optical disks, flash memory, USB devices provided with non-volatile memory, networked storage devices, and so on. In addition, devices implementing methods according to these disclosures can comprise hardware, firmware and/or software, and can take any of a variety of form factors. Typical examples of such form factors include laptops, smart phones, small form factor personal computers, personal digital assistants, and so on. Functionality described herein also can be embodied in peripherals or add-in cards. Such functionality can also be implemented on a circuit board among different chips or different processes executing in a single device, by way of further example. Instructions, media for conveying such instructions, computing resources for executing them, and other structures for supporting such computing resources are means for providing the functions described in these disclosures. Accordingly this description is to be taken only by way of example and not to otherwise limit the scope of the embodiments herein. Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the embodiments herein.