Intelligent data acquisition for gyrocompass applications

Abstract

A method for performing gyroscopic azimuth angle measurements includes estimating a pitch angle and a roll angle of a gyroscopic inclinometer in a wellbore; determining a measurement duration for each of the plurality of gyroscope measurements from the estimated pitch and roll angles; making each of the plurality of gyroscopic measurements at the determined measurement duration when the gyroscopes are disposed at a corresponding plurality of rotational positions in the tool housing; and calculating an azimuth angle of the wellbore from the plurality of gyroscopic measurements.

Description

Cross-reference paragraphs

[0001] This application claims the benefit of U.S. Provisional Application No. 18 / 453,369, filed August 22, 2023, entitled “SMART DATA ACQUISITION FORGYROCOMPASS APPLICATIONS,” the disclosure of which is incorporated herein by reference. Technical Field

[0002] The disclosed implementation scheme generally relates to downhole gyroscopic surveying methods, and more specifically to an intelligent data gyroscopic data acquisition method for improving the accuracy of downhole surveying operations. Background Technology

[0003] In conventional drilling and measurement-while-drilling (MWD) operations, wellbore inclination and azimuth are typically measured at multiple longitudinally discrete points along the wellbore axis. These discrete measurements are combined to form a wellbore inclination and used to calculate the three-dimensional well path (e.g., using the minimum curvature assumption or other curvature assumptions). Wellbore inclination is typically derived (calculated) from measurements of the Earth's gravitational field using accelerometers. Wellbore azimuth can be obtained by combining measurements of the Earth's gravitational and magnetic fields using accelerometers and magnetometers.

[0004] In some drilling operations, magnetic interference from nearby wells or magnetic ore bodies can cause errors in the calculated magnetic azimuth. The wellbore azimuth can also be obtained (calculated) by combining accelerometer measurements with gyroscopic measurements of the Earth's rotation. While this gyroscopic azimuth measurement method is commercially available, there is still room for improvement, such as increasing measurement accuracy. Summary of the Invention

[0005] A method and system for measuring gyro azimuth angles are disclosed. An exemplary method includes: deploying a gyro-based inclinometer tool in a wellbore, the gyro-based inclinometer tool including at least one gyroscope configured to rotate within and relative to a tool housing; estimating the pitch and roll angles of the gyro-based inclinometer tool in the wellbore; determining a measurement duration for each of a plurality of gyro measurements based on the estimated pitch and roll angles; performing each of the plurality of gyro measurements for the determined measurement duration while the gyroscope is positioned at corresponding plurality of rotational positions within the tool housing; and calculating the azimuth angle of the wellbore based on the plurality of gyro measurements.

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

[0007] To gain a more complete understanding of the disclosed subject matter and its advantages, reference is now made to the following description taken in conjunction with the accompanying drawings: Figure 1 A drilling rig is depicted, which includes an exemplary gyro-based inclinometer tool.

[0008] Figure 2 This is a cross-section of an exemplary gyroscope inclinometer tool.

[0009] Figure 3A and Figure 3B (Collectively referred to as Figure 3) depicts the exemplary coordinate system used in this paper.

[0010] Figure 4A , Figure 4B and Figure 4C (Collectively referred to as Figure 4) depicts a contour map showing the sensitivity of the azimuth angle to changes in the X, Y, and Z axis gyroscope measurements.

[0011] Figure 5 A flowchart is depicted for an exemplary method for performing gyro inclinometer measurements.

[0012] Figure 6 A flowchart is depicted for an exemplary method for determining the acquisition time of each of a plurality of gyroscope measurements. Detailed Implementation

[0013] The disclosed method for performing gyro-based inclinometer measurements includes: deploying a gyro-based inclinometer tool in a wellbore, the gyro-based inclinometer tool including at least one gyroscope configured to rotate within and relative to the tool housing; estimating the pitch and roll angles of the gyro-based inclinometer tool in the wellbore; determining a measurement duration for each of a plurality of gyroscope measurements based on the estimated pitch and roll angles; performing each of the plurality of gyroscope measurements for the determined measurement duration while the gyroscope is positioned at corresponding plurality of rotational positions within the tool housing; and calculating the azimuth angle of the wellbore based on the plurality of gyroscope measurements.

[0014] The exemplary embodiments disclosed herein offer several technical advantages and improvements over the prior art. For example, the disclosed embodiments can provide more accurate gyro azimuth measurements during drilling operations. Furthermore, the disclosed embodiments advantageously enable efficient adjustment of gyroscope measurements throughout the drilling operation, thereby enabling high-precision gyro azimuth measurements throughout the entire drilling operation.

[0015] Figure 1 A drilling rig 10 is depicted suitable for implementing the various methodologies disclosed herein. A semi-submersible drilling platform 12 is positioned above an oil or gas layer located below the seabed 16. A subsea guide pipe 18 extends from the deck 20 of the platform 12 to the wellhead assembly 22. The platform may include a derrick and lifting equipment for raising and lowering a drill string 30, as shown, which extends into the wellbore 40 and includes a drill bit 32 and a gyro-guided directional drilling tool 50. The drill string 30 may also include a downhole drilling motor, a downhole telemetry system, a rotary steerable tool, and one or more other measurement-while-drilling (MWD) or logging-while-drilling (LWD) tools, which include various sensors for sensing downhole characteristics of the wellbore and surrounding formation. The disclosed embodiments are not limited in these respects.

[0016] As described in more detail below, exemplary embodiments of the gyro-based inclinometer tool 50 may include a gyro sensor device with a single gyro sensor or a gyro sensor device with two gyro sensors (a dual-sensor device). The gyro sensor may include virtually any gyro sensor suitable for measurement in a drill string, such as a microelectromechanical system (MEMS) gyroscope. As is known to those skilled in the art, such MEMS gyroscopes are manufactured using integrated circuit fabrication technology and are widely used in a variety of fields, including spacecraft, aircraft, underwater devices, motor vehicles, gaming devices, and smartphones.

[0017] Those skilled in the art will understand that Figure 1 The deployment shown is merely an example. It will be further understood that the disclosed implementation is not limited to, for example, [examples of deployments]. Figure 1 The semi-submersible platform 12 shown is used in conjunction with this embodiment. The disclosed embodiment is equally well suited for any type of underground drilling operation, whether offshore or onshore.

[0018] Figure 2 An exemplary embodiment of a gyroscope inclinometer tool 50 is depicted. In the depicted exemplary embodiment, a gyroscope sensor housing 60 is configured to rotate within a tool body 52. ​​The sensor housing 60 may include one or two gyroscopes 62 (e.g., inductive MEMS gyroscopes) deployed therein. One or more accelerometers 64 (with the measurement axis of at least one accelerometer aligned with the measurement axis of the gyroscope 62) may also be deployed on or within the gyroscope sensor housing 60.

[0019] The gyro inclinometer tool 50 also includes an electric motor 70 (e.g., a stepper motor) deployed in the tool body 52 and configured to sequentially rotate the gyroscope sensor housing 60 (and the gyroscope 62) through multiple measurement positions (e.g., three or six measurement positions). An electronic controller 80 may also be deployed in the tool body (or another location on the drill string) and configured to control the operation of the electric motor 70 and the gyroscope 62. As described in more detail below, the controller may also be configured to perform the disclosed method for performing gyro inclinometer measurements.

[0020] The gyro-based inclinometer tool 50 can be advantageously used for static inclinometer measurements (e.g., wellbore azimuth measurements) in the wellbore. Static inclinometer measurements are typically performed after drilling has been temporarily halted (e.g., when a new section of drill pipe is added to the drill string) and the drill bit has been lifted off the bottom. Such static measurements are typically performed at measurement depth intervals ranging from approximately 30 feet to approximately 90 feet (e.g., approximately 10 meters to approximately 30 meters) during drilling operations. During static inclinometer operations, the gyroscope sensor housing 60 (and the gyroscope 62) can be rotated sequentially (using the motor 70) to multiple different rotation orientations (e.g., rotating to two or four orientations in an embodiment including a dual gyroscope sensor unit, or rotating to three or six orientations in an embodiment including a single gyroscope). Gyroscope measurements can be performed at each rotation orientation. Accelerometer measurements can also be performed at each rotation orientation. The gyroscope and accelerometer measurements can then be combined and processed to calculate the wellbore azimuth.

[0021] Figure 3A and Figure 3B(Collectively referred to as Figure 3) illustrates an exemplary coordinate system used to represent gyro inclinometer measurements. For simplicity, the industry typically uses the Global North-East Ground (NED) coordinate system (where North and East refer to the north and east directions in the horizontal plane at the Earth's surface, and "down" refers to the direction directly pointing towards the Earth's center of gravity). A commonly used tool coordinate system is also depicted, comprising an x-axis, y-axis, and z-axis, where the x-axis coincides with the longitudinal axis of the measuring tool and points downhole. The y-axis and z-axis are orthogonal to the x-axis and to each other, and together define the transverse YZ plane. As further illustrated, rotations about the x-axis, y-axis, and z-axis are referred to herein as the roll angle (φ), pitch angle (θ), and azimuth angle (ψ), respectively. The azimuth angle is defined as the angle between the projection of the Earth's rotation vector onto the horizontal plane and the projection along the x-direction (tool axis). The pitch angle is defined as the angle between the x-axis and the horizontal plane, while the roll angle is defined as the angle between the y-axis and the horizontal plane in the YZ plane. It should be understood that the roll angle φ is also known in the industry as the tool face angle, while the pitch angle θ is related to the tilt angle (the pitch angle is defined as the tilt angle minus ninety degrees). Those skilled in the art should readily understand that mathematical transformations can be used to convert measurements from one coordinate system to another, for example, from the depicted tool coordinate system to the NED coordinate system.

[0022] Continue to refer to Figure 3 and refer again. Figure 2 It should be understood that the multiple rotational orientations used for gyroscope measurements do not necessarily coincide with the x, y, and z axes that define the tool coordinate system. In an exemplary embodiment, multiple rotational orientations can be selected to define an alternative tool coordinate system with u, v, and w axes. Measurements performed in this UVW coordinate system can be easily converted to the tool coordinate system (XYZ coordinate system) using mathematical transformations known to those skilled in the art.

[0023] In a non-limiting embodiment, the gyroscope inclinometer 50 can be configured to perform six gyroscope measurements along the +u, -u, +v, -v, +w, and -w axes (directions) in the UVW coordinate system. These six axes can be provided by rotating the gyroscope sensor housing 60 in the YZ plane at 60-degree intervals. More specifically, when the +u axis is orthogonally projected onto the YZ plane, it is tilted 60° relative to the -z axis. When the -u axis is orthogonally projected onto the YZ plane, it is tilted 60° relative to the +z axis. When the +v axis is orthogonally projected onto the YZ plane, it is tilted 60° relative to the -z axis. When the -v axis is orthogonally projected onto the YZ plane, it is tilted 60° relative to the +z axis. When the +w axis is orthogonally projected onto the YZ plane, it coincides with the +z axis. When the -w axis is orthogonally projected onto the YZ plane, it coincides with the -z axis. Further details regarding this exemplary gyro inclinometer implementation can be found in U.S. Patents 8,528,220 and 10,982,530.

[0024] It should be understood that to calculate or estimate the azimuth angle, at least three separate (independent) measurements of gravitational acceleration and Earth's rotational angular velocity are required. When these measurements are taken in the direction of the tool (XYZ) coordinate system, the gyroscope measurements are used... This indicates that the accelerometer measurement value is used... This indicates that measurements performed in the UVW coordinate system are also represented. and This indicates that measurements from gyroscopes and accelerometers can be taken in any independent coordinate system (e.g., the UVW coordinate system) and then transformed to the tool coordinate system (XYZ coordinate system) using a rotation matrix. Such mathematical techniques are well known to those skilled in the art.

[0025] The coordinate transformation matrix from the NED coordinate system to the XYZ coordinate system is represented based on the Euler angle definition. Since rotations defined by Euler angles should be described in the order of azimuth, pitch, and roll, the coordinate transformation matrix... Become By definition, the direction of Earth's gravity (g) is parallel to the downward direction of NED, such that: therefore: The Earth's rotation component (Ω) can be described by the following matrix relationship: in Indicates the latitude of the measurement (or observation), and This represents the angular velocity of the Earth's rotation (e.g., (degrees / hour).

[0026] In the XYZ (tool) coordinate system, the angular velocity of the Earth's rotation can be given, for example, as follows: Based on the above, azimuth angle The calculation can be performed using one or more of the following three independent equations, where , , and Techniques known to a person of ordinary skill can be used, and as in Equation 1 Determined based on accelerometer measurements as indicated above.

[0027] As can be seen from the above equations, the sensitivity of the calculated azimuth angle clearly depends on the pitch and roll angles of the gyro inclinometer tool (e.g., tool 50). Since each measurement provided by the accelerometer has a certain degree of noise, the measurement error of each accelerometer measurement can be attenuated or amplified according to trigonometric functions related to the pitch and roll angles. For example, in In the equation, when the roll angle At that time, the azimuth angle is opposite to Very sensitive, and to Insensitive (or with very low sensitivity), because and Similarly, when the roll angle At that time, the azimuth angle is opposite to Very sensitive, and to Insensitive (or with very low sensitivity).

[0028] Figure 4A , Figure 4B and Figure 4C (Collectively referred to as Figure 4) depicts a contour map, showing the azimuth pairs based on Equation 5. (4A) (4B) and (4C) Sensitivity to changes. In these exemplary contour plots, sensitivity is defined as the azimuth angle versus the gyroscope measurement component. , or The partial derivatives are as indicated. Note that sensitivity is clearly related to both pitch and roll angles. For example, azimuth to... The absolute value of the sensitivity to change is independent of the roll angle, and reaches its maximum value at a pitch angle of zero and at a pitch angle of... The azimuth is close to zero at degrees. and The sensitivity of is mathematically more complex.

[0029] In exemplary embodiments where the gyro inclinometer tool has one or two different gyro sensors (i.e., using one or two different gyro sensors to obtain at least three independent measurements), the total acquisition duration (time) must be allocated to accommodate measurements in three (or more) different measurement directions (e.g., along the six axes mentioned above). Traditionally, the total acquisition time is evenly distributed, resulting in gyro measurements having the same noise level along each measurement axis. One aspect of the disclosed embodiments is the recognition that the accuracy of the calculated azimuth angle can be improved by modifying the acquisition duration of each gyro measurement. In particular, it is recognized that more sensitive gyro measurement components (e.g., The sampling time of ) can be longer than that of gyroscope measurement components with lower sensitivity (e.g., The length is increased to reduce noise in the most sensitive components. It is further recognized that the specific components with high or low sensitivity depend on the pitch and roll angles of the gyro measurement tool, and may also depend on the azimuth and wellbore latitude of the measurement.

[0030] Figure 5 A flowchart of an exemplary method 100 for performing gyro-based inclinometer measurements (e.g., azimuth measurements) is depicted. At 102, accelerometer measurements (e.g., independent accelerometer measurements taken at other locations on the tool or drill string) are evaluated to estimate the pitch and roll angles of the gyro-based inclinometer tool. The estimated pitch and roll angles can be evaluated at 104 to determine the measurement duration for each gyro measurement (e.g., along each of the three or six axes in the UVW coordinate system). It should be understood that these measurement durations are generally not equal to each other. Although not depicted in this exemplary embodiment, it should be understood that the determination at 104 may further utilize previous (or recent) azimuth measurements and the latitude of the drilling operation (e.g., the latitude of the drilling platform or rig). A gyroscope measurement is then performed at 106 using the measurement duration determined at 104. For example, a first gyro measurement can be performed for a first determined duration when the gyroscope sensor housing is in a first rotational position. The sensor housing can then be rotated to a second position, and a second gyroscope measurement is performed for a second determined time. This process is repeated until six or more gyroscope measurements are completed. A corresponding accelerometer measurement can also be performed while the gyroscope measurement is being performed at position 106. Then, at position 108, these accelerometer and gyroscope measurements can be used to calculate the azimuth angle, for example, using one or more of equations (3)-(5).

[0031] To minimize the overhead associated with the estimation in 102, it may be advantageous to use any other readily available information prior to the survey. For example, many downhole tools use multiple accelerometers / vibration sensors. These sensors are typically used to monitor the vibration conditions of the survey and drilling environment, but can also be used to estimate the pitch and roll angles of the survey tool (with sufficient accuracy to determine the measurement duration of each gyro measurement at 104).

[0032] In an exemplary embodiment, the gyroscope inclinometer tool 50 ( Figure 1 and Figure 2 This may include a triaxial accelerometer array, where two accelerometers include low-cost vibration sensors and one accelerometer is a high-precision accelerometer for inclinometer measurements. Because the high-precision accelerometer is indexed to a specific position just before the inclinometer measurement begins, the three-dimensional gravity vector can be determined, for example, as follows: in and This represents accelerometer measurements obtained from low-cost vibration sensors. This indicates the accelerometer measurement value obtained from a high-precision accelerometer, and This represents the rotation matrix from the coordinate system of the triaxial accelerometer group to the tool (XYZ) coordinate system. The components are related to the alignment position of the high-precision accelerometer. The pitch and roll angles can then be calculated based on the estimated gravity vector, for example, using equation (1).

[0033] Continue to refer to Figure 5 and further reference Figure 6 The flowchart depicts an exemplary method 110 for determining the measurement duration in 104. At 112, the method can be selected to correlate with azimuth and gyroscope measurements. , and One or more of the equations (Equations 3-5). In some exemplary embodiments, the equations can be pre-selected, for example, based on well planning and the expected wellbore azimuth range. In other exemplary embodiments, the equations can be based on the estimated pitch and roll angles obtained in 102, or on previously measured azimuth values. Alternatively, an equation (or system of equations) can be chosen based on another estimate of the expected azimuth value. The wellbore latitude can also be considered, especially for... and Equation. An evaluation at 114 can assess the previously determined azimuth angle. To estimate the gyroscope measurement vector (e.g., , and or , and ).

[0034] At point 116, one or more partial derivatives of the azimuth angle with respect to the gyroscope measurement can be calculated, for example, using the pitch and roll angles obtained at 102 and the estimated gyroscope measurement vector obtained at 114. Such partial derivatives may include, for example... , and or , and Partial derivatives can be calculated using mathematical expressions obtained by differentiating one or more of equations 3-5. Such expressions can take the following forms: or in , and or , and This represents the mathematical equations that relate the partial derivatives to the pitch and roll angles, as well as the gyroscope measurement vectors. It should be understood that the calculations at 116 can alternatively and / or additionally use polynomial (or other) mathematical approximations of these equations, or can be interpolated using a pre-calculated lookup table of partial derivatives.

[0035] Continue to refer to Figure 6 The calculated partial derivative can then be evaluated at 118 to determine the measurement duration for each of the gyroscope measurements (e.g., under each of the aforementioned rotational orientations of the gyroscope). In one exemplary embodiment, the duration can be proportional to the partial derivative calculated at 116. For example, , and The duration of the measurement can be compared with , and proportional (or , and The duration of the measurement can be compared with , and (proportional). This can be expressed mathematically in, for example, the following form: in Indicates the gyroscope measurement components Duration (e.g., , and (duration of measurement) This represents the total time available for gyroscope measurements. Indicates the azimuth angle relative to the gyroscope measurement component. The partial derivatives of, and This represents the sum of the partial derivatives of each component (e.g., , and sum).

[0036] It should be understood that the methods described herein can be configured to be used via one or more controllers deployed downhole (e.g., Figure 2 The controller 80 in the diagram is used for implementation. A suitable controller may include, for example, a programmable processor, such as a digital signal processor or other microprocessor or microcontroller, and processor-readable or computer-readable program code embodying logic. For example, a suitable processor may be used to execute the above-described... Figure 5and Figure 6 The described method implementation (or various steps within the method implementation) may optionally include other controllable components such as sensors (e.g., temperature sensors), data storage devices, power supplies, timers, etc. The controller may also be configured to communicate electronically with accelerometers and magnetometers. A suitable controller may also optionally communicate with other instruments in the drill string (e.g., telemetry systems such as those communicating with the ground). A suitable controller may also optionally include volatile or non-volatile memory or data storage devices.

[0037] Further reference Figure 5 and Figure 6 The methods disclosed herein should be understood to allow for the storage of calculated inclination parameters (e.g., calculated azimuth) in a downhole memory and / or transmission to the surface, for example, via mud pulse telemetry, electromagnetic telemetry, wired drill pipe, or other telemetry technologies. The drilling direction can be changed based on the calculated azimuth. For example, in many drilling operations, the wellbore (or a portion thereof) is drilled according to a drilling plan, such as according to a predetermined direction (e.g., defined by the wellbore inclination and wellbore azimuth) or a predetermined curvature. In some embodiments, the calculated azimuth can be compared to a desired azimuth. For example, the drilling direction can be changed to meet the drilling plan, or when the difference between the calculated and desired azimuth (or the azimuth change) exceeds a predetermined threshold. Such a change in drilling direction can be implemented, for example, via actuation of a guiding element in a directional tool.

[0038] While the intelligent data acquisition for gyrocompass acquisition and some of its advantages have been described in detail, it should be understood that various changes, substitutions and alterations may be made herein without departing from the spirit and scope of this disclosure.

Claims

1. A method for surveying a wellbore, the method comprising: deploying a gyro survey tool in the wellbore, the gyro survey tool including at least one gyroscope configured to rotate in a tool housing and relative to the tool housing; estimating a pitch angle and a roll angle of the gyro survey tool in the wellbore; determining a measurement duration for each of a plurality of gyroscope measurements as a function of the estimated pitch angle and roll angle; taking each of the plurality of gyroscope measurements for the determined measurement duration as the gyroscope is disposed at a corresponding plurality of rotational positions in the tool housing; and calculating an azimuth angle of the wellbore from the plurality of gyroscope measurements.

2. The method of claim 1, wherein the measurement duration for the plurality of gyroscope measurements is proportional to a sensitivity of the azimuth angle to each of the plurality of gyroscope measurements.

3. The method of claim 1, wherein the measurement duration for the plurality of gyroscope measurements are not equal to one another.

4. The method of claim 1, wherein: the gyro survey tool includes a single-axis gyroscope configured to rotate in the tool housing between at least a first rotational position, a second rotational position, and a third rotational position; the determining the measurement duration includes determining the measurement duration for each of at least a first gyroscope measurement, a second gyroscope measurement, and a third gyroscope measurement taken at the at least first rotational position, second rotational position, and third rotational position as a function of the estimated pitch angle and roll angle; and the taking each of the plurality of gyroscope measurements includes taking at least a first gyroscope measurement, a second gyroscope measurement, and a third gyroscope measurement at the at least first rotational position, second rotational position, and third rotational position using the determined duration.

5. The method of claim 1, wherein the pitch angle and the roll angle are calculated from accelerometer measurements.

6. The method of claim 5, wherein the accelerometer measurements are taken using a three-axis accelerometer package including a first vibration sensor and a second vibration sensor and a single high-precision accelerometer.

7. The method of claim 1, wherein the determining the measurement duration includes determining the measurement duration for each of the plurality of gyroscope measurements as a function of the estimated pitch angle and roll angle and a previously calculated azimuth angle.

8. The method of claim 7, wherein the determining the measurement duration further includes: estimating a first component, a second component, and a third component of a gyroscope measurement vector as a function of the previously calculated azimuth angle; evaluating the estimated first component, second component, and third component of the gyroscope measurement vector and the pitch angle and the roll angle to calculate partial derivatives of the azimuth angle with respect to the first component, the second component, and the third component of the gyroscope measurement vector; and determining the measurement duration as a function of the partial derivatives. ​ ​ ​ 9. The method of claim 8, wherein the measurement duration is proportional to the partial derivative.

10. The method of claim 1, wherein the evaluating the gyroscope measurements further comprises evaluating an estimated pitch angle and roll angle or a previously estimated azimuth angle to select one or more equations for calculating the azimuth angle.

11. A gyroscopic survey tool, comprising: at least one gyroscope disposed in a tool housing and configured to rotate relative to the tool housing; and a processor configured to: receive or estimate a pitch angle and a roll angle of the gyroscopic survey tool in a wellbore; determine a measurement duration for each of a plurality of gyroscope measurements from the estimated pitch angle and roll angle; take each of the plurality of gyroscope measurements at the determined measurement duration as the gyroscope is disposed at a corresponding plurality of rotational positions in the tool housing; and calculate an azimuth angle of the wellbore from the plurality of gyroscope measurements.

12. The gyroscopic survey tool of claim 11, wherein: the at least one gyroscope is a single-axis gyroscope configured to rotate in the tool housing between at least a first rotational position, a second rotational position, and a third rotational position; and the processor is configured to: determine the measurement duration for each of at least a first gyroscope measurement, a second gyroscope measurement, and a third gyroscope measurement taken at the at least first rotational position, second rotational position, and third rotational position from the estimated pitch angle and roll angle; and take at least a first gyroscope measurement, a second gyroscope measurement, and a third gyroscope measurement at the at least first rotational position, second rotational position, and third rotational position using the determined durations.

13. The gyroscopic survey tool of claim 11, wherein the determining the measurement duration comprises determining the measurement duration for each of the plurality of gyroscope measurements from the estimated pitch angle and roll angle and a previously calculated azimuth angle.

14. The gyroscopic survey tool of claim 13, wherein the determining the measurement duration further comprises: estimating a first component, a second component, and a third component of a gyroscope measurement vector from the previously calculated azimuth angle; evaluating the estimated first component, second component, and third component of the gyroscope measurement vector and the pitch angle and the roll angle to calculate a partial derivative of the azimuth angle with respect to the first component, the second component, and the third component of the gyroscope measurement vector; and determining the measurement duration from the partial derivative.

15. The gyroscopic survey tool of claim 14, wherein the measurement duration is proportional to the partial derivative.

16. A method for surveying a wellbore, the method comprising: deploying a gyroscopic survey tool in the wellbore, the gyroscopic survey tool comprising a single-axis gyroscope configured to rotate in a tool housing and rotate relative to the tool housing; ​ ​ ​ estimating a pitch angle and a roll angle of the gyroscopic survey tool in the wellbore; determining at least a first measurement duration, a second measurement duration, and a third measurement duration corresponding to at least a first gyroscope measurement, a second gyroscope measurement, and a third gyroscope measurement in accordance with the estimated pitch angle and roll angle; taking each of the at least first gyroscope measurement, second gyroscope measurement, and third gyroscope measurement for the determined at least first measurement duration, second measurement duration, and third measurement duration when the gyroscope is disposed at a corresponding at least first rotational position, second rotational position, and third rotational position in the tool housing; and calculating an azimuth angle of the wellbore from the at least first gyroscope measurement, second gyroscope measurement, and third gyroscope measurement.

17. The method of claim 16, wherein the pitch angle and the roll angle are calculated from accelerometer measurements.

18. The method of claim 16, wherein the determining includes determining the at least first measurement duration, second measurement duration, and third measurement duration for each of the corresponding at least first gyroscope measurement, second gyroscope measurement, and third gyroscope measurement from the estimated pitch angle and roll angle and a previously calculated azimuth angle.

19. The method of claim 18, wherein the determining the at least first measurement duration, second measurement duration, and third measurement duration further includes: estimating a first component, a second component, and a third component of a gyroscope measurement vector from the previously calculated azimuth angle; evaluating the estimated first component, second component, and third component of the gyroscope measurement vector and the pitch angle and the roll angle to calculate partial derivatives of the azimuth angle with respect to the first component, the second component, and the third component of the gyroscope measurement vector; and determining the at least first measurement duration, second measurement duration, and third measurement duration from the partial derivatives.

20. The method of claim 19, wherein the at least first measurement duration, second measurement duration, and third measurement duration are proportional to the partial derivatives.

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