Variational h-coil calibration method for tri-axial magnetometer

Abstract

A directional tool and a method of surveying a wellbore with the directional tool in a borehole string are provided. A first sensor of the directional tool is disposed in a non-uniform ambient magnetic field. A first applied magnetic field is applied to the first sensor. A first measurement is obtained at the first sensor while the first sensor is disposed in the non-uniform ambient magnetic field and the first applied magnetic field is applied. A second applied magnetic field is applied to the first sensor. A second measurement is obtained at the first sensor while the first sensor is disposed in the non-uniform ambient magnetic field and the second applied magnetic field is applied. The directional tool is calibrated based on the first measurement and the second measurement. Downhole survey measurements are obtained in the wellbore with the calibrated directional tool.

Description

[0001] Cross Reference to Related Applications

[0002] This application claims the benefit of earlier filing date of U.S. Application Serial No. 62 / 935,284 filed November 14, 2019, the entire disclosure of which is incorporated herein by reference. BACKGROUND

[0003] In the resource recovery industry, magnetic field sensors such as fluxgate magnetometers can be used in downhole applications such as surveying, drill string trajectory correction, etc. The accuracy of these measurements depends on a properly calibrated magnetometer. The calibration process of a magnetometer is performed against natural or ambient magnetic fields. These fields can be inhomogeneous due to steel structures or magnetic components near the calibration location, which leads to difficulties in the calibration process. Therefore, there is a need to develop a method for calibrating a magnetometer in areas of unknown and / or inhomogeneous ambient magnetic fields. SUMMARY

[0004] Disclosed herein is a method of surveying a wellbore with a directional tool in a drill string. A first sensor of the directional tool is disposed in an inhomogeneous ambient magnetic field. A first applied magnetic field is applied to the first sensor. A first measurement is obtained at the first sensor while the first sensor is disposed in the inhomogeneous ambient magnetic field and the first applied magnetic field is applied. A second applied magnetic field is applied to the first sensor. A second measurement is obtained at the first sensor while the first sensor is disposed in the inhomogeneous ambient magnetic field and the second applied magnetic field is applied. The directional tool is calibrated based on the first measurement and the second measurement. Downhole survey measurements are obtained in the wellbore with the calibrated directional tool.

[0005] Also disclosed herein is a system for a directional tool for surveying a wellbore in a drill string. The directional tool includes a first sensor that is calibrated by: disposing the first sensor in an inhomogeneous ambient magnetic field; applying a first applied magnetic field to the first sensor; obtaining a first measurement at the first sensor while the first sensor is disposed in the inhomogeneous ambient magnetic field and the first applied magnetic field is applied; applying a second applied magnetic field to the first sensor; obtaining a second measurement at the first sensor while the first sensor is disposed in the inhomogeneous ambient magnetic field and the second applied magnetic field is applied; and calibrating the directional tool based on the first measurement and the second measurement. Downhole survey measurements are obtained with the calibrated directional tool disposed in the wellbore. BRIEF DESCRIPTION OF DRAWINGS

[0006] The following description should not be read as limiting in any way. With reference to the drawings, like elements are numbered alike:

[0007] Figure 1A borehole assembly in an illustrative embodiment is shown;

[0008] Figure 2 An orientation tool or probe is illustrated, showing an orientation tool or probe coordinate system and a sensor coordinate system of a sensor mounted within the orientation tool or probe;

[0009] Figure 3 An illustrative magnetic measurement orientation tool or probe that can be used for surveying and geosteering of a drill string is shown, along with a reference orientation tool or probe and a calibration assembly;

[0010] Figure 4 A relationship between a z-axis of a sensor and an axis of an orientation tool or probe coordinate system is shown;

[0011] Figure 5 A flowchart illustrating a method of determining an misalignment angle of a sensor is shown; and

[0012] Figure 6 A flowchart illustrating a method of determining a sensor bias is shown. DETAILED DESCRIPTION

[0013] A detailed description of one or more embodiments of the devices and methods disclosed herein is presented with reference to the accompanying drawings, by way of example and not limitation.

[0014] In the present disclosure, an orientation tool or probe including a primary (uncalibrated) sensor of a downhole tool is calibrated with respect to a reference (calibrated) sensor. The sensors can be characterized by a scaling parameter, such that the sensor response can be calculated from the scaling parameter and an applied field. There is a known angle, e.g., a measured angle, between the orientation tool or probe and the reference sensor. For example, the known angle between the orientation tool or probe and the reference sensor can be 0°, such that the axis of the orientation tool or probe and the axis of the reference sensor are aligned along the same orientation tool or probe axis. The primary sensor is misaligned from the axis of the orientation tool or probe, with a misalignment angle determined during a calibration process. In addition to an ambient magnetic field, such as a natural or background magnetic field that can be non-uniform, an applied magnetic field is applied to both the primary sensor and the reference sensor. In the context of the present disclosure, a non-uniform magnetic field includes an anisotropic magnetic field, e.g., an anisotropic magnetic field having a uniform total magnetic field strength. The non-uniform magnetic field can be non-time-varying at least for a time required for a measurement to perform a calibration process as described below. The scaling parameter and an optional bias of the primary sensor are determined using magnetic measurements with the applied magnetic field, which in turn can be used to determine the misalignment angle. Downhole magnetic field measurements at the primary sensor can then be corrected based on the scaling parameter, the optional bias, and the misalignment angle.

[0015] REFERENCE Figure 1A drilling assembly 100 is shown in an illustrative embodiment. The drilling assembly 100 includes a drill string 102 extending from a rig 104 into a wellbore 106 formed in a formation 108. The drill string 102 includes a drill bit 110 at a bottom end for drilling the wellbore 106. The drill bit 110 can be rotated by rotation of the drill string 102 from the rig 104 at a surface location 130 and / or by rotation of a downhole motor (not shown). The drill string 102 includes a hollow interior bore 114, and an annulus is formed between an outer surface of the drill string 102 and a wall 118 of the wellbore 106.

[0016] In operation, a drilling mud 112 is pumped downhole from a mud pit 120 at the surface location 130 via a pump 122 at the surface location 130 through the interior bore 114 and out of the drill string 102 at the drill bit 110. The pump 122 is typically located in a standpipe 124 that delivers the drilling mud 112 from the mud pit 120 to the top of the drill string 102. Once the drilling mud 112 exits the drill string 102 at the drill bit 110, the drilling mud 112 returns to the surface location 130 via the annulus 116. At the surface location 130, the drilling mud 112 is returned to the mud pit 120 via a flowback line 126.

[0017] The drill string 112 also includes a control unit 150 for controlling various operations of the drill string 102. The control unit 150 includes a processor 152 and a memory storage 154 including various programs and instructions 156 that, when accessed by the processor 152, enable the processor 152 to perform various operations disclosed herein. The control unit 150 can be located at the surface location 130 or at a downhole location along the drill string 102, such as a downhole control unit 150a.

[0018] The drill string 112 also includes a steering unit 140 adapted to change the drilling direction of the drill string 102. In various embodiments, the downhole control unit 150a activates the steering unit 140 based on survey information. The drill string 102 also includes a tool, such as a directional tool, which can be a drill tool or can be a probe 142 included in a drill tool, and can include survey instruments. The directional tool or probe 142 can include an inclinometer. In various embodiments, the directional tool or probe 142 includes a set of magnetometers that respond to and obtain measurements of the earth's magnetic field. The magnetometers can be single- or multi-axial, such as two- or three-axial magnetometers. In various embodiments, the magnetometers are fluxgate sensors, Hall effect magnetometers, magnetoresistive sensors, superconducting quantum interference devices (SQUIDs), MEMS, Lorentz force type magnetometers, atomic gas magnetometers, atomic interferometers, fiber optic magnetometers, or any other magnetometer device known in the art. The measurements of the magnetic field can be used at the downhole control unit 150a or at the surface control unit 150 in order to determine the drilling direction of the drill string 102, and thus activate the steering unit 140 in order to change the direction of the drill string 102 according to the drilling direction determined from the measurements.

[0019] In alternative embodiments, the directional tool or probe 142 is included in a wireline string (not shown) that is configured to be lowered into the wellbore 106 for downhole survey measurements while the directional tool or probe 142 is downhole in the wellbore 106. The magnetometers of the directional tool or probe 142 can be calibrated to provide accurate downhole measurements prior to use in a drilling string such as the wireline string or the drill string 102. Methods for calibrating the magnetometers of the directional tool or probe 142 are discussed below.

[0020] In one embodiment, the directional tool or probe 142 includes a combination of one two-axial sensor and one single-axial sensor. In various embodiments, the directional tool or probe 142 includes two two-axial magnetometers, where one two-axial magnetometer is always used as a two-axial magnetometer, and the other two-axial magnetometer is used as a two-axial or single-axial magnetometer depending on the requirements of the directional tool or probe. In further embodiments, the directional tool or probe 142 can include one or two three-axial magnetometers. Two magnetometers aligned and oriented similarly along a chosen direction can be used to determine the magnetic field gradient along the aligned direction.

[0021] In one embodiment, it can be assumed that the magnetometer calibrated with the methods described herein has a response H m which can then be defined by a first order polynomial, which can then be written as shown in equation (1):

[0022] H m = aH + b equation (1)

[0023] where a (also referred to as an "a matrix") includes one or more scale parameters and misalignment coefficients of the sensor, and (Also referred to as a "b vector") includes one or more bias coefficients of the magnetometer. However, the magnetometer response that can be defined by a first order polynomial is merely an example and should not be understood as limiting. While the method is illustrated herein using a response that can be defined by a first order polynomial, it should be understood that the method can be readily applied and extended to magnetometer responses that are defined, interpolated, or described by higher order polynomials or other functions, such as exponential functions, logarithmic functions, step functions, such as tables, or any combination thereof. The higher order polynomials or other functions used to define, interpolate, or describe the magnetometer response can include more than one scale parameter per magnetic field component. For example, the higher order polynomials or other functions used to define, interpolate, or describe the magnetometer response can include two, five, ten, or more scale parameters per magnetic field component.

[0024] Figure 2 A directional tool or probe 142 is shown, showing a directional tool or probe coordinate system 202 and a sensor coordinate system 204 of a magnetometer mounted within the directional tool or probe 142. The directional tool or probe coordinate system 202 is indicated by using a prime vector (x', y', z'). The z' axis is parallel to a longitudinal axis 210 of the directional tool or probe 142, and the x' and y' axes are oriented within a plane that is transverse (typically perpendicular) to the longitudinal axis 210. In the example shown, the x' axis is perpendicular to z' and points out of the page, while the y' axis is perpendicular to z' (longitudinal axis 210) and x' and lies within the page. Figure 2

[0025] The sensor coordinate system 204 is indicated by using a non-prime vector (x, y, z). The sensor coordinate system 204 is shown as misaligned with the directional tool or probe coordinate system 202 in order to reflect the misalignment problem addressed by the present invention. When mounting a sensor in a mechanical assembly of a directional tool or probe 142, small axis misalignments make it difficult to align the axes of the sensor with the axes of the directional tool or probe. This misalignment between a selected directional tool or probe axis and a selected sensor axis is represented by Θ 工具轴线′-传感器轴线 (i.e., the angle between the two axes). Thus, the projection of a magnetic field component in an axis of the directional tool or probe into an axis of the sensor is represented as a so-called misalignment coefficient, which is the cosine of the misalignment angle (i.e., cos(Θ 工具轴线′-传感器轴线 )). This cosine term is part of the a component of the a matrix of equation (1) (i.e., a 工具轴线′-传感器轴线 ), representing scale and misalignment errors. In particular, the coefficient a 工具轴线′-传感器轴线 of the a matrix can be written as shown in equation (2): ​

[0026] α 工具轴线′-传感器轴线 = α o cos(θ 工具轴线′-传感器轴线 ) Equation (2)

[0027] where a0is a scale factor based on sensor sensitivity and environmental disturbances, and θ 工具轴线-传感器轴线 is the misalignment angle between the sensor axis and the axis of the orientation tool or probe.

[0028] Response of a three-axis arrangement of magnetometers may be expressed as shown in Equation (3):

[0029]

[0030] where is the magnetic field within the (x', y', z') coordinate system of the orientation tool or probe. Equation (3) applies to single-axis or multi-axis arrangements of magnetometers, such as two-axis, three-axis, or even more axis arrangements of magnetometers. and The number of vector components of and is equal to the number of axes of the multi-axis arrangement of magnetometers, while the number of matrix elements of matrix a is equal to the square of the number of axes of the multi-axis arrangement of magnetometers. For example, for a single-axis arrangement of magnetometers, and The number of vector components of and is equal to 1, while the number of matrix elements of matrix a is also equal to 1. In other words, for a single-axis arrangement, and a is a simple scalar. For a two-axis arrangement of magnetometers, and The number of vector components of and is equal to 2, while the number of matrix elements of matrix a is equal to 4. For a three-axis arrangement of magnetometers, and The number of vector components of and is equal to 3, while the number of matrix elements of matrix a is equal to 9. The following derivation is made for a three-axis arrangement of magnetometers for illustrative purposes only and not by way of limitation. The following discussion can also be readily applied to other single-axis or multi-axis arrangements of magnetometers. For a three-axis arrangement of magnetometers, Equation (3) can be rewritten in expanded form as shown in Equation (4):

[0031]

[0032] where is the response of the sensor on the sensor x-axis, is the response of the sensor on the sensor y-axis, and is the response of the sensor on the sensor z-axis. Coefficients β x , β y , and βz is the scale factor along the respective x, y, and z axes of the sensor.

[0033] Using equation (2), the alpha matrix in equation (4) can be decomposed into functions of its scale factors and misalignment angles as follows:

[0034]

[0035] which can also be written as equation (6):

[0036]

[0037] As an illustrative example, is the scale factor along the y axis, and is the angle between the directional tool or probe axis y' and the sensor axis x. The coefficient illustrates the projection of the field along the y' axis of the directional tool or probe with respect to the x axis of the sensor. Thus, for example, the coefficient of the alpha matrix between the directional tool or probe y' axis and the sensor z axis can be written as the product of the scale factor along the sensor z axis and the angle between the directional tool or probe y' axis and the sensor z axis, as shown in equation (7):

[0038]

[0039] Due to the misalignment between the directional tool or probe and the sensor, the measurement along a selected axis of the sensor, such as the z axis, i.e., carries information from all three components of the magnetic field referenced along the directional tool or probe axis, i.e., H x , H y , and H z Equations (8), (9), and (10) represent the calculations along the bottom row of equation (4) to For example, the decomposition of the sensor magnetic field components of the sensor with respect to the magnetic field in the directional tool or probe axis is described as:

[0040]

[0041]

[0042] and

[0043]

[0044] Typically, the misalignment angle (Θ z′-z ) between the z' axis of the directional tool or probe and the z axis of the sensor is a small angle (e.g., less than ±10 degrees, such as less than ±5 degrees, for example less than ±2 degrees). Thus, for The contribution of the magnetic field component perpendicular to the longitudinal axis 210 (i.e., component H'x x ) is significantly smaller than the contribution of the magnetic field component parallel to the longitudinal axis (i.e., component H'z y ). z

[0045] For example, when the angle between the z-axis and the z'-axis is small (e.g., less than ±10 degrees, such as less than ±5 degrees, for example less than ±2 degrees), the angle θ x′-z and θ y′-z of equation (10) is close to 90 degrees. In this case, the following approximations of equations (11) and (12) hold:

[0046] cos(θ x′-z ) << cos(θ z′-z ) equation (11)

[0047] cos(θ y′-z ) << cos(θ z′-z ) equation (12)

[0048] Thus, equations (13) and (14) are true:

[0049] α x′-z << α z′-z equation (13)

[0050] α y′-z << α z′-z equation (14)

[0051] Figure 3 ​An exemplary first orientation tool or probe 300, which can be used for surveying and geosteering of a drill string such as drill string 102, is shown. First orientation tool or probe 300 is mounted within a calibration assembly 320. First orientation tool or probe 300 includes a first sensor 302 (also referred to herein as a "primary sensor") to be calibrated. A second sensor 304 (also referred to herein as a "reference sensor") is included in calibration assembly 320. First sensor 302 is axially separated from second sensor 304 by a selected separation distance. For a three-axis arrangement of magnetometers, first sensor 302 and second sensor 304 are three-axis sensors characterized by xl, yl, zl coordinate system and x2, y2, z2 coordinate system, respectively. There can be an unknown misalignment angle between a longitudinal axis z' of first orientation tool or probe and the xl, yl, zl coordinate system of first sensor, such that none of the xl, yl, zl axes of first sensor is pointing in the direction of longitudinal axis z' of first orientation tool or probe. In addition, longitudinal axis z' of first orientation tool or probe 300 and the x2, y2, z2 coordinate system of reference sensor can be separated by a certain separation angle, such that at least one of the xl, y2, and z2 axes and the z' axis are separated by a separation angle that can be measured and / or selected. For example, longitudinal axis z' of first orientation tool or probe 300 and the z2 axis of reference sensor can be separated by a separation angle of 0°, such as the z2 axis of reference sensor being oriented or substantially oriented along longitudinal axis z' of first orientation tool or probe 300, or longitudinal axis z' of first orientation tool or probe 300 and the z2 axis of reference sensor can be separated by a separation angle of 20 degrees, 30 degrees, 45 degrees, or any other suitable separation angle. In one embodiment, second sensor 304 is included in a second orientation tool or probe. Advantageously, first orientation tool or probe 300 and the second orientation tool or probe are of the same or similar kind, e.g., have the same or similar dimensions, materials, and / or design. This configuration of first orientation tool or probe and second orientation tool of the same or similar kind will ensure that the magnetic interference caused by first orientation tool or probe will be the same or similar to the magnetic interference caused by the second orientation tool or probe. In one embodiment, the second orientation tool or probe has been calibrated, and the above-mentioned separation angle is defined by the angle between the longitudinal axis of first orientation tool or probe and the longitudinal axis of the second orientation tool or probe. In addition, an optional barrier 306 can be provided between first sensor 302 and second sensor 304 to isolate first sensor 302 from second sensor 304. First sensor 302 and second sensor 304 typically experience different magnetic fields due to their axial separation and due to possibly non-uniform ambient magnetic fields.

[0052] In one embodiment, the first and second orientation tools or probes 300, 400 can include other orientation sensors, such as accelerometers / gravity meters (not shown) or gyroscopes. Such other orientation sensors can be used to at least partially align the first and second orientation tools or probes 300, 400 to adjust or measure the separation angle between the first and second orientation tools or probes, such as a separation angle between the first and second orientation tools or probes of less than 10 degrees, such as less than 5 degrees or even less than 2 degrees. For example, by using at least one of the high side of the first orientation tool or probe 300 or the high side of the second orientation tool or probe as a reference point, the first and second orientation tools or probes 300, 400 can be at least partially aligned with each other. This can be performed by aligning one axis of the accelerometers / gravity meters or gyroscopes (not shown), which can be aligned opposite the gravity field of the reference and primary sensors in the orientation tools or probes 300, 400, respectively. However, once placed in position, there is typically still an angular misalignment between the z1 axis of the first sensor 302 and the longitudinal axis of the first orientation tool or probe, and similarly between the z2 axis of the second sensor 304 and the longitudinal axis of the second orientation tool or probe.

[0053] Figure 3A pair of coils for introducing an applied magnetic field to the first and second sensors is also shown. Alternatively, one or more permanent magnets can be utilized to apply an applied magnetic field to the first and second sensors. The pair of coils can be electrically coupled to a power source 315 that provides current through the coils in order to generate one or more magnetic fields at the first and second sensors. Additionally, measurements from the pair of coils can be sent to the processor 310 that performs various calculations disclosed herein for calibrating the primary sensors within the first directional tool or probe, determining misalignment angles, scale parameters, sensor bias, etc. The primary coil 312 can be used to apply an applied magnetic field to the first sensor 302 and the reference coil 314 can be used to apply an applied magnetic field to the second sensor 304. In various operations, the primary coil 312 and the reference coil 314 apply the same applied magnetic field to the first sensor 302 and the second sensor 304, respectively. The applied magnetic field can be controlled, for example, by a control algorithm. In various embodiments, the primary coil 312 and the reference coil 314 can be a single coil that extends along the length of the first directional tool or probe and / or the second directional tool or probe 300 to cover both the first sensor 302 and the second sensor 304. The primary coil 312 and the reference coil 314 are shown oriented to produce an applied magnetic field along the longitudinal axis z' of the first directional tool or probe or the first directional tool or probe and the second directional tool or probe. In further embodiments discussed below, the primary coil 312 and the reference coil 314 can be oriented to produce magnetic fields along the x' and y' axes of the first directional tool or probe.

[0054] To calibrate the primary sensors 302 to the reference sensors 304, a first set of magnetic measurements is obtained with a first applied magnetic field, and then a second set of magnetic measurements is obtained once a second applied magnetic field is applied. In one embodiment, the first applied magnetic field or the second applied magnetic field can be zero. For example, if the first applied magnetic field is zero, the first set of magnetic measurements is obtained without an applied magnetic field, and then the second set of magnetic measurements is obtained once a second non-zero applied magnetic field is applied. Equations (3) and (4) generally describe the relationship between the magnetic field measurements in the directional tool or probe coordinate system and the magnetic field in the coordinate system of each sensor. Thus, application of these equations provides sufficient information to determine the scale parameter, misalignment angle, and bias coefficient, as shown below. Advantageously, such measurements can be taken at different temperatures, for example by using heating and / or cooling equipment, such as a heating chamber and / or a cooling chamber. Determining the calibration parameters, such as the scale parameter, misalignment angle, and bias coefficient, at various temperatures will allow for determination and application of calibration parameters as a function of temperature. For example, if a first set of calibration parameters is determined at a first temperature, it can be applied to measurements taken at temperatures represented by the first temperature, for example measurements falling within the same temperature interval as the first temperature. If a second set of calibration parameters is determined at a second temperature, it can be applied to measurements taken at temperatures represented by the second temperature, for example measurements falling within the same temperature interval as the second temperature, and so on. If the calibration parameters depend significantly on temperature, such temperature-dependent calibration allows for more accurate measurements to be obtained. Similarly, in some directional tools or probes, the installed wires can carry large currents at run-time that interfere with the local magnetic field at the sensor locations. If this is not taken into account, such current-carrying wires can cause systematic measurement errors that are not accounted for in the calibration method. Thus, one or more wires (not shown) or other conductors can be added to the calibration assembly 320, adapted to carry selected currents, and thus can be used to simulate the interfering magnetic fields caused by currents through the first sensors 302 at run-time. For example, one or more wires can be added within the directional tool or probe, located at the same or similar locations, and carrying the same or similar currents as at run-time. Determining the calibration parameters, such as the scale parameter, misalignment angle, and bias coefficient, at various currents in the one or more wires or conductors will allow for determination and application of calibration parameters as a function of current and / or distance between the wires or conductors and the sensors. For example, if a first set of calibration parameters is determined at a first current / distance resulting in a first interfering magnetic field, it can be applied to measurements taken at interfering magnetic fields represented by the first interfering magnetic field, for example measurements falling within the same interfering magnetic field interval as the first interfering magnetic field.If the second set of calibration parameters is determined at a second current / distance that results in a second interfering magnetic field, they can be applied to measurements made at interfering magnetic fields represented by the second interfering magnetic field, e.g. measurements falling within the same interfering magnetic field interval as the second interfering magnetic field. If the calibration parameters are significantly dependent on interfering magnetic fields that can occur at runtime, such interfering magnetic field dependent calibration allows to obtain more accurate measurement results.

[0055] In the following, it is assumed that a first set of magnetic measurements is obtained without applying an external magnetic field, and then a set of magnetic measurements is obtained once a non-zero external magnetic field is applied. As mentioned above, this is a special case in which a first set of magnetic measurements is obtained with a first external magnetic field applied, and then a second set of magnetic measurements is obtained once a second external magnetic field is applied. For the case in which a first set of magnetic measurements is obtained without applying an external magnetic field, and then a set of magnetic measurements is obtained once a non-zero external magnetic field is applied, inverting equation (3) gives the following equation (15):

[0056]

[0057] Similarly, inverting equation (4) gives the following equation (16):

[0058]

[0059] The calibration method is used to determine the matrix a and the vector b using the following equations (17) and (18): The components of the vector b are determined using the following equations (19) to (21): At a first time (t = to), the ambient magnetic field is measured at both the primary sensor 302 and the reference sensor 304. Due to the inhomogeneity of the ambient magnetic field, the ambient magnetic field at the primary sensor 302 and the reference sensor 304 can be different. In general, the primary sensor 302 will measure the ambient magnetic field whereas the reference sensor 304 measures the ambient magnetic field At a second time (t = ti), a known external magnetic field H is applied via the coils 312, 314 c and a second set of measurements is made at the primary sensor 302 and the reference sensor 304.

[0060] At a first time (t = to), only the ambient magnetic field is measured. At the first sensor 302, the raw measurement M1 x , M1 y and M1 z are defined as shown in equation (17):

[0061]

[0062] where H1' x , H1' y and H1'z are the x', y', and z' components of the true magnetic field at the location of the first sensor 302.

[0063] Since the second sensor 304 has been previously calibrated, the measurements M2 x , M2 y , and M2 z are equal to H2' x , H2' y , and H2' z , the x', y', and z' components of the true magnetic field at the location of the second sensor 304, as shown in equation (18):

[0064]

[0065] At a second time period (t = ti), a current Ic is applied to one or more of the coils 312, 314 to produce an applied magnetic field The current Ic can be controlled to produce a controlled applied magnetic field. The applied magnetic field may be aligned with an axis of the first orientation tool or probe, such as the z' axis of the first orientation tool or probe. Thus, is applicable. When the applied magnetic field is applied to the first sensor, the output value is as shown in equation (19):

[0066]

[0067] When the applied magnetic field is applied to the second sensor 304, the output value is as shown in equation (20):

[0068]

[0069] The scaling parameter a along the z axis z′-z may be determined using the calculations discussed below with respect to equations (21) through (24). The scaling parameter is first determined using measurements from the reference sensor. From equation (17):

[0070]

[0071] From equation (19) (where ), it follows that:

[0072]

[0073] Combining equation (21) and equation (22) results in:

[0074]

[0075] And therefore

[0076]

[0077] At time t = to (equation (18)) the measurements from the reference sensor give the following equations (25) to (27):

[0078]

[0079]

[0080]

[0081] Similarly, at time t = ti (equation (20)) the measurements from the reference sensor give the following equation (28):

[0082]

[0083] Combining equations (25) and (28) gives:

[0084]

[0085] Thus:

[0086]

[0087] Equation (30) determines the z' component of the applied magnetic field, which can then be used in equation (24) to determine the value of a z′-z Alternatively, the value of a This can be done by analytical formulae known in the art or by numerical simulation of the magnetic field at the first sensor position.

[0088] The coils 312, 314 can now be reoriented to produce an applied magnetic field along the y' axis, and a new set of measurements recorded at time t = t2. At time t = t2, the applied magnetic field is applied to the primary sensor 302 and the reference sensor 304 along the y' axis, so that From the third row of the matrix of equation (19), it can be determined that:

[0089]

[0090] Thus, combining equations (17) and (32) gives:

[0091]

[0092] And therefore

[0093]

[0094] The value of a can be calculated from equation (34) because all other variables are known. Specifically, from equation (20) it is known that: y′-z

[0095]

[0096] Substituting equation (26) into equation (35) gives:

[0097]

[0098] giving

[0099]

[0100] Equation (37) can be substituted into equation (34) to determine a y′-z .

[0101] The coils 312, 314 can now be reoriented to produce an applied magnetic field along the x' axis, and a new set of measurements is made at time t = t3. At time t = t3, the applied magnetic field is applied to the primary sensor 302 and the reference sensor 304 along the direction of the x' axis, so that From equation (19), it can be determined that:

[0102]

[0103] Combining equation (17) and equation (38) gives:

[0104]

[0105] Therefore,

[0106]

[0107] The value of a can be calculated in equation (40) because all other variables are known. Specifically, from equation (20) it is known that: x′-z

[0108]

[0109] Substituting equation (27) into equation (41) gives:

[0110]

[0111] Therefore,

[0112]

[0113] ​​Equation (43) can be substituted into equation (40) to determine a x′-z .

[0114] The steps shown in equations (15) to (43) can be used to similarly derive all missing coefficients of the matrix a as defined in equation 4. While the above description regarding calibration is limited to sensors having a response that can be defined, interpolated or described by a first order polynomial, sometimes also referred to as "linear sensors", the same approach can be readily applied to sensors using a higher order polynomial or other function for defining, interpolating or describing the magnetometer response and each magnetic field component can include more than one scaling parameter. In this case, the number of measurements to be made under various applied magnetic fields has to be increased accordingly to result in a corresponding set of equations that can be solved analytically or numerically, for example by numerical inversion, lookup functions or similar methods.

[0115] To determine the bias coefficient along the z-axis, the magnetic fields along the y' and x' axes are made zero or substantially zero. To obtain negligible magnetic fields along the y' and x' axes, the directional tool or probe is oriented with the z' axis parallel to the earth's magnetic field. For illustrative purposes, this is performed at time t = t4. According to equation (17), for time t = t4, it can be derived that:

[0116]

[0117]

[0118]

[0119] At time t = t5, the z' axis of the first directional tool or probe is aligned with the earth's magnetic field and is rotated 180 degrees from its position at time t = t4. The resulting magnetic fields at time t = t5 are as follows:

[0120]

[0121]

[0122]

[0123] Adding equation (46) and equation (49) gives:

[0124]

[0125] Thus:

[0126]

[0127] In various embodiments, the calibration assembly 320 rotates with the first orientation tool or probe 300 between t = t4 and t = t5. However, this can be a disadvantage if the calibration assembly 320 comprises magnetic material. Advantageously, the calibration assembly 320 does not comprise magnetic material. Additionally or alternatively, the calibration assembly 320 can not rotate with the first orientation tool or probe between t = t4 and t = t5. In various aspects, it is also beneficial when the first orientation tool or probe rotates around the position of the first sensor 302. To achieve rotation of the first orientation tool or probe around the position of the first sensor 302, the first orientation tool or probe can be rotatably mounted such that the rotation pivot passes through the center of the first sensor 302, or through the first sensor close to the center of the first sensor, e.g. at a distance of no more than 50 cm, e.g. less than 20 cm, such as less than 10 cm.

[0128] Figure 4 The relationship between the z-axis of the sensor and the x', y', z' axes of the orientation tool or probe coordinate system is shown. Since the x', y', and z' axes of the orientation tool or probe and the x, y, and z axes of the sensor are mutually orthogonal, it is possible to express the misalignment with only two unknown angles, such as Φ and Ψ or γ and θ z z As Figure 4 shown and described below.

[0129] The magnetic field components in the sensor coordinate system can be written using the magnetic components in the orientation tool or probe coordinate system with equations (7) to (9) disclosed herein. The misalignment angle θ z′-z can be rewritten in terms of the complementary angle. For example, the misalignment angle θ z′-z can be written in terms of the angle Φ between the z-axis 402 of the sensor and the projection 404 of the z-axis of the sensor onto the x'-y' plane.

[0130]

[0131] Also in terms of Figure 4 :

[0132]

[0133] Also in terms of Figure 4 , the angle θ x′-z can be determined:

[0134] cos(θ x′-z ) = cos(Ψ) * cos(Φ) equation (54)

[0135] and by combining this with equation (52) and equation (53), this leads to

[0136] cos(θ​x′-z ) = cos(90° - γ) * cos(90° - θ z′-z ). Equation (55)

[0137] Using the trigonometric identity, Equation (55) can be rewritten as:

[0138] cos(θ x′-z ) = (cos(90°) * cos(γ) + sin(90°) * sin(γ)) * (cos(90°) * cos(θ z′-z ) + sin(90°) * sin(θ z′-z )) Equation (56)

[0139] Thus, we have:

[0140] cos(θ x′-z ) = sin(γ) * sin(θ z′-z ). Equation (57)

[0141] With respect to the angle θ x′-z , we can determine: Figure 4

[0142] cos(θ y′-z ) = cos(γ) * cos(Φ) Equation (58)

[0143] Thus, using Equation (52),

[0144] cos(θ y′-z ) = cos(γ) * cos(90° - θ z′-z ) Equation (59)

[0145] And using the trigonometric identity

[0146] cos(θ y′-z ) = cos(γ) * sin(θ z′-z ). Equation (60)

[0147] Using the results of Equation (57) and Equation (60), Equation (10) can be rewritten as:

[0148]

[0149] Thus,

[0150]

[0151]

[0152]

[0153] By solving for the three parameters​ γ and θ z′-z Solving equations (62) to (64) allows determination of the misalignment angle and scale factor along the sensor z-axis. Corresponding analysis can be performed to determine scale factors and other misalignment angles along the sensor y-axis and sensor x-axis.

[0154] Figure 5 A flowchart 500 illustrating a method of determining a misalignment angle of a sensor in an embodiment is shown. In block 502, a first orientation tool or probe is disposed in a non-uniform magnetic field, such as the Earth's magnetic field, for example, the ambient Earth's magnetic field disturbed by a magnetic material and / or source. The first orientation tool or probe includes a first sensor at a misalignment angle from an axis of the first orientation tool or probe. A second sensor is disposed in the non-uniform magnetic field at a location where the non-uniform magnetic field can have a different strength compared to the magnetic field strength of the non-uniform magnetic field at the location of the first sensor. The second sensor can be disposed within a second orientation tool or probe that can have been previously calibrated. In block 504, a first magnetic measurement is obtained at the first sensor and a second magnetic measurement is obtained at the second sensor. In block 506, an applied magnetic field is applied to the first sensor and the second sensor. In block 508, a third magnetic measurement is obtained at the first sensor and a fourth magnetic measurement is obtained at the second sensor while the applied magnetic field is applied. In block 510, a first magnetic difference is determined between the third magnetic measurement and the first magnetic measurement and a second magnetic difference is determined between the fourth magnetic measurement and the second magnetic measurement. In block 512, a scale parameter relating the axis of the first sensor to the axis of the second sensor is determined from a ratio of the first magnetic difference to the second magnetic difference. In block 514, the misalignment angle of the first sensor is determined from the scale parameter.

[0155] Figure 6 A flowchart 600 illustrating a method of determining a sensor bias is shown. In block 602, an orientation tool or probe including at least a first sensor is disposed in a non-uniform magnetic field (i.e., the Earth's magnetic field) with a selected axis (e.g., the z-axis) of the first sensor or the orientation tool or probe aligned with the Earth's magnetic field. In block 604, a first magnetic measurement is obtained at the first sensor. In block 606, the first sensor is rotated 180 degrees about a rotation axis substantially perpendicular to the selected axis. Preferably, the rotation axis passes through the first sensor or is at a close distance (e.g., no more than 50 cm, for example, less than 20 cm, such as less than 10 cm) from the sensor. In block 608, a second magnetic measurement is obtained at the first sensor. In block 610, a bias of the first sensor along the selected axis is determined from the first magnetic measurement and the second magnetic measurement.

[0156] Some embodiments of the foregoing disclosure are shown below:

[0157] Implementation 1 : A method of surveying a wellbore with a directional tool in a bore string. A first sensor of the directional tool is disposed in a non-uniform ambient magnetic field. A first applied magnetic field is applied to the first sensor. A first measurement is obtained at the first sensor while the first sensor is disposed in the non-uniform ambient magnetic field and the first applied magnetic field is applied. A second applied magnetic field is applied to the first sensor. A second measurement is obtained at the first sensor while the first sensor is disposed in the non-uniform ambient magnetic field and the second applied magnetic field is applied. The directional tool is calibrated based on the first measurement and the second measurement. Downhole survey measurements are obtained in the wellbore with the calibrated directional tool.

[0158] Implementation 2: The method of any previous implementation, wherein at least one of the first applied magnetic field and the second applied magnetic field is substantially zero.

[0159] Implementation 3: The method of any previous implementation, wherein the bore string is a drill string, and further comprising drilling the wellbore with the drill string using the downhole survey measurements.

[0160] Implementation 4: The method of any previous implementation, further comprising disposing a second sensor in the non-uniform ambient magnetic field, applying the first applied magnetic field to the second sensor, obtaining a third measurement at the second sensor while the second sensor is disposed in the non-uniform ambient magnetic field and the first applied magnetic field is applied, applying the second applied magnetic field to the second sensor, obtaining a fourth measurement at the second sensor while the second sensor is disposed in the non-uniform ambient magnetic field and the second applied magnetic field is applied, and calibrating the directional tool based on the first measurement, the second measurement, the third measurement, and the fourth measurement.

[0161] Implementation 5: The method of any previous implementation, wherein the second sensor is a calibrated sensor.

[0162] Implementation 6: The method of any previous implementation, wherein calibrating comprises determining one or more calibration parameters that are a function of at least one of a sensitivity of the first sensor and a misalignment of the first sensor relative to the directional tool.

[0163] Embodiment 7: The method of any of the preceding embodiments, wherein the orientation tool has a first longitudinal axis, further comprising determining a sensor reference angle between the first longitudinal axis and an axis of the second sensor, and calibrating the orientation tool based on the determined tool reference angle.

[0164] Embodiment 8: The method of any previous embodiment, wherein the orientation tool has a first longitudinal axis and the second sensor is in a calibrated reference orientation tool having a second longitudinal axis, further comprising determining a tool reference angle between the first longitudinal axis and the second longitudinal axis, and calibrating the orientation tool based on the determined tool reference angle.

[0165] Embodiment 9: The method of any previous embodiment, wherein the tool reference angle is less than 10 degrees.

[0166] Embodiment 10: The method of any previous embodiment, wherein the orientation tool and the reference orientation tool have the same design in at least one of size, magnetic source, and magnetic material distribution.

[0167] Embodiment 11: The method of any previous embodiment, wherein the orientation tool has a first longitudinal axis, and the first applied magnetic field and the second applied magnetic field are substantially aligned with the first longitudinal axis.

[0168] Embodiment 12: The method of any previous embodiment, wherein the first sensor has a response to the first applied magnetic field or the second applied magnetic field defined by a first order polynomial.

[0169] Embodiment 13: The method of any previous embodiment, further comprising applying an applied magnetic field along at least one of the x-axis of the orientation tool, the y-axis of the orientation tool, and the z-axis of the orientation tool.

[0170] Embodiment 14: The method of any previous embodiment, wherein the orientation tool has a first longitudinal axis, the method further comprising determining a bias of the first sensor from measurements of the first sensor obtained with the first longitudinal axis aligned parallel to the Earth’s magnetic field and measurements of the first sensor obtained with the first longitudinal axis aligned anti-parallel to the Earth’s magnetic field.

[0171] Embodiment 15: The method of any previous embodiment, further comprising determining a misalignment of the first sensor relative to the orientation tool from the one or more calibration parameters.

[0172] Embodiment 16: The method of any previous embodiment, further comprising determining a magnetic gradient in the wellbore using the calibrated first sensor.

[0173] Embodiment 17: The method of any previous embodiment, wherein calibrating comprises determining one or more calibration parameters, the one or more calibration parameters being a function of temperature, and wherein the calibration parameters are determined for a first temperature and a second temperature.

[0174] Embodiment 18: The method of any previous embodiment, wherein the orientation tool comprises a wire configured to carry an electrical current, and wherein calibrating comprises determining one or more calibration parameters, wherein the calibration parameters are a function of the electrical current, and wherein the calibration parameters are determined for a first electrical current through the wire and a second electrical current through the wire.

[0175] Embodiment 19: An orientation tool for surveying a wellbore in a bore string. The system comprises a first sensor, the first sensor being calibrated by: disposing the first sensor in a non-uniform ambient magnetic field, applying a first applied magnetic field to the first sensor, obtaining a first measurement at the first sensor while the first sensor is disposed within the non-uniform ambient magnetic field and the first applied magnetic field is applied, applying a second applied magnetic field to the first sensor, obtaining a second measurement at the first sensor while the first sensor is disposed within the non-uniform ambient magnetic field and the second applied magnetic field is applied, and calibrating the orientation tool based on the first measurement and the second measurement. Downhole survey measurements are obtained with the calibrated orientation tool disposed in the wellbore.

[0176] Embodiment 20: The orientation tool of any previous embodiment, wherein at least one of the first applied magnetic field and the second applied magnetic field is substantially zero.

[0177] Embodiment 21: The orientation tool of any previous embodiment, wherein the bore string is a drill string, and further comprising drilling the wellbore with the drill string using the downhole survey measurements.

[0178] Embodiment 22: The directional tool of any previous embodiment, further comprising disposing a second sensor in the non-uniform ambient magnetic field, applying the first applied magnetic field to the second sensor, obtaining a third measurement at the second sensor while the second sensor is disposed within the non-uniform ambient magnetic field and the first applied magnetic field is applied, applying the second applied magnetic field to the second sensor, obtaining a fourth measurement at the second sensor while the second sensor is disposed within the non-uniform ambient magnetic field and the second applied magnetic field is applied, and calibrating the directional tool based on the first measurement, the second measurement, the third measurement, and the fourth measurement.

[0179] Embodiment 23: The directional tool of any previous embodiment, wherein the second sensor is a calibrated sensor.

[0180] Embodiment 24: The directional tool of any previous embodiment, wherein calibrating comprises determining one or more calibration parameters that are a function of at least one of a sensitivity of the first sensor and a misalignment of the first sensor relative to the directional tool.

[0181] Embodiment 25: The directional tool of any of the preceding embodiments, wherein the directional tool has a first longitudinal axis, further comprising determining a sensor reference angle between the first longitudinal axis and an axis of the second sensor, and calibrating the directional tool based on the determined tool reference angle.

[0182] Embodiment 26: The directional tool of any previous embodiment, wherein the directional tool has a first longitudinal axis and the second sensor is in a calibrated reference directional tool having a second longitudinal axis, further comprising determining a tool reference angle between the first longitudinal axis and the second longitudinal axis, and calibrating the directional tool based on the determined tool reference angle.

[0183] Embodiment 27: The directional tool of any previous embodiment, wherein the tool reference angle is less than 10 degrees.

[0184] In the context of describing the application (particularly in the context of the appended claims), the use of the terms “one,” “a,” and “the” and similar referents is to be construed as covering both the singular and the plural, unless otherwise indicated herein or clearly contradicted by context. Further, it should be noted that the terms “first,” “second,” and the like, herein do not denote any order, quantity, or importance, but rather are used to distinguish one element from another. The modifier “about” used in connection with a quantity is inclusive of the stated value and has the meaning dictated by the context (e.g., it includes the degree of error associated with a particular measurement).

[0185] The teachings of the present disclosure can be used in a variety of well operations. These operations can involve treating a formation, a fluid residing in the formation, a wellbore, and / or equipment in the wellbore, such as production tubing, with one or more treatment agents. The treatment agents can be in the form of liquids, gases, solids, semi-solids, and mixtures thereof. Exemplary treatment agents include, but are not limited to, fracturing fluids, acids, steam, water, brine, corrosion inhibitors, cementing agents, permeability modifiers, drilling muds, emulsifiers, demulsifiers, tracers, flow improvers, and the like. Exemplary well operations include, but are not limited to, hydraulic fracturing, stimulation, tracer injection, cleaning, acidizing, steam injection, water injection, cementing, and the like.

[0186] While the application has been described with reference to one or more example embodiments, it will be understood by those skilled in the art that various changes can be made and equivalents can be substituted for elements thereof without departing from the scope of the application. In addition, many modifications can be made to adapt a particular situation or material to the teachings of the application without departing from the essential scope thereof. Therefore, it is intended that the application not be limited to the particular embodiment disclosed as the best or only mode contemplated for carrying out this application, but that the application will include all embodiments falling within the scope of the claims. Also, in the drawings and the specification, there have been disclosed exemplary embodiments of the application and, although specific terms are employed, they are used in a generic and descriptive sense only and not for purposes of limitation, otherwise, the scope of the application thus not to be construed as being limited to the specific embodiments set forth herein but to include all embodiments falling within the scope of the claims.

Claims

1. A method of surveying a wellbore (106) with a directional tool (142) in a bore string (102), the method comprising: setting a first sensor (302) of the directional tool (142) in a non-uniform ambient magnetic field prior to use of the directional tool in the bore string, the first sensor (302) being a primary sensor, a primary coil (312) being disposed about the first sensor, wherein the non-uniform ambient magnetic field comprises a surrounding earth magnetic field disturbed by a magnetic material or a magnetic source; applying a first applied magnetic field to the first sensor (302); obtaining a first measurement at the first sensor (302) while the first sensor (302) is set in the non-uniform ambient magnetic field and the first applied magnetic field is applied; applying a second applied magnetic field to the first sensor (302); obtaining a second measurement at the first sensor (302) while the first sensor (302) is set in the non-uniform ambient magnetic field and the second applied magnetic field is applied; setting a second sensor (304) in the non-uniform ambient magnetic field, the second sensor (304) being a reference sensor, a reference coil (314) being disposed about the second sensor; wherein a barrier is provided between the first sensor and the second sensor to isolate the first sensor and the second sensor axially apart by a selected distance; applying the first applied magnetic field to the second sensor (304); obtaining a third measurement at the second sensor (304) while the second sensor (304) is set in the non-uniform ambient magnetic field and the first applied magnetic field is applied; applying the second applied magnetic field to the second sensor (304); obtaining a fourth measurement at the second sensor while the second sensor (304) is set in the non-uniform ambient magnetic field and the second applied magnetic field is applied; calibrating the directional tool (142) based on the first measurement, the second measurement, the third measurement, and the fourth measurement; placing the directional tool in a borehole; and obtaining downhole survey measurements with the calibrated directional tool (142) in the wellbore (106).

2. The method of claim 1, wherein the second sensor (304) is a calibrated sensor.

3. The method of claim 1, wherein calibrating includes determining one or more calibration parameters that are a function of at least one of a sensitivity of the first sensor (302) and a misalignment of the first sensor (302) relative to the directional tool (142).

4. The method of claim 1, wherein the directional tool (142) has a first longitudinal axis, further comprising determining a sensor reference angle between the first longitudinal axis and an axis of the second sensor (304), and calibrating the directional tool (142) based on the determined tool reference angle.

5. The method of claim 1, wherein the orientation tool (142) has a first longitudinal axis, and the second sensor (304) is located in a calibrated reference orientation tool having a second longitudinal axis, further comprising determining a tool reference angle between the first longitudinal axis and the second longitudinal axis, and calibrating the orientation tool based on the determined tool reference angle.

6. The method of claim 1, wherein the orientation tool (142) has a first longitudinal axis, and the first applied magnetic field and the second applied magnetic field are substantially aligned with the first longitudinal axis.

7. The method of claim 1, further comprising applying an applied magnetic field along at least one of an x-axis of the orientation tool (142), a y-axis of the orientation tool (142), and a z-axis of the orientation tool (142).

8. The method of claim 1, wherein the directional tool (142) has a first longitudinal axis, further comprising: determining a bias of the first sensor (302) from measurements of the first sensor (302) obtained with the first longitudinal axis aligned parallel to the earth's magnetic field and measurements of the first sensor (302) obtained with the first longitudinal axis aligned anti-parallel to the earth's magnetic field.

9. The method of claim 3, further comprising determining a misalignment of the first sensor (302) relative to the orientation tool (142) from the one or more calibration parameters.

10. The method of claim 1, further comprising determining a magnetic gradient in the wellbore (106) using the calibrated first sensor (302).

11. The method of claim 1, wherein calibrating comprises determining one or more calibration parameters, the one or more calibration parameters being a function of temperature, and wherein the calibration parameters are determined for a first temperature and a second temperature.

12. The method of claim 1, wherein the orientation tool (142) comprises a wire configured to carry a current, and wherein calibrating comprises determining one or more calibration parameters, wherein the calibration parameters are a function of the current, and wherein the calibration parameters are determined for a first current through the wire and a second current through the wire.

13. An orientation tool (142) for surveying a wellbore (106) in a borehole string (102), the orientation tool comprising: a first sensor (302), the first sensor (302) being a primary sensor, a primary coil (312) being disposed about the first sensor, the first sensor being calibrated prior to use of the orientation tool in the borehole string by: disposing the first sensor (302) in a non-uniform ambient magnetic field, wherein the non-uniform ambient magnetic field comprises a surrounding earth's magnetic field disturbed by a magnetic material or a magnetic source; applying a first applied magnetic field to the first sensor (302); obtaining first measurements at the first sensor (302) while the first sensor (302) is disposed within the non-uniform ambient magnetic field and the first applied magnetic field is applied; applying a second applied magnetic field to the first sensor (302); obtaining a second measurement at the first sensor (302) while the first sensor (302) is disposed within the non-uniform ambient magnetic field and the second applied magnetic field is applied; the orientation tool further comprises: a second sensor (304) disposed within the non-uniform ambient magnetic field, wherein the second sensor (304) is a reference sensor, a reference coil (314) is disposed about the second sensor; wherein a barrier is disposed between the first sensor and the second sensor to isolate the first sensor and the second sensor axially a selected distance apart; the second sensor (304) is applied with the first applied magnetic field; obtaining a third measurement at the second sensor (304) while the second sensor (304) is disposed within the non-uniform ambient magnetic field and the first applied magnetic field is applied; the second sensor (304) is applied with the second applied magnetic field; obtaining a fourth measurement at the second sensor (304) while the second sensor (304) is disposed within the non-uniform ambient magnetic field and the second applied magnetic field is applied; and calibrating the orientation tool (142) based on the first measurement, the second measurement, the third measurement, and the fourth measurement; wherein downhole survey measurements are obtained with the calibrated orientation tool (142) disposed within the wellbore (106).

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