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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

This paper proposes an approach for measuring the azimuth angle and tilt angle of underground drilling tools with a MEMS three-axis accelerometer and a three-axis fluxgate sensor. A mathematical model of well logging attitude angle is deduced based on combining space coordinate transformations and algebraic equations. In addition, a system implementation plan of the inclinometer is given in this paper, which features low cost, small volume and integration. Aiming at the sensor and assembly errors, this paper analyses the sources of errors, and establishes two mathematical models of errors and calculates related parameters to achieve sensor calibration. The results show that this scheme can obtain a stable and high precision azimuth angle and tilt angle of drilling tools, with the deviation of the former less than ±1.4° and the deviation of the latter less than ±0.1°.

In oil, gas or geological exploration well logging work, acquiring the posture and orientation of the drilling tool in real-time [

Consequently this paper proposes a scheme to develop a measuring system for well logging attitude by using a MEMS three-axis accelerometer and a three-axis fluxgate sensor which has small size [

For directional well logging attitude measurement, the guiding parts mainly include a sensor module, signal acquisition module, microcontroller module and communication interface (SPI, SCI). All these component are assembled in a probe casing with small size and diameter (φ35 mm × 260 mm), and a PC works as the remote control and display device. The mechanical configuration of a logging tool can be expressed as shown in

A picture of the designed measuring probe is shown in

The main function of the measuring probe based on accelerometers and fluxgate sensors designed in this paper is to measure the azimuth angle and tilt angle of the well logging tool. The system principle block diagram of the hardware for the measuring device is shown in

The equipment used for measuring the attitude of a directional well logging tool is usually called a well logging inclinometer. Acquiring the posture and orientation uses the gravity field and magnetic field which have relative stability characteristics. Under different orientations, the fluxgate sensor and acceleration sensor data output will have different values. By 3D coordinate rotation and transformation, the current attitude angle and azimuth angle of the equipment can be uniquely determined.

As shown in _{b}/Y_{b}/Z_{b}_{b}_{b}

A device can always transform a fixed location to the current location through a rotation matrix. As shown in _{0}y_{0}z_{0}_{0} to the coordinates o_{1}y_{1}z_{1}_{1}_{2}y_{2}z_{2}_{2}_{b}Y′_{b}Z′_{b} which are the device body coordinates.

Therefore the rotation matrix can be expressed as _{b}Y′_{b}Z′_{b}_{0}y_{0}z_{0}

Let _{x}_{y}_{z}_{x}_{y}_{z}_{x}_{0} = _{y}_{0} = 0, _{z}_{0} = +1

Therefore,

In the local horizontal plane, the values of the fluxgate sensor in the E/N/U direction can be calculated as _{x}_{0} = 0, _{y}_{0} = _{z}_{0} =

And

Here _{x}_{x}_{y}_{y}_{z}_{z}.

The acceleration sensors and fluxgate sensors used in this paper have been strictly calibrated before they leave the factory, and their accuracy has a certain guarantee. However, in considering the overall measurement equipment, after the device is assembled in the mechanical aspects, this will cause new errors due to the inevitable mechanical installation axial misalignment, circuit effects, hard-iron interference,

The actual device axis (Z-axis) is defined as the reference axis for calibration. Taking the errors of the sensors and the types of errors after completion of the sensors assembly into unified consideration and fusion processing, the errors of the system after assembly are mainly the result of four aspects: (1) Misalignment error is defined as the angles between the sensor sensing axes and the device body axes, caused by manufacturing and installation; (2) Hard-iron interference magnetic field is normally generated by ferromagnetic materials with permanent magnetic fields that are part of the device structure. These materials could be permanent magnets, magnetized iron or steel; (3) A soft-iron interference magnetic field is generated by the uncertain magnetically soft materials surrounding the device or the items inside current carrying traces on the PCB. For some platforms, hard-iron interference is the primary source of error and soft-iron distortion is minimal or non-existent; (4) The scale factor error is defined as the mismatch of the sensitivity of the sensor sensing axes. Ideally, the three-axis sensors that make up the triad are identical. In reality, however, this may not be the case. Each sensor channel may have different sensitivities. Calibration is designed to reduce these errors.

To calibrate these errors, the existing least square method is, through the establishment of multi-parameter equation, used to collect multiple samples to calculate the calibration parameters. However, they have the following shortcomings. First, some perform the error correction incompletely. For example, it only corrects two or three of the four errors. Second, the number of samples limits the accuracy of the parameters. Third, a variety of established equations are not simple and clear with complicated solving processes for the parameter equations. Additionally, the ellipsoidal model is also established in some papers to achieve the magnetic calibration, but it involves a complex parameter solving process and adopts a simplified approximation to replace the parameter values, which cannot fully represent the types of errors. This paper establishes a comprehensive error model based on the above four errors, and uses the least square method to calculate a calibration matrix. A simple and practical calibration process is thus designed.

The error model of the accelerometer can be expressed as follows [

Here [_{m}_{3×3} is a 3 × 3 misalignment matrix between the accelerometer sensing axes and the device body axes; _{i}(_{i}(_{10}∼_{33} are the calibration parameters, _{x}_{0}, _{y}_{0}, _{z}_{0} are raw measurements and _{x}_{y}_{z}_{10} to _{33}, and with any given normalized values in a position, the raw measurements can be obtained. For example, at _{b}_{x} A_{y} A_{z}_{x}_{0}, _{y}_{0} and _{z}_{0} can be collected. According to the standard turntable, we choose 10 positions with _{b}_{b}_{b}_{x}_{y}_{z}_{x}_{y}_{z}_{x}_{0}, _{y}_{0} and _{z}_{0}. The calibration parameter matrix

If the raw data of accelerometer is [_{x}_{1} _{y}_{1} _{z}_{1}], the calibrated data which be used to calculate the attitude angle can be expressed as [_{x}_{2} _{y}_{2} _{z}_{2}] = [_{x}_{1} _{y}_{1} _{z}_{1} 1]·

The calculation process of the accelerometer calibration parameters is shown in

The relationship between the normalized data _{x}_{y}_{z}_{x0}, _{y0}, _{z0} can be expressed as

Here [_{m}_{m}_{i}(_{m}_{i}(_{s}] is a 3 × 3 matrix caused by soft-iron distortion. The goal of the magnetic sensor calibration is to determine the parameters from _{10}_{33}

Here _{0}, _{0}, _{0} are the offsets _{m}_{i}(

Here _{0},_{0},_{0} are the offsets _{m}_{i}(_{x}_{y}_{z} a

Then:

The least square method can be applied to determine the parameters

Then:

Let:

Then

Therefore:

Let:

Up to now, _{m}_{i}(_{m}_{i}(_{s}]_{3×3} matrix caused by soft-iron distortion have been determined.

Let _{m}_{×3} = [_{2z} _{2z} _{2z}] be the Z_{b} down rotation circle data after scale factor, hard-iron and soft-iron correction:

Then:

So the normalized rotation vector for Z_{b} down rotation is:

Similarly, the normalized rotation vectors _{x}_{y}_{b}_{b}

So the parameters from _{10} to _{33} can be calculated by

For attitude measurement of exploring casinga in production and practice, a non-magnetic and omnibearing standard turntable, which can display and inspect the tilt angle (ranging 0 to ±90°) and the azimuth (ranging 0 to 360°), is often used as the test platform. Moreover the standard turntable is strictly adjusted by precise third party calibration instruments before the experiment, and then we can examine the resulting precision of the angle measurement based on the turntable. The adjusted turntable can guarantee the tilt angle is 0° and the azimuth is 0° when it is at the zero position, and the reading error of the turntable calibration is within ±0.1°; That is, the experimental turntable guarantees the tile angle scale indicates 0° with the exploring casing is vertically direct to the ground, and it also guarantees the azimuth scale indicates 0° with the exploring casing is directed to the magnetic north. The system adopts the output value of the final calculated measurement result by comparing the current value of the turntable calibration as the relative error for measurements, which is a conventional method for cylindrical, probe tubular underground inclinometer devices, this method is simple, easily used, and also able to test the measurement precision.

The test calibration and experiment platform is shown in

There is a standard method to show the fluxgate calibration results. As shown in

The turntable is use to test the designed inclinometer. The inclinometer needs keep the same center with the turntable. Taking eight tilt angles (3°, 15°, 30°, 60°, −3°, −15°, −30°, −60°), and rotating eight azimuth angles, respectively (0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°) at each tilt angle, we then record and save the current measurement result values. Comparing these values with the standard tilt angle and azimuth angle, _{0} is the tilt angle calculated with no accelerometer calibration, _{0} is the deviation between _{0} and the stander value of tilt and

_{0} is the azimuth angle calculated after fluxgate sensor calibration using the traditional ellipse matching error compensation algorithm [_{0} and the stander values of azimuth and _{0}, _{0} and

The two forms of data above show that the azimuth angle error with traditional compensation will reach ±4° and the tilt angle error with no calibration will reach ±0.4°. As for the errors of the accelerometer (scale factor error, misalignment error, external disturbance) and fluxgate sensor (hard-iron interference, soft-iron interference, scale factor error, misalignment error), the azimuth error is less than ±1.4° and tilt angle error is less than ±0.1° after calibration by the proposed method, so we can state that the calibration method improves the accuracy of the attitude angle and is proved to be effective, so it can be applied to actual well logging work.

Based on the design of a measuring system for well logging attitude, this paper proposes a concise formula for attitude angle calculation, and establishes mathematical models to calibrate errors. Taking the error of the sensor itself and the four main types of errors after completion of the sensor assembly into unified consideration and fusion processing, a comprehensive error model has been established. Based on this model, a simple and practical calibration process is designed, which can be completed using a standard test turntable. Furthermore, it can complete the calibration of equipment errors, including installation errors and sensor errors, so the measurement accuracy can then be improved. The test results show that these schemes are effective and highly precise. The designed measuring equipment has utility in engineering applications and has the characteristics of small size, high integration, low-cost and easy adaptation to other devices. The measuring deviation of azimuth angle and tilt angle of drilling tools are thus greatly reduced. The calibration parameters can be calculated in advance and be used in the soft system to acquire the attitude angle of directional well logging.

The authors would like to acknowledge the reviewers for their constructive and helpful suggestions. The authors thank the editor for English verification.

The authors declare no conflict of interest.

The configuration of a logging tool.

The measuring probe.

The system principle block diagram.

Attitude angle diagram.

The process of coordinate transformation.

The calculation process of accelerometer calibration parameter.

The calculation process of fluxgate sensor calibration parameters.

The calibration and experiment platform.

Comparision of fluxgate output data before and after calibration.

The tilt angle measurement data.

_{0} |
||||
---|---|---|---|---|

3° | 2.76° | −0.24° | 2.91° | −0.09 |

15° | 14.73° | −0.27° | 14.93° | −0.07 |

30° | 29.71° | −0.29° | 29.98° | −0.02 |

60° | 59.62° | −0.38° | 59.95° | −0.05 |

−3° | −3.28° | −0.28° | −3.03° | −0.03 |

−15° | −15.33° | −0.33° | −15.06° | −0.06 |

−30° | −30.37° | −0.37° | −30.08° | −0.08 |

−60° | −60.40° | −0.40° | −60.03° | −0.03 |

The azimuth angle measurement data.

_{0} |
||||
---|---|---|---|---|

0° | 0.85° | 0.85° | 0.57° | 0.57° |

45° | 41.73° | −3.27° | 45.14° | 0.14° |

90° | 87.89° | −2.12° | 91.35° | 1.35° |

135° | 131.45° | −3.55° | 136.19° | 1.19° |

180° | 184.11° | 4.11° | 181.36° | 1.36° |

225° | 228.05° | 3.05° | 223.98° | −1.02° |

270° | 273.4° | 3.4° | 269.12° | −0.88° |

315° | 318.20° | 3.2° | 313.81° | −1.19° |