UNIVERSITY OF CALGARY. Reduction of Wellbore Positional Uncertainty During Directional Drilling. Zahra Hadavand A THESIS

Transcription

1 UNIVERSITY OF CALGARY Reduction of Wellbore Positional Uncertainty During Directional Drilling by Zahra Hadavand A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN GEOMATICS ENGINEERING CALGARY, ALBERTA JANUARY, 2015 Zahra Hadavand 2015

2 Abstract Magnetic measurement errors significantly affect the wellbore positional accuracy in directional drilling operations taken by Measurement While Drilling (MWD) sensors. Therefore this research has provided a general overview of error compensation models for magnetic surveys and elaborated the most accurate calibration methods of hard- and soft-iron, as well as multiple-survey correction for compensating drilling assembly magnetic interference to solve the problem of wellbore positional uncertainty and provide accurate surveying solution downhole. The robustness of hard- and soft-iron calibration algorithm was validated through an iterative least-squares estimator initialized using a two-step linear solution. A case study of a well profile, a simulated well profile and a set of experimental data are utilized to perform a comparison study. The comparison analysis outcomes imply that position accuracy gained by multistation analysis surpasses hard- and soft-iron compensation results. Utilization of multiple-survey correction in conjunction with real-time geomagnetic referencing to monitor geomagnetic disturbances, such as diurnal effects, as well as changes in the local field by providing updated components of reference geomagnetic field, provide superior accuracy. ii

3 Acknowledgements I would like to express my gratitude to my supervisors, Dr. Michael Sideris and Dr. Jeong Woo Kim for their support on this research project over the past two and a half years. I am deeply thankful to my supervisor, Dr. Sideris for his professional supervision, critical discussions, guidance and encouragements. I would like also to thank Dr. Kim, my co-supervisor, for proposing this research project, for his continuous support, and immeasurable contributions throughout my studies. I would like to thank Dr. Kim for the time he offered to facilitate this research project by providing access to the surveying equipment available at the Laboratory of the Department of Geomatics Engineering at the University of Calgary. I thank the students in the Micro Engineering Dynamics and Automation Laboratory in department of Mechanical & Manufacturing Engineering at the University of Calgary for the collection of the MEMS sensors experimental data. I would thank Dr. Simon Park, and Dr. Mohamed Elhabiby for serving on my examination committee. I am really thankful of Department of Geomatics Engineering, University of Calgary for the giving me the chance to pursue my studies in the Master of Science program. iii

4 Dedication To my father and my mother for their unlimited moral support and continuous encouragements, You have been a constant source of love, encouragement and inspiration. Words will never say how grateful I am to you iv

5 Table of Contents Abstract... ii Acknowledgements... iii Dedication... iv Table of Contents...v List of Tables... vii List of Symbols and Abbreviations... xi CHAPTER ONE: INTRODUCTION Problem statement Borehole Azimuth Uncertainty Geomagnetic Referencing Uncertainty Thesis Objectives Thesis Outline...7 CHAPTER TWO: REVIEW OF DIRECTIONAL DRILLING CONCEPTS AND THEORY Wellbore Depth and Heading Review of Sources and Magnitude of Geomagnetic Field Variations Review of Global Magnetic Models Measuring Crustal Anomalies Using In-Field Referencing (IFR) Technique Interpolated IFR (IIFR) Theory of Drillstring Magnetic Error Field Ferromagnetic Materials, Hard-Iron and Soft-Iron Interference Surveying of Boreholes Heading Calculation Review of the Principles of the MWD Magnetic Surveying Technology Horizontal Wells Azimuth Previous Studies Magnetic Forward Modeling of Drillstring Standard Method Short Collar Method or Conventional Magnetic Survey (Single Survey) Multi-Station Analysis (MSA) Non-Magnetic Surveys Summary...30 CHAPTER THREE: METHODOLOGY MSA Correction Model Hard-Iron and Soft-Iron Magnetic Interference Calibration Static Hard-Iron Interference Coefficients Soft-Iron Interference Coefficients Relating the Locus of Magnetometer Measurements to Calibration Coefficients Calibration Model Symmetric Constrait Least-Squares Estimation...47 v

6 3.2.7 Establishing Initial Conditions Step 1: Hard-Iron Offset estimation Step 2: Solving Ellipsoid Fit Matrix by an Optimal Ellipsoid Fit to the Data Corrected for Hard Iron Biases Well path Design and Planning Summary...58 CHAPTER FOUR: RESULTS AND ANALYSIS Simulation Studies for Validation of the Hard and Soft-Iron Iterative Algorithm Experimental Investigations Laboratory Experiment Experimental Setup Turntable Setup Data Collection Procedure for Magnetometer Calibration Heading Formula Correction of the Diurnal Variations Calibration Coefficients Simulated Wellbore A Case Study Summary CHAPTER FIVE: CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH Summary and Conclusions Recommendations for Future Research Cautions of Hard-Iron and Soft-iron Calibration Cautions of MSA Technique REFERENCES APPENDIX A: SIMULATED WELLBORE vi

7 List of Tables Table 4-1. The ellipsoid of simulated data Table 4-2. Parameters solved for magnetometer calibration simulations Table 4-3. Features of 3-axis accelerometer and 3-axis magnetometer MEMS based sensors Table 4-4. Turn table setup for stationary data acquisition Table 4-5. Diurnal correction at laboratory Table 4-6. Parameters in the magnetometer calibration experiment Table 4-7. A comparative summary of headings calibrated by different methods with respect to the nominal heading inputs Table 4-8. Geomagnetic referencing values applied for the simulated wellbore Table 4-9. The ellipsoid of simulated data Table Calibration parameters solved for simulated wellbore Table Comparative wellbore trajectory results of all correction methods Table Geomagnetic referencing values Table Calibration parameters solved for the case study Table Comparative wellbore trajectory results of all correction methods vii

8 List of Figures and Illustrations Figure 2-1. Arrangement of sensors in an MWD tool... 8 Figure 2-2. A Perspective view of Earth-fixed and instrument-fixed orthogonal axes which denote BHA directions in three dimensions Figure 2-3. Horizontal component of error vector Figure 2-4. East/west component of error vector Figure 2-5. Conventional correction by minimum distance Figure 3-1. Representation of the geometry of the tangential method Figure 3-2. Representation of the geometry of the minimum curvature method Figure 4-1. Sphere locus with each circle of data points corresponding to magnetic field measurements made by the sensor rotation at highside 90, Figure 4-2. Sphere locus with each circle of data points corresponding to magnetic field measurements made by the sensor rotation at inclination 90, Figure 4-3. A schematic illustrating the disturbed data lying on an ellipsoid Figure 4-4. Histogram of the magnetometer output error based on real data of a case study Figure 4-5. Case #I: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the least-squares iterations Figure 4-6. Case #II: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the least-squares iterations Figure 4-7. Case #III: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the least-squares iterations Figure 4-8. Case #IV: Divergence of hard-iron (in mgauss) and soft-iron (unit-less) estimates for the least-squares iterations Figure 4-9. Case #V: Divergence of hard-iron (in mgauss) and soft-iron (unit-less) estimates for the least-squares iterations Figure Case #VI: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the least-squares iterations Figure Experimental setup of MEMS integrated sensors on turn table at 45 inclination.. 74 Figure Inclination set up for each test viii

9 Figure The observations of the geomagnetic field strength follow a 24 hour periodic trend Figure Geomagnetic field intensity in the frequency domain Figure Geomagnetic field intensity in the time domain Figure Portion of the ellipsoid representing the locus of magnetometer measurements from laboratory experimental data Figure Headings calibrated by MSA versus hard and soft iron (georeferncing obtained by IGRF model corrected for diurnal effects) Figure Headings calibrated by MSA versus hard and soft iron (georeferncing obtained by IGRF model without diurnal corrections) Figure Simulated wellbore horizontal profile Figure Portion of the ellipsoid representing the locus of magnetometer measurements from BUILD section of the simulated wellbore Figure Portion of the ellipsoid representing the locus of magnetometer measurements from LATERAL section of the simulated wellbore Figure Conventional correction is unstable in LATERAL section Figure Conventional correction instability based on inclination Figure Calculated field strength by calibrated measurements Figure Calculated field direction by calibrated measurements Figure Case #I: Wellbore pictorial view of the simulated wellbore in horizontal plane (no error) Figure Case #II: Wellbore pictorial view of the simulated wellbore in horizontal plane (random normally distributed noise of 0.3 mgauss) Figure Case #III: Wellbore pictorial view of the simulated wellbore in horizontal plane (random normally distributed noise of 6 mgauss) Figure Conventional correction is unstable in LATERAL section Figure Zoom1 of Figure Figure Zoom2 of Figure Figure Conventional correction instability based on inclination ix

10 Figure Calculated field strength by calibrated measurements Figure Calculated field direction by calibrated measurements Figure Wellbore pictorial view in horizontal plane by minimum curvature x

11 th List of Symbols and Abbreviations Symbol AZ AZ 1 AZ 2 B B B P B N, B E, B V B V(n), B h(n) B V (ref), B h(ref) B x, B y, and B z B xcorr(n), B ycorr(n), B zcorr(n) B xm(n), B ym(n), B zm(n) DIP DL g Description borehole azimuth azimuth angle at upper survey point azimuth angle at lower survey point geomagnetic vector strength of geomagnetic field magnetic field measured at a survey point geomagnetic components along Earth s coordinate frame vertical and horizontal components of magnetic field at np survey station reference value of vertical and horizontal components of geomagnetic field geomagnetic components along instrument-fixed coordinate frame corrected magnetic components at np instrument-fixed coordinate frame measured magnetic components at np instrument-fixed coordinate frame dip angle of geomagnetic vector dog-leg curvature magnitude of gravity vector th survey station in survey station in th xi

12 g G x, G y, G z HS I I 1 I 2 MD N RF TVD U NEV U XYZ V V x,v y, and V z W AZ B x, B y B z gravity vector gravity components along instrument-fixed coordinate frame borehole highside angle borehole inclination inclination angle at upper survey point inclination angle at lower survey point measured depth number of surveys ratio factor for minimum curvature true vertical depth unit vectors in Earth s coordinate frame unit vectors in instrument-fixed coordinate frame hard-iron vector components of hard-iron vector along instrument-fixed coordinate frame soft-iron matrix change in parameter borehole azimuth error drillstring magnetic error field in cross-axial direction drillstring magnetic error field in axial direction ε x, ε y, ε z V small perturbations of B x, B y, B z variance xii

13 Abbreviation BGGM BHA HDGM IFR IGRF IIFR MEMS mgauss MSA MWD NMDC NOAA nt SSA WBM Description British Global Geomagnetic Model Bottom-Hole-Assembly High Definition Geomagnetic Model In-Field Referencing International Geomagnetic Reference Field Interpolated IFR Micro Electro-Mechanical Systems miligauss Multi-Station Analysis Measurement While Drilling Non-Magnetic Drill Collars National Oceanic and Atmospheric Administration nanotesla Single Station Analysis Wellbore Mapping xiii

14 Chapter One: Introduction Directional drilling is the technology of directing a wellbore along a predefined trajectory leading to a subsurface target (Bourgoyne et al. 2005). In recent years, directional drilling technology has gained more attention than vertical drilling in global oil and gas industries. The reason is that horizontal (deviated) wells, whose borehole intentionally departs from vertical by a significant extent over at least part of its depth (Russell and Russell 2003), have higher oil and gas deliverability since they have larger contact area with oil and gas reservoirs (Joshi and Ding 1991). This in turn significantly reduces the cost and time of drilling operation since a cluster of deviated wells drilled from a single drilling rig allows a wider area to be tapped at one time without the need for relocation of the rig which is expensive and time-consuming. Therefore drilling horizontal wells can reduce the number of wells required and minimize surface disturbance, which is important in environmentally sensitive areas. However, suitable control of the borehole deviation as it is drilled requires precise knowledge of the instantaneous depth and heading of the wellbore. Therefore, obtaining accurate measurements of depth, inclination and azimuth of wellbore is a fundamental requirement for the drilling engineer to be all the time aware of the drilling bit direction. Depth is acquired by drill pipe measurements, while inclination and azimuth are achieved from gravitational and magnetic field measurements. Horizontal drilling operations in the oil industry utilize the measurement while drilling (MWD) technique. MWD incorporates a package of sensors including a tri-axial magnetometer and a tri-axial accelerometer mounted in three mutually orthogonal directions inserted within a downhole probe. The sensors monitor the position and the orientation of the bottom-hole-assembly (BHA) during drilling by instantaneous measuring of magnetic and gravity conditions while the BHA is completely stationary. 1

15 A perpendicular pair or an orthogonal triad of accelerometers measure the Earth s gravity field to determine the BHA inclination and tool face angles while the magnetometers measure the geomagnetic components to determine the BHA azimuth at some predetermined survey stations along the wellbore path. In a directional survey of wellbore, many sources of uncertainty can degrade accuracy, including gravity model errors, depth errors, sensor calibration, instrument misalignment, BHA bending, centralization errors, and environmental magnetic error sources. This thesis focuses on the wellbore magnetic directional survey since the main difficulty in making an accurate positional survey of wellbore is largely driven by uncertainty resulting from environmental magnetic error sources which are caused by two major error sources, the un-modeled geomagnetic field variations and drillstring magnetism interference induced by ferrous and steel materials around the drilling rig. The best insurance against the geomagnetic referencing uncertainty is a site survey to map the crustal anomalies (local magnetic parameters) using In-Field Referencing (IFR) and remove geomagnetic disturbances using the Interpolated IFR (IIFR) method. Magnetic interference of drilling assembly is compensated through various methods such as a multiple-survey correction in order to reduce positional survey uncertainty. Reduced separation between adjacent wells is allowed as a result of the overall reduced position uncertainty (Lowdon and Chia 2003). In recent years, the oil companies and the drilling contractors have shown a great deal of interest in research investigations of possible error sources in directional drilling magnetic surveys. A drilling engineer s ability to determine the borehole trajectory depends on the accumulation of errors from wellhead to total path. In modern magnetic surveys with MWD tools, the 2

16 combined effects of accumulated error may reach values of 1% of the measured well depth, which could be unacceptably large for long wellbores (Buchanan et al. 2013). To place wellbores accurately when using MWD surveying tools, the modern industry has promoted the development of rigorous mathematical procedures for compensating various error sources. As a result, the general wellbore positional accuracies available in the industry are of the order of 0.5% of the wellbore horizontal displacement. 1.1 Problem statement The Wellbore Positional accuracy in directional drilling operations taken by Measurement While Drilling (MWD) sensors decreases with the increase of the inclination from the vertical. From experiments, it is evident that, at small inclinations, the influence of the drilling assembly interfering field in the azimuth can often be neglected while at high inclinations, the error in the azimuth is significant. As a result, horizontal wells which are frequently employed in the oil and gas industry represent a challenge in directional surveying (Grindrod and Wolff 1983). This study is concerned with the magnetic surveying of boreholes, and relates more particularly but, not exclusively to determining the corrected azimuth of a horizontal well. Several error sources affect the accuracy of the magnetic surveys and can be summarized as follows: Borehole Azimuth Uncertainty Since in conventional magnetic instruments the azimuth read by the compass is determined by the horizontal component of the local magnetic field, all magnetic surveys are subject to azimuth uncertainty if the horizontal component of the local magnetic field observed by the instrument at the borehole location is not aligned with the expected magnetic north direction whose declination is obtained from main-field geomagnetic models or the IFR technique (Brooks et al. 1998). The sources of error in azimuth can be categorized as follows (Scott and MacDonald 1979): 3

17 (i) The massive amount of ferrous and steel materials around the drilling rig have a deleterious impact on the magnetometer measurements taken by MWD sensors (Thorogood and Knott 1990). Drilling assembly magnetic error field is a common phenomenon, as there is a desire to get the survey information as close to the bit as possible. (ii) The magnetic surveying sensors are installed at a distance behind the drill bit due to the additional weight imposed by Non-Magnetic Drill Collars (NMDC) on the bit (Conti 1989). Consequently, Conti (1989) and Rehm et al. (1989) have evaluated that the sensors might not be capable of monitoring some rotational motions experienced only by the drill bit assembly and thus the overall reliability of the magnetic survey is affected. Another source of error in magnetic surveys is misalignment of the survey tool s axis with the borehole axis. The cause of this could be bending of the drill collars within the borehole or poor centralization of the tool within the drill collar (Carden and Grace 2007). The azimuth errors caused by instrument misalignment are usually small in comparison with others and their effect tends to be randomized as the toolface angle changes between surveys (Brooks et al. 1998). (iii) Sensor calibration errors, which are the errors in accelerometer and magnetometer readings (and gyro readings), cause the measurements to be imprecise and consequently, there is uncertainty in the azimuth calculated from these measurements (Brooks et al. 1998). In this study, effects of temperature and pressure were considered negligible. The calibration of the magnetometer is more complicated because there are error sources not only from instrumentation but also from the magnetic deviations on the probe which was classified as the first error source above. 4

18 1.1.2 Geomagnetic Referencing Uncertainty The geomagnetic field declination is normally determined by estimations of the geomagnetic field obtained from global and regional models of the main field, such as the International Geomagnetic Reference Field (IGRF) (Russell et al. 1995). Russell et al. (1995) indicated that the geomagnetic field for any location at any time, calculated only from a main-field model includes significant error. These models do not consider short term magnetic variations of geologic sources and geomagnetic disturbances such as diurnal variations which are potentially large and thus lead to considerable uncertainty in declination which is a major contributor to azimuth uncertainty. The best insurance against crustal anomalies is a site survey to measure the local magnetic parameters in real-time using IFR in order to map the local anomalies as corrections to one of the global models. Diurnal variations can be corrected using IIFR method. Since variations of the geomagnetic field are quite significant with respect to the performance capabilities of the magnetic survey tools, geomagnetic referencing uncertainty raises a global drilling problem whenever magnetic survey tools are employed (Wright 1988). Cheatham et al. (1992) and Thorogood (1990) have investigated that the declination uncertainty and the drillstring magnetization interference associated with the surrounding magnetic environment are systematic over a group of surveys and thus dominate the overall uncertainty in the determination of wellbore orientation. Recent trends in the drilling industry tend to establish several horizontal wells from the same platform (Anon 1999; Njaerheim et al. 1998). This necessitates the reduction of magnetic survey uncertainty by the utilization of a reliable error model so as to correct the BHA position and orientation within the severe downhole drilling conditions and avoid collision with adjacent wells. 5

19 1.2 Thesis Objectives Within the context of using magnetic error correction models for the purpose of reducing wellbore position uncertainty, the main research objectives are as follows: Execute multistation analysis as well as utilize hard and soft-iron algorithm for calibration of magnetometers to compensate the drilling assembly magnetic disturbances through real, experimental and simulated results. Estimate the applicability of the magnetic compensation methods, including singlesurvey analysis, multiple-survey analysis, as well as hard- and soft-iron calibration, by comparative evaluation of respective results in order to be able to identify the most accurate magnetic compensation solution for drilling assembly magnetic interference and reach the desired target. Analyze experimental results to investigate whether there is a noticeable improvement in survey accuracy when the effects of time varying disturbances of geomagnetic field such as diurnal variations are reduced through adaptive filtering of the wellsite raw magnetic data. It may be implied that the position accuracy of all correction methods can be improved by mapping the crustal magnetic field of the drilling area. Correct the case study wellbore trajectory by applying the most accurate magnetic compensation solution for drillstring-induced interference, and combine the results with realtime geomagnetic referencing (accounting for the influence of the crustal field, as well as secular variations in the main magnetic field). Afterward, the achieved positional accuracy is compared with the available wellbore positional accuracy in the industry. 6

20 1.3 Thesis Outline Chapter 2 provides background information necessary for understanding the concepts discussed in Chapter 3 and 4. Chapter 3 discusses the MSA correction model as well as the hardiron and soft-iron magnetic interference calibration model, and examines the most accurate well path planning method applied in the oil industry to achieve the corrected wellbore trajectory. Chapter4 evaluates the proposed methods through the results of a case study, simulation analysis and experimental investigations. Finally, Chapter 5 provides the main conclusions with respect to the stated thesis objectives and also provides recommendations for future investigations. 7

21 Chapter Two: REVIEW of DIRECTIONAL DRILLING CONCEPTS and THEORY 2.1 Wellbore Depth and Heading While the depth of the BHA can be determined from the surface simply by counting the number of standard-length tubes coupled into the drillstring, determination of the BHA heading requires downhole measurements (Russell and Russell 2003). Russell et al. (1978) denoted the word heading as the vertical direction and horizontal direction in which the BHA is pointing. The vertical direction is referred to as inclination and the horizontal direction is referred to as azimuth. The combination of inclination and azimuth at any point down the borehole is the borehole heading at that point. For the purpose of directional analysis, any length of the borehole path can be considered as straight. The inclination at any point along the borehole path is the angle of the instrument s longitudinal axis with respect to the direction of the Earth s gravity vector when the instrumental axis is aligned with the borehole path at that point. In other words, inclination is the deviation of the longitudinal axis of the BHA from the true vertical. Azimuth is the angle between the vertical plane containing the instrument longitudinal axis and a reference vertical plane, which may be magnetically or gyroscopically defined (Figure 2-1 and Figure 2-2). Figure 2-1. Arrangement of sensors in an MWD tool 8

22 This study is concerned with the measurement of the azimuth defined by a magnetic reference vertical plane, containing a defined magnetic north (Russell and Russell 1991). The horizontal angle from the defined magnetic north clockwise to the vertical plane including the borehole axis is hereafter simply referred to as azimuth. When the defined magnetic north contains the geomagnetic main field vector at the instrument location, the corresponding azimuth, referred to as absolute azimuth or corrected azimuth, is the azimuth value required in directional drilling process. However, in practice, the measured local magnetic field is deviated from the geomagnetic main field (Russell and Russell 2003). The process of estimating these magnetic distorting errors and removing them from the magnetometer measurements is the subject of this research. The azimuth of wellbore is measured from magnetic north initially but is usually corrected to the geographic north to make accurate maps of directional drilling. A spatial survey of the path of a borehole is usually derived from a series of measurements of an azimuth and an inclination made at successive stations along the path and the distance between these stations are accurately known (Russell 1989). 2.2 Review of Sources and Magnitude of Geomagnetic Field Variations The geomagnetic field at any location is defined in terms of three components of a vector, including the field strength, the declination angle defined as the direction of the geomagnetic north relative to geographic (true) north, and the dip angle defined as the dip angle of the geomagnetic vector measured downwards from the horizontal (University of Highlands and Island 2012). According to Wright (1988) and Ripka (2001), the geomagnetic field is used as a north reference from which the wellbore direction is computed. Afterward, the geomagnetic north is referenced to the geographic north form a knowledge of the declination angle. A 9

23 knowledge of the sources and magnitude of geomagnetic field variations helps our understanding of the magnetic survey accuracy problem. A concise description of the geomagnetic field is, therefore, appropriate here. The geomagnetic field at any point on the Earth s surface is the result of the principal sources of magnetism as follows: (i) The main field originating from the enormous magnetic core at the heart of the Earth accounts for about 98-99% of the field strength at most places at most times. (ii) The Earth s core field is not stationary and has been turbulent over the course of history resulting in a magnetic vector that is constantly changing. This change, referred to as the secular variation, is very rapid in geological time scales. (iii) Diurnal magnetic field variations and magnetic storms are caused by varying solar wind and electric currents flowing external to the Earth s surface and interacting with the main field (Wolf and dewardt 1981). Fields created by the magnetization of rocks and minerals in the Earth s crust, typically found in deep basement formations in the vicinity of drilling (crustal anomalies) (Bourgoyne et al. 2005). Geomagnetic field is a function of time and geographical location (Ozyagcilar 2012c) and can be modeled with reasonable accuracy using the global geomagnetic reference field models Review of Global Magnetic Models In order to keep track of the secular variation (long wavelengths) in the magnetic field of the Earth core, several global magnetic models are maintained to provide prediction models. International organizations such as INTERMAGNET collate data from observatories scattered throughout the world to model the intensity and attitude of the geomagnetic field (University of Highlands and Island 2012). For instance, every year, the data is sent to the British Geological 10

24 Survey in Edinburg where this data is entered to a computer model called the British Global Geomagnetic Model (BGGM). Higher-order models take into account more localized crustal effects (short wavelengths) by using a higher order function to model the observed variations in the Earth field (University of Highlands and Island 2012). The lower order models, such as the International Geomagnetic Reference Field (IGRF), are freely accessible over the internet whereas the higher order models require an annual license. This research applies the IGRF model coefficients produced by the participating members of the IAGA Working Group V-MOD (Finlay et al. 2010). Geomagnetic referencing is now a well-developed service and various techniques have been used in the industry for the purpose of measuring and predicting the geomagnetic field at the wellsite Measuring Crustal Anomalies Using In-Field Referencing (IFR) Technique One significant source of error in the determination of the geomagnetic reference field is crustal variations. The global models can only resolve longer wavelength variations in the geomagnetic field and cannot be expected to account for localized crustal anomalies (University of Highlands and Island 2012). In order to correct for the crustal anomalies, the geomagnetic field has to be measured on site. IFR is the name given to the novel technique of measuring the local geomagnetic field elements, including field strength, dip angle and declination in real-time, routinely made at magnetic observatories in the vicinity of the drilling activity while the interference from the rig and drilling hardware, and other man-made sources of magnetic interference should be avoided. The field strength is measured by a Caesium or proton precision magnetometer. Declination and dip angle measurements are made by a non-magnetic theodolite with a fluxgate magnetometer mounted on its telescope. The measurement of declination angle is made against a 11

25 true north. The true north can be determined by means of astronomical observations or by using a high-accuracy north-seeking gyro mounted on the theodolite (Russell et al. 1995). Once the IFR measurements of the geomagnetic field have been taken, contoured maps and digital data files are produced and can be viewed with a computer software. This allows the MWD contractor to view the data and interpolate suitable geomagnetic field values at any point within the oilfield (University of Highlands and Island 2012). The crustal corrections vary only on geological time scales and therefore can be considered as fixed over the lifetime of the field. On the other hand, the global model (such as IGRF) tracks very well the time variation in the overall geomagnetic field. As a result, combining the global model and the IFR crustal corrections provide the MWD contractor with the most accurate estimate of the geomagnetic field at the rig (University of Highlands and Island 2012). IFR significantly reduces declination uncertainty and improves the accuracy of magnetic surveys by monitoring changes in the local geomagnetic field during surveys and therefore providing updated components of the reference field (Russell et al. 1995) Interpolated IFR (IIFR) IIFR is a method of correcting for time variant disturbances of the geomagnetic field in a way that a reference station is installed on the surface at or near the wellsite to sense geomagnetic disturbances such as diurnal variations (Lowdon and Chia 2003). The variations observed at this surface reference station can be applied to the downhole data, which will experience similar variation (University of Highlands and Island 2012). Experimental results have shown that time-variable disturbances experienced by observatories even a long way apart follow similar trends. The comparison of the observations made at a fixed observatory with derived observations interpolated from other observatories several hundreds of 12

26 kilometers away from the drill site show a good match. The data are interpolated from one or more locations to another. The readings observed at the nearby stations are effectively weighted by the proximity to the drill site. This is not always practical and requires a magnetically clean site with power supply nearby and some method of transmitting the data in real-time from the temporary observatory (University of Highlands and Island 2012). IIFR is a patented method and can be used under license from the inventors (Russell et al. 1995). 2.3 Theory of Drillstring Magnetic Error Field The measurements of magnetic vectors are susceptible to distortion arising from inherent magnetism of ferrous materials incorporated in the drillstring and BHA (Cheatham et al. 1992). By convention, this magnetic field interference is divided into remnant hard-iron offset and induced soft-iron distortions. At the bit and above and below the sensors, the magnetic flux is forced to leave the steel, i.e. magnetic poles occur at the ends of the steel sections which construct a dipole. A magnetic error field is produced by the dipole at the compass location. This magnetic error field will interact with the Earth s total field to produce a resultant field. The compass will respond to the horizontal component of the resultant field (Scott and MacDonald 1979). Drillstring magnetic error field in the axial direction, B z, exceeds the cross axial magnetic error field. The reason is that the ferromagnetic portions of the drillstring are displaced axially from the instrument (Brooks 1997) and the drillstring is long and slender, and is rotated in the geomagnetic field (Brooks 1997). 13

27 2.4 Ferromagnetic Materials, Hard-Iron and Soft-Iron Interference Iron, cobalt, nickel and their alloys referred to as ferromagnets can maintain permanent magnetic field and are the predominant sources to generate static hard-iron fields on the probe in the proximity of the magnetometers. Static hard-iron biases are constant or slowly time-varying fields (Ozyagcilar 2012c). The same ferromagnetic materials, when normally unmagnetized and lack a permanent field, will generate their own magnetic field through the induction of a temporary soft-iron induced magnetization as they are repeatedly strained in the presence of any external field, whether the hard-iron or the geomagnetic field, during drilling operations (Ozyagcilar 2012b). In the absence of the external field there is no soft-iron field (Ozyagcilar 2012c). This generated field is affected by both the magnitude and direction of the external magnetic field. In a moving vehicle, the orientation of the geomagnetic field relative to the vehicle changes continuously. Thus, the resulting soft-iron errors are time varying. The ability of a material to develop an induced soft-iron field in response to an external field is proportional to its relative magnetic permeability. Magnetic interference can be minimized by avoiding materials with high relative permeability and strongly magnetized ferromagnetic components wherever possible and selecting alternatives and also placing the magnetometer as far away as possible from such components (Brooks et al. 1998). The geomagnetic field is distorted by the hard-iron and soft-iron interference and the magnetometer located in the vicinity of the magnets will sense the sum of the geomagnetic field, permanent hard-iron, and induced soft-iron field, and will compute an erroneous azimuth (Ozyagcilar 2012c). The magnitude of hard-iron interference can exceed 1000μT and can saturate the magnetometer since the operating range of the magnetometer cannot accommodate the sum of the geomagnetic and interference fields. Thus, in practice, it is essential to accurately 14

28 estimate and subtract the hard-iron offset through correction methods of drilling assembly corrupting magnetic field. 2.5 Surveying of Boreholes The heading measurements are made using three accelerometers which are preferably orthogonal to one another and are set up at any suitable known arrangement of the three orthogonal axes to sense the components of the Earth s gravity in the same three mutually orthogonal directions as the magnetometers sense the components of the local magnetic field (Helm 1991). The instrumentation sensor package, including accelerometers and magnetometers, is aligned in the survey tool s coordinate system with the orthogonal set of instrument-fixed axes; so that these three orthogonal axes define the alignment of the instrumentation relative to the BHA (Thorogood and Knott 1990). Since both the accelerometers and magnetometers are fixed on the probe, their readings change according to the orientation of the probe. With three accelerometers mounted orthogonally, it is always possible to figure out which way is down, and with three magnetometers it is always possible to figure out which way is the magnetic north. The set of Earth-fixed axes (N, E, V) shown in Figure 2-2 is delineated with V being in the direction of the Earth s gravity vector and N being in the direction of the horizontal component of the geomagnetic main field which points horizontally to the magnetic north axis, and the E axis, orthogonal to the V and N axes, being at right angles clockwise in the horizontal plane as viewed from above; i.e., the E axis is an east-pointing axis. The set of instrument-fixed axes (X, Y, Z) is delineated with the Z axis lying along the longitudinal axis of the BHA, in a direction towards the bottom of the assembly, and X and Y axes lying in the BHA cross-axial plane, 15

29 perpendicular to the borehole axis, while the Y axis stands at right angles to the X axis in a clock wise direction as viewed from above. Figure 2-2. A Perspective view of Earth-fixed and instrument-fixed orthogonal axes which denote BHA directions in three dimensions (modified from Russell and Russell 2003). The set of instrument-fixed orthogonal axes (X, Y, Z) are related to the Earth-fixed set of axes (N, E, V) through a set of angular rotations of azimuth (AZ), inclination (I), and, toolface or highside (HS) angles as shown in Figure 2-2 (for convenience of calculations, a hypothetical origin, O, is deemed to exist at the center of the sensor package). HS, a further angle required to determine the borehole orientation, is a clockwise angle measured in the cross-axial plane of borehole from a vertical plane including the gravity vector to the Y axis. The transformation of a 16

30 unit vector observed in the survey tool s coordinate system to the Earth s coordinate system enables the determination of the borehole orientation (Russell and Russell 2003). At certain predetermined surveying stations, while the BHA is completely stationary, the undistorted sensor readings of the gravity and magnetic field components measured along the direction of the orthogonal set of instrument-fixed coordinate frame are denoted by (G x, G y, G z ) and (B x, B y, B z ), respectively, and are mathematically processed to calculate the corrected inclination, highside, and azimuth of borehole along the borehole path at the point at which the readings were taken. The BHA position is then computed by assuming certain trajectory between the surveying stations (Russell and Russell 1979). These calculations, which are performed by the computing unit of the drilling assembly, are well-known in the literature and were well discussed by different researchers. Based on the installation of orthogonal axes mentioned in this section, Russell and Russell (1978), Russell (1987), and Walters (1986) showed that the inclination (I), the highside (HS) and the azimuth (AZ) can be determined as discussed below. 2.6 Heading Calculation The transformation between unit vectors observed in the survey tool s coordinate system (X, Y, Z) and the Earth s coordinate system (N, E, V) is performed by the vector Equation (2.1): U NEV = {ΑZ} {I} {HS} U XYZ, (2.1) where U N, U E and U V are unit vectors in the N, E and V directions, U X, U Y and U Z are unit vectors in the X, Y and Z directions, respectively, and {ΑZ}, {I}, {HS} represent the rotation matrices according to Russell and Russell (1978): cos AZ sin AZ 0 {AZ} = sin AZ cos AZ 0, (2.2)

31 cos I 0 sin I {I} = 0 1 0, (2.3) sin I 0 cos I cos HS sin HS 0 {HS} = sin HS cos HS 0. (2.4) The vector operation for a transformation in the reverse direction can be written as: U XYZ = {HS} T {I} T {ΑZ} T U. NEV (2.5) The first step is to calculate the borehole inclination angle and highside angle. Operating the vector Equation (2.5) on the Earth s gravity vector results in Equation (2.6): G x cos HS sin HS 0 cos I 0 sin I cos AZ sin AZ 0 0 G y = sin HS cos HS sin AZ cos AZ 0 0, (2.6 ) G z sin I 0 cos I g where g is the magnitude of gravity derived as the square root of the sum of the individual squares of gravity vector and the gravity vector is defined as: g = g U V = G x U X + G y U Y + G z U Z. (2.7) It is assumed that the probe is not undergoing any acceleration except for the Earth s gravity field. In the absence of external forces, in static state, the accelerometer experiences only the Earth gravity with components G x, G y, G z, which are therefore a function of the gravity magnitude and the probe orientation only. This study is also based on the assumption that the gravity measurements G x, G y, and G z are substantially identical to the respective actual Earth s gravity field (because accelerometers are not affected by magnetic interference). Equations (2.8) through (2.10) provide gravity field components in the (X, Y, Z) frame: G x = g cos HS sin I, (2.8) G y = g sin I sin HS, (2.9) 18

32 G z = g cos I. (2.10) Thus, the highside angle HS can be determined from: G y tan HS =. (2.11) G x The inclination angle can be determined from: 2 G x 2 + G y sin I, (2.12) cos I = G z Or G z cos I =. (2.13) 2 G x 2 + G y 2 + G z Based on the above equations, it is obvious that the inclination and highside angles are functions of only the gravity field components. The next step is to calculate the borehole azimuth. The vector expression of the geomagnetic field in Earth-fixed and instrument-fixed frames are denoted as: B = B N U N + B E U E + B V U V = B x U X + B y U Y + B z U Z, (2.14) where B N, B E and B V are the geomagnetic field components in (N, E, V) frame. Operating the vector Equation (2.1) on the magnetic field vector results in Equation (2.15): B N B cos(dip) B E = 0 B V B sin(dip) cos AZ sin AZ 0 cos I 0 sin I cos HS sin HS 0 B x = sin AZ cos AZ sin HS cos HS 0 B y, (2.15) sin I 0 cos I B z 19

33 2 where strength of geomagnetic field, B, is obtained as, B 2 1/2 N + B V. DIP is the dip angle of the geomagnetic vector measured downwards from the horizontal. There is no requirement to know the details of B or the DIP in order to calculate the azimuth since these cancel in the angle calculations. Equation (2.15) yields magnetic field components in the (N, E, V) frame as follows: B N = cos AZ cos I B x cos HS B y sin HS + B z sin I sin AZ B x sin HS + B y cos HS, (2.16) B E = sin AZ cos I B x cos HS B y sin HS + B z sin I + cos AZ B x sin HS + B y cos HS, (2.17) B V = sin I B x cos HS B y sin HS + B z cos I. (2.18) The calculated azimuth at the instrument location is the azimuth with respect to the Earth s magnetic north direction if the local magnetic field vector measured at the instrument location is solely that of the geomagnetic field (Russell and Russell 1979). Under this condition, the equations here ignore any magnetic field interference effects thus B E is zero and then the azimuth is derived from Equation (2.17) by: sin AZ B x sin HS + B y cos HS cos AZ = cos I B x cos HS B y sin HS + B z sin I. (2.19) The azimuth angle is derived as a function of the inclination angle, the highside angle and the magnetic field components B x, B y, and B z. Therefore the azimuth is a function of both the accelerometer and magnetometer measurements. Substituting the above inclination and highside equations into the above azimuth equation results in the following equation, which is used to 20

34 convert from three orthogonal accelerations and three orthogonal magnetic field measurements to the wellbore azimuth: sin AZ B x G y B y G x G x + G y + G z cos AZ = 2 G z B x G x +B y G y + B z G x + G 2 y. (2.20) If the X-Y plane of the body coordinate system is level (i.e., the probe remains flat), only the magnetometer readings are required to compute the borehole azimuth with respect to magnetic north using the arctangent of the ratio of the two horizontal magnetic field components (Gebre- Egziabher and Elkaim 2006): B y AZ = tan 1. (2.21) B x In general, the probe will have an arbitrary orientation and therefore the X-Y plane can be leveled analytically by measuring the inclination and highside angles of the probe (Gebre- Egziabher and Elkaim 2006). Post analysis of the results made by Russell and Russell (1978) showed that the coordinate system of the sensor package (instrument-fixed coordinate system) may be set up at any suitable known arrangements of the three orthogonal axes, and different axes arrangements lead to different azimuth formulas (Helm 1991). Therefore, care should be taken when reading raw data files and identifying the axes. 2.7 Review of the Principles of the MWD Magnetic Surveying Technology Conti et al. (1989) showed that the directional drilling process should include MWD equipment, in addition to the conventional drilling assembly. The well is drilled according to the designed well profile to hit the desired target safely and efficiently. Information about the location of the BHA and its direction inside the wellbore is determined by use of an MWD tool 21

35 (Bourgoyne et al. 2005). In current directional drilling applications, the MWD tool incorporates a package of sensors which includes a set of three orthogonal accelerometers and a set of three orthogonal magnetometers inserted within a downhole probe to take instantaneous measurements of magnetic and gravity conditions at some predetermined survey stations along the wellbore path (with regular intervals of e.g., 10 m) while the BHA is completely stationary (Thorogood 1990). In addition, the MWD tool contains a transmitter module that sends these measurement data to the surface while drilling. Interpretation of this downhole stationary survey data provides azimuth, inclination, and toolface angles of the drill bit at a given measured depth for each survey station. Coordinates of the wellbore trajectory can then be computed using these measurements and the previous surveying station values for the inclination, azimuth, and distance (Thorogood, 1990). The accelerometer measurements are first processed to compute the inclination and toolface angles of the drill bit. The azimuth is then determined using the computed inclination and toolface angles and the magnetometer measurements (Russell and Russell 1979). Present MWD tools employ fluxgate saturation induction magnetometers (Bourgoyne et al. 2005). After completing the drilling procedure, wellbore mapping (WBM) of the established wells is performed for the purpose of quality assurance. WBM determines the wellbore trajectory and direction as a function of depth, and compares it to the planned trajectory and direction (Bourgoyne et al. 2005). 2.8 Horizontal Wells Azimuth The borehole inclination is determined by use of the gravitational measurements alone, while the borehole azimuth is determined from both the gravitational and magnetic measurements. 22

36 Since the accelerometers are not affected by magnetic interference, inclination errors are very small compared to azimuth errors. On the other hand, the calculation of borehole azimuth is especially susceptible to magnetic interference from the drilling assembly. The drillstring magnetic error field does not necessarily mean an azimuth error will occur. Grindrod and Wolff (1983) proved that no compass error results from a vertical assembly or one which is drilling in north or south magnetic direction. The reason is as follows: (i) The conventional magnetic compass placed near the magnetic body aligns itself according to the horizontal component of the resultant field produced from interaction of the Earth s total field and the error field of the magnetic body interference. This resultant field is the vectorial sum of the geomagnetic field and the drillstring error field (Grindrod and Wolff 1983). (ii) It was mathematically proved that drillstring magnetic error field in axial direction exceeds cross axial direction Therefore simple vector addition in Equation (2.22) shows that the azimuth error equals the ratio of the east-west component of the drillstring error vector and the Earth s horizontal field as shown in Figure 2-3 and Figure 2-4: B z sin I sin AZ AZ =, (2.22) B cos(dip) where AZ = Borehole Azimuth error, B z = drillstring magnetic error field in axial direction, I = Borehole inclination, AZ= Borehole azimuth, DIP= dip angle of geomagnetic vector, B = Strength of geomagnetic field, B z. sin I = Horizontal component of the drillstring error vector, B z. sin I. sin AZ = East/West component of the drillstring error vector, 23

37 B N = B. cos(dip) = Horizontal component of geomagnetic field, However, as the borehole direction and inclination change, errors will occur. This means that the compass azimuth error increases with borehole inclination and also with a more easterly or westerly direction of the borehole. Therefore the azimuth uncertainty will particularly occur for wells drilled in an east-west direction (Grindrod and Wolff 1983). Figure 2-3. Horizontal component of error vector (modified from Grindrod and Wolff 1983) Figure 2-4. East/west component of error vector (modified from Grindrod and Wolff 1983) 2.9 Previous Studies 24

38 The problem of drilling assembly magnetic interference has been investigated extensively in the literature. An overview of different methods that can be implemented for the correction of this corrupting magnetic error field is provided here Magnetic Forward Modeling of Drillstring The magnitude of error field, B z, produced by the drillstring cylinder is modeled by a dipole moment along the axis of the cylinder. The application of classical magnetic theory, together with a better understanding of the changes in the magnetic properties of the drilling assembly as drilling progresses, provides a knowledge of magnetic moment, size and direction of error field, which enables us to make good estimates of the drilling assembly s magnetic effects on the survey accuracy for the particular geographic location (Scott and MacDonald 1979). Scott and MacDonald (1979) made use of field data from a magnetic survey operation to investigate magnetic conditions and proposed a procedure for quantifying magnetic pole strength changes during drilling. The strength of a magnetic pole is defined as equal to the magnetic flux that leaves or enters the steel at the pole position (Grindrod and Wolff 1983). It is noted that the pole strengths are proportional to the component of the geomagnetic field (magnetizing field) in the axis of the borehole, and this component is dependent on the local magnetic dip angle, inclination, and direction of the borehole (Scott and MacDonald 1979). This fact is useful to predict magnetic pole strength changes during the drilling process. This method is not practical since the pole strength of dipole varies with a large number of factors Standard Method Russel & Roesler (1985) and Grindord & Wolf (1983) reported that drilling assembly magnetic distortion could be mitigated but never entirely eliminated by locating the magnetic survey instruments within a non-magnetic section of drillstring called Non-Magnetic Drill 25

39 Collars (NMDC) extending between the upper and lower ferromagnetic drillstring sections. This method brings the magnetic distortion down to an acceptable level if the NMDC is sufficiently long to isolate the instrument from magnetic effects caused by the proximity of the magnetic sections of the drilling equipment, the stabilizers, bit, etc., around the instrument (Russell and Russell 2003). Since such special non-magnetic drillstring sections are relatively expensive, it is required to introduce sufficient lengths of NMDC and compass spacing into BHA. Russell and Russell (2002) reported that such forms of passive error correction are economically unacceptable since the length of NMDC increases significantly with increased mass of magnetic components of BHA and drillstring, and this leads to high cost in wells which use such heavier equipment Short Collar Method or Conventional Magnetic Survey (Single Survey) This method is called short collar because the shorter length of NMDC can be used in the field without sacrificing the accuracy of the directional survey (Cheatham et al. 1992). In the literature, the short collar method, referred to as conventional magnetic survey or Single Survey Analysis (SSA), processes each survey station independently for magnetic error compensation (Brooks et al. 1998). In the SSA method, the corrupting magnetic effect of drillstring is considered to be aligned axially and thereby leaving the lateral magnetic components B x and B y uncorrupted, i.e., they only contain the geomagnetic field (Russell and Russell 2003). The magnetic error field is then derived by the use of the uncorrupted measurements of B x and B y, and an independent estimate of one component or combination of components of the local geomagnetic field obtained from an external reference source or from measurements at or near the site of the well (Brooks et al. 1998). 26

40 The limitation of this calculation correction method is that there is an inherent calculation error due to the availability of only the uncorrupted cross-axial magnetic components. This method thus tends to lose accuracy in borehole attitudes in which the direction of independent estimate is perpendicular to the axial direction of drillstring and therefore contributes little or no axial information (Brooks 1997). As a result, single survey methods result in poor accuracy in borehole attitudes approaching horizontal east-west, and the error in the calculation of corrected azimuth may greatly exceed the error in the measurement of the raw azimuth. In other words, the error in the calculation of corrected azimuth by this method is dependent on the attitude of the instrument because the direction of B z is defined by the set of azimuth and inclination of the borehole (Russell and Russell 2003) Some of the important works already done by researchers on SSA method are shortly explained here. For instance, an approach is that if the magnitude of the true geomagnetic field B is known together with some knowledge of the sign of the component B z, then B z is calculated from equation (2.23) and substituted in to equation (2.19) to yield the absolute azimuth angle (Russell 1987): B z = B 2 2 B x B 2 y 1/2. (2.23) If the vertical component of the true geomagnetic field, B V, is known, then B z can be calculated from equation (2.24), B z = (B V g B x G x B y G y )/G z, (2.24) Various single directional survey methods have therefore been published, which ignore small transverse bias errors and seek to determine axial magnetometer bias errors. It should be 27

41 mentioned here that there are other types of SSA computational procedures published by other researchers which seek to determine both axial and transverse Multi-Station Analysis (MSA) Conventional magnetic correction methods assume the error field to be aligned with the z- axis. Therefore the correct z-component of the local magnetic field is considered as unknown and thus the unknown z-component leaves a single degree of freedom between the components of the local field. Figure 2-5 indicates these components while each point along the curve represents a unique z-axis bias and its corresponding azimuth value (Brooks et al. 1998). The unknown z-component is solved by z-axis bias corresponding to the point on the curve which minimizes the vector distance to the externally-estimated value of reference local geomagnetic field (Brooks et al. 1998). Therefore the result is the point at which a perpendicular line from the reference point meets the curve, as shown on Figure

42 Figure 2-5. Conventional correction by minimum distance (Brooks et al, 1998) In this type of correction, the accuracy degrades in attitudes approaching horizontal east-west (Brooks et al. 1998). The multiple-survey magnetic correction algorithm developed by Brooks (1997) generalizes the said minimum distance method to a number of surveys through defining the magnetic error vector in terms of parameters which are common for all surveys in a group and minimizing the variance (distance) among computed and central values of local field (Brooks et al. 1998). Since the tool is fixed with respect to the drillstring, the magnetic error field is fixed with respect to the tool s coordinate system (Brooks 1997). The major advantage of the MSA over the SSA method is that the MSA method is not limited by orientation, and can be reliable in all orientations. MSA is an attitude-independent technique and, unlike conventional corrections, makes use of the axial magnetometer measurements while 29

43 it still results in greater accuracy of azimuth even in the critical attitudes near horizontal eastwest (Brooks 1997) Non-Magnetic Surveys Alternatively, gyroscopic surveys are not subject to the adverse effects of magnetic fields (Uttecht and dewadrt 1983). Therefore, wellbore positional uncertainty tends to be greater for magnetic surveys than for high accuracy gyro systems, and gyros are reported to have the best accuracy for wellbore directional surveys. However, there are shortcomings associated with Gyro surveys. Gyro surveys are much more susceptible to high temperatures than the magnetic surveys. Due to the complex procedure of directional drilling and the severe downhole vibration and shock forces, gyroscopic instruments cannot be employed for directional operations for the entire drilling process. Each time the gyroscope reference tool is needed; drilling operator has to stop drilling to run the gyro to get the well path survey data (McElhinney et al. 2000). The gyroscope is pulled out of the well as soon as the surveys are taken. Directional drilling can then commence relying on the magnetic based MWD tool in the BHA. A considerable delay time is incurred by following this process Summary The drill bit direction and orientation during the drilling process is determined by accelerometer and magnetometer sensors. Geomagnetic field variations and magnetization of nearby structures of the drilling rig all have a deleterious effect on the overall accuracy of the surveying process. Drilling operators utilize expensive nonmagnetic drill collars along with reliable error models to reduce magnetic survey uncertainty so as to avoid collision with adjacent wells. 30

44 Comparing the applicability, advantages and disadvantages of the aforementioned approaches in the literature for the magnetic error correction, we conclude that the multi-station analysis is the most reliable approach for drilling assembly magnetic compensation in order to provide position uncertainties with acceptable confidence levels. Therefore, the methodology section that follows provides a detailed description of the MSA approach. Furthermore, the hard- and softiron magnetic calibration is examined and investigated for the directional drilling application. 31

45 th Chapter Three: METHODOLOGY This section describes the methodology for MSA correction model as well as the hard- and soft-iron model to achieve the objectives of this thesis. The sensor readings of the local gravity and the corrupted local magnetic field components at each survey station are measured along instrument-fixed coordinate frame and entered to the error compensation model of the MSA or the hard- and soft-iron to solve for magnetic disturbances. Three components of the geomagnetic vector, including the field strength, the declination angle, and the dip angle at the location of drilling operation are acquired from an external reference source such as IGRF model freely over the internet in order to add to the above models. Eventually, the corrected magnetic field measurements are used in the wellknown azimuth expressions such as (2.19) and (2.20) to derive the corrected borehole azimuth along the borehole path at the point at which the readings were taken. The BHA position is then computed by assuming certain trajectory between the surveying stations. 3.1 MSA Correction Model The MSA algorithm assumes common error components to all surveys in a group and solves for these unknown biases by minimizing the variance of the computed magnetic field values about the central (reference) value of the local field to obtain calibration values. The central values may be either independent constants obtained from an external source of the local magnetic field or the mean value of the computed local magnetic field (Brooks et al. 1998). Where the common cross-axial and axial magnetic bias components B x, B y, and B z are affecting the measured components B xm(n), B ym(n), and B zm(n) at np survey station in the (X, Y, Z) frame respectively, the corrected values are calculated by: B xcorr(n) = B xm(n) B x, (3.1) 32

46 np th th B ycorr(n) = B ym(n) B y, (3.2) B zcorr(n) = B zm(n) B z. (3.3) The vertical and horizontal components of the true geomagnetic field acquired from an external reference source (such as IGRF) at the location of the borehole are denoted as B V (ref), B, respectively. Moreover, the vertical component of the local magnetic field at the h(ref) survey station denoted as B V(n) is computed by the vector dot product: B. g B V =. (3.4) g By substituting Equations (2.7), (2.14) for the np field is obtained from: survey station the computed value of local B xcorr(n) G x(n) + B ycorr(n) G y(n) + B zcorr(n) G z(n) B V(n) =, (3.5) 2 2 G x(n) + G y(n) + G z(n) B h(n) = B xcorr(n) 2 + B ycorr(n) 2 + B zcorr(n) 2 B V(n). (3.6) Values of the computed magnetic field B V(n) and B h(n) for a survey group at a range of n = 1,, N will typically exhibit some scatter with respect to the reference value of B V (ref) and B h, therefore, reflecting the varying direction of drillstring magnetization error (Brooks (ref) 1997). This scatter, formulated as variance (distance) among computed magnetic field values and the reference local field value over N surveys, is expressed as (Brooks et al. 1998): 1 N 2 2 V = (N 1) B h(n) B h (ref) + B V(n) B V. (3.7) (ref) n=1 The unknown biases are solved for by minimizing this scatter through minimizing the variance, V, expressed in equation (3.7). This can be accomplished by differentiating equation (3.7) with respect to the small unknown biases and setting the results to zero. 33

47 The differentiations are nonlinear with respect to unknown biases. An approximate solution can therefore be found by linearizing the differentiations and solving for the unknown biases by an iterative technique such as Newton s method in which successive approximations to the unknown biases are found. The updated bias estimates are replaced with previous estimates to refine the values of the computed magnetic field for the next iteration. The computation process has been investigated in detail in U.S. pat. Nos. 5,623,407 to Brooks and are also explained as following. MSA Computation: From equation (3.7) where the small perturbations of B x, B y, and B z can be denoted as ε x, ε y, and ε z, differentiations give: V F ε x, ε y, ε z = = ε x N B h(n) B h (ref) = B h(n) B h (ref) ε x ε x n=1 + B V(n) B V (ref) B V(n) B V (ref) = 0, (3.8) ε x ε x V G ε x, ε y, ε z = = ε y N B h(n) B h (ref) = B h(n) B h (ref) ε y ε y n=1 B V(n) B V (ref) + B V(n) B V = 0, (3.9) (ref) ε y ε y 34

48 V H ε x, ε y, ε z = = ε z N B h(n) B h (ref) = B h(n) B h (ref) ε z ε z n=1 B V(n) B V (ref) + B V(n) B V (ref) = 0. (3.10) ε z ε z The differentiations F, G, and H are nonlinear with respect to ε x, ε y, and ε z. An approximate solution can therefore be found by linearizing equations (3.8) through (3.10) by an iterative technique such as Newton s method. The linearized form of F, G, and H denoted as f, g, and h are, f = a 1 (ε x ε x ) + b 1 ε y ε y + c 1 (ε z ε z ) + F ε x, ε y, ε z = 0, (3.11) g = a 2 (ε x ε x ) + b 2 ε y ε y + c 2 (ε z ε z ) + G ε x, ε y, ε z = 0, (3.12) h = a 3 (ε x ε x ) + b 3 ε y ε y + c 3 (ε z ε z ) + H ε x, ε y, ε z = 0, (3.13) where F ε x, ε y, ε z F ε x, ε y, ε z F ε x, ε y, ε z a 1 = ; b 1 = ; c 1 =, (3.14) ε x ε y ε z G ε x, ε y, ε z G ε x, ε y, ε z G ε x, ε y, ε z a 2 = ; b 2 = ; c 2 =, (3.15) ε x ε y ε z H ε x, ε y, ε z H ε x, ε y, ε z H ε x, ε y, ε z a 3 = ; b 3 = ; c 3 =. (3.16) ε x ε y ε z The primed error terms ε x, ε y, and ε z represent the previous estimates of these values. The linearized equations (3.11) through (3.13) can be solved for the unknowns ε x, ε y, and ε z by 35

49 iterative technique of Newton s method in which successive approximations to ε x, ε y, and ε z are found by (Brooks et al. 1998): ε x ε x ε y ε y ε z ε z 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z V ε x, ε y, ε z ε 2 x ε x ε y ε x ε z ε x 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z V ε x, ε y, ε z =, (3.17) ε y ε x ε 2 y ε y ε z ε y 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z V ε x, ε y, ε z ε z ε x ε z ε y ε 2 z ε z 1 ε x ε y ε z ε x = ε y ε z 1 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z V ε x, ε y, ε z ε 2 x ε x ε y ε x ε z ε x 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z V ε x, ε y, ε z, (3.18) ε y ε x ε 2 y ε y ε z ε y 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z 2 V ε x, ε y, ε z V ε x, ε y, ε z ε z ε x ε z ε y ε 2 z ε z 1 ε x ε x a 1 b 1 c 1 F ε x, ε y, ε z ε y = ε y a 2 b 2 c 2 G ε x, ε y, ε z. (3.19) ε z ε z a 3 b 3 c 3 H ε x, ε y, ε z 36

50 The updated bias estimates of ε x, ε y, and ε z obtained from equation (3.19) is replaced with previous estimates of ε x, ε y, and ε z in equations (3.1) through (3.3) to refine the values of B xcorr(n), B ycorr(n), and B zcorr(n) for the next iteration. A suitable convergence criterion is used to determine whether further iterations are needed. The stopping criterion for the iteration can be defined as the change between successive values of ε x, ε y, and ε z falling below a predefined minimum limit or a specified number of iterations having been performed (Brooks et al. 1998). The derivatives a 1 through c 3 can be obtained by use of the chain rule. In the case where the central values are independent constants obtained from an external source of the local magnetic field, a 1 is derived by: N a 1 = N B h (ref) 2 B h(n). (3.20) ε 2 x n=1 In the case where the central values are the mean values of the computed local magnetic field which are not constant, the coefficient a 1 is derived more complicated as: N 2 B h(n) ε 2 B x h(n) B V(n) n=1 a 1 = N B h(n), (3.21) ε x N + ε x N + N n=1 where 2 2 B h(n) 1 G B xcorr(n) + B V(n) G x(n) x(n) g = ε 2 1, (3.22) x B h(n) g B h(n) N 2 37

51 B h(n) ε x x(n) B xcorr(n) + B V(n) G g =, (3.23) B h(n) B V(n) ε x G x(n) =. (3.24) g Similar calculations can derive the remaining coefficients b 1 through c 3. Upon completion of the iteration, the compensated magnetic field vectors which are now more closely grouped than the primary scatter are used in well-known azimuth expressions such as (2.19) and (2.20) to derive the corrected borehole azimuth (Brooks 1997). 3.2 Hard-Iron and Soft-Iron Magnetic Interference Calibration A magnetometer senses the geomagnetic field plus magnetic field interference generated by ferromagnetic materials on the probe. By convention, this magnetic field interference is divided into static (fixed) hard-iron offset and induced soft-iron distortions. A mathematical calibration algorithm was developed by Ozyagcilar (2012) which is available via Freescale application document number of AN4246 at This algorithm calibrates the magnetometers in the magnetic field domain to estimate magnetometer output errors and remove the hard-iron and soft-iron interference from the magnetometer readings taken under different probe orientations, allowing the geomagnetic field components to be measured accurately (Ozyagcilar 2012c). The calibration problem is solved through the transformation of the locus of magnetometer measurements from the surface of an ellipsoid displaced from the origin to the surface of a sphere located at the origin Static Hard-Iron Interference Coefficients Since the magnetometer and all components on the probe are in fixed positions with respect to each other and they rotate together, the hard-iron effect is independent of the probe orientation 38

52 and is therefore modeled as a fixed additive magnetic vector which rotates with the probe. Since any zero field offset in the magnetometer factory calibration is also independent of the probe orientation, it simply appears as a fixed additive vector to the hard-iron component and is calibrated and removed at the same time. Both additive vectors are combined as a hard-iron vector V with components of V x,v y, and V z adding to the true magnetometer sensor output (Ozyagcilar 2012a). Therefore, the standard hard-iron estimation algorithms compute the sum of any intrinsic zero fields offset within the magnetometer sensor itself plus permanent magnetic fields within the probe generated by magnetized ferromagnetic materials (Ozyagcilar 2012a) Soft-Iron Interference Coefficients Soft-iron effects are more complex to model than hard-iron effects since the induced soft-iron magnetic field depends on the orientation of the probe relative to the geomagnetic field (Ozyagcilar 2012c). The assumption is that the induced soft-iron field is linearly related to the inducing local geomagnetic field measured in the rotated probe (Ozyagcilar 2012b). This linear relationship is mathematically expressed by the 3 3 matrix W Soft. The components of W Soft are the constants of proportionality between the inducing local magnetic field and the induced soft-iron field. For example, the component in the first raw of the second column of W Soft represents the effective coefficient relating the induced field generated in the x-direction in response to an inducing field in the y-direction. Thus, W Soft can be a full matrix. The magnetometer is normally calibrated by the company to have approximately equal gain in all three axes. Any remaining differences in the gain of each axis can be modeled by a diagonal 3 3 gain matrix W Gain which is also referred to as scale factor error. Another 3 3 matrix, W NonOrthog, referred to as misalignment error is used to model: 39

53 (i) The rotation of magnetometer relative to the instrument-fixed coordinate system (X, Y, Z) (ii) The lack of perfect orthogonality between sensor axes (Ozyagcilar 2012b) Since the misalignment between the two axes is normally very small (but not negligible), W NonOrthog can be modelled as the following symmetric matrix (Gebre-Egziabher et al. 2001): 1 ε z ε y W NonOrthog = ε z 1 ε x. (3.25) ε y ε x 1 The three independent parameters ε x, ε y, and ε z defining the matrix W NonOrthog represent small rotations about the body axes of the vehicle that will bring the platform axes into perfect alignment with the body axes. The linear soft-iron model is derived from the product of these three independent matrices which results in nine independent elements of a single 3 by 3 softiron matrix W defined as: W = W NonOrthog W Gain W Soft. (3.26) The process of calibrating a triad of magnetometers involves estimating the hard-iron vector, V, and the soft-iron matrix, W, defined above Relating the Locus of Magnetometer Measurements to Calibration Coefficients In complete absence of hard-iron and soft-iron interference, a magnetometer will measure the uncorrupted geomagnetic field and thus the vector magnitude of the measured field will equal the magnitude of the geomagnetic field. As a result, at different probe orientations, the measured magnetic field components along the instrument-fixed coordinate system (X, Y, Z) will be different but the vector magnitude will not change. Therefore the locus of the magnetometer measurements under arbitrary orientation changes will lie on the surface of a sphere in the space 40

54 of magnetic measurements centered at the zero field with radius equal to the geomagnetic field strength. This sphere locus is the fundamental idea behind calibration in the magnetic field domain. In the presence of hard-iron effects, the hard-iron field adds a fixed vector offset to all measurements and displaces the locus of magnetic measurements by an amount equal to the hard-iron offset so that the sphere is now centered at the hard-iron offset but still has radius equal to the geomagnetic field strength (Ozyagcilar 2012c). Soft-iron, misalignment, and scale factor errors distort the sphere locus to an ellipsoid centered at the hard-iron offset with rotated major and minor axes. The following equations indicate the ellipsoidal locus Calibration Model Equation (2.15) defines the true magnetic field observed in the absence of hard and soft-iron effects. Incorporating the hard-iron offset V and the soft-iron matrix W into the inverse form of equation (2.15) yields the magnetometer measurement affected by both hard-iron and soft-iron distortions as illustrated in equation (3.27), where B P denotes the distorted magnetometer measured at a survey point. B Px B P = B Py = B Pz cos HS sin HS 0 cos I 0 sin I cos AZ sin AZ 0 cos(dip) W sin HS cos HS sin AZ cos AZ 0 B sin I 0 cos I sin(dip) V x V y. (3.27) V z 41

55 In a strong hard and soft-iron environment, the locus of magnetometer measurements under arbitrary rotation of the probe lie on a surface which can be derived as (Ozyagcilar 2012b) W 1 B p V T W 1 B p V = B p V T {W 1 } T W 1 B p V = B 2, (3.28) substituting from equation (3.27) and denoting: cos HS sin HS 0 cos I 0 sin I cos AZ sin AZ 0 sin HS cos HS sin AZ cos AZ 0 = Γ, (3.29) sin I 0 cos I results in: cos(dip) W 1 B p V = Γ B 0. (3.30) sin(dip) Therefore, it is proved that: cos(dip) W 1 B p V T W 1 B p V = Γ B 0 sin(dip) T cos(dip) Γ B 0 = B 2. (3.31) sin(dip) In general, the locus of the vector B p lying on the surface of an ellipsoid with center coordinate of the vector V is expressed as: B p V T A B p V = const, (3.32) Where matrix A must be symmetric. Equation (3.31) and (3.32) are similar since it can be easily proved that the matrix {W 1 } T W 1 is symmetric as A T = [{W 1 } T W 1 ] T = {W 1 } T W 1 = A. Thus, in a strong hard and soft-iron environment, the locus of raw magnetometer measurements forms the surface of an ellipsoid defined by: B p V T {W 1 } T W 1 B p V = B 2. (3.33) 42

56 The ellipsoid is centered at the hard-iron offset V. Its size is defined by the geomagnetic field strength B and its shape is determined by the matrix {W 1 } T W 1 which is transposed square of the inverse soft-iron matrix W 1. In the absence of soft-iron and misalignment errors, the diagonal 3 3 scale factor matrix W Gain distorts the sphere locus along preferred axes with differing gains along each axis. The measurement locus then becomes an ellipsoid centered at the hard-iron offset, with the major and minor axes magnitudes determined by the scale factor errors, sfx, sfy and sfz along the instrument-fixed coordinate frame. This can be expressed mathematically as follows: (1 + sfx) 0 0 W Gain = 0 (1 + sfy) 0, (3.34) 0 0 (1 + sfz) sfx 1 1 W Gain = sfy, (3.35) sfz B p V T W Gain 1 T W Gain 1 B p V = B 2, (3.36) sfx 2 1 B p V T 0 0 B p V = B 2. (3.37) 1 + sfy sfz Mathematically, the locus of measurements is described by the following equation: B Px V x B Py V y + B Pz V z + = B 2. (3.38) 1 + sfx 1 + sfy 1 + sfz 43

57 Soft-iron and misalignment errors will modify the error-free sphere locus into an ellipsoid but also rotate the major and minor axes of the ellipsoid. Thus, the ellipsoid does not need to be aligned with the axes of the magnetometer and the ellipsoid can be non-spherical. Magnetometer measurements subject to both hard-iron and soft-iron distortions lying on the surface of an ellipsoid can be modeled by nine ellipsoid parameters comprising the three parameters which model the hard-iron offset and six parameters which model the soft-iron matrix. The calibration algorithm that will be developed is nothing more than a parameter estimation problem. The algorithm is an attempt to fit the best ellipsoid in least-squares sense to the measured magnetometer data. The calibration algorithm consists of mathematically removing hard-iron and soft-iron interference from the magnetometer readings by determining the parameters of an ellipsoid that best fits the data collected from a magnetometer triad (Gebre- Egziabher et al. 2001). After the nine model parameters are known, the magnetometer measurements are transformed from the surface of ellipsoid to the surface of a sphere centered at the origin. This transformation removes the hard-iron and soft- iron interference and then the calibrated measurements are used to compute an accurate azimuth (Ozyagcilar 2012b) Symmetric Constrait The linear ellipsoid model of B p V T {W 1 } T W 1 B p V = B 2 is solved for the transposed square of the inverse soft-iron matrix {W 1 } T W 1 and the hard-iron vector V by optimum fitting of the ellipsoid to the measured data points. The ellipsoid fit provides the matrix {W 1 } T W 1 whereas the calculation of the calibrated magnetic field measurement B P Cal according to equation (3.39) requires the inverse soft-iron matrix W 1 : 44

58 B Px V x B P Cal = W 1 B Py V y. (3.39) B Pz V z Although it is trivial to compute the ellipsoid fit matrix {W 1 } T W 1 from the inverse softiron matrix W 1, there is no unique solution to the inverse problem of computing W 1 from the matrix {W 1 } T W 1. The simplest solution is to impose a symmetric constraint onto the inverse soft-iron matrix W 1. As proved in equation (3.32), the matrix {W 1 } T W 1 is symmetric with only six independent coefficients while the soft-iron matrix W has nine independent elements. This means that three degrees of freedom are lost. Physically, it can be understood as a result of the loss of angle information in the ellipsoidal locus of the measurements constructed in the mathematical model which is a function of the magnetometer measurements only (Ozyagcilar 2012b). To solve this problem, a constraint is imposed that the inverse soft-iron matrix W 1 also be symmetric with six degrees of freedom. The reading B p is calibrated by an estimated hard-iron offset V Cal and an estimated soft-iron matrix W Cal, then the resulting corrected magnetic field measurement B P is given by: Cal cos(dip) B P Cal = W Cal 1 B p V Cal = W Cal 1 W Γ B 0 + V V Cal. (3.40) sin(dip) If the calibration algorithm correctly estimates the hard and soft-iron coefficients, then the corrected locus of reading lie on the surface of a sphere centered at the origin. Therefore in equation (3.40), it is necessary that W Cal 1 W = I and V Cal = V thus resulting in (B P Cal )T (B PCal ) = B 2. (3.41) 45

59 Mathematically an arbitrary rotation can be incorporated in the estimated inverse soft-iron 1 matrix W Cal as W Cal 1 W = R(ς) where R(ς) is the arbitrary rotation matrix by angle ς. Considering that the transpose of any rotation matrix is identical to rotation by the inverse angle leads to R(ς) T = R( ς). Substituting V Cal = V in equation (3.40) yields: cos(dip) B P Cal = W Cal 1 B p V Cal = W Cal 1 W Γ B 0, (3.42) sin(dip) cos(dip) cos(dip) 1 1 (B P ) = W Cal W Γ B 0 W Cal W Γ B 0, (3.43) Cal )T (B PCal sin(dip) sin(dip) T then substituting W Cal 1. W = R(ς) yields T cos(dip) cos(dip) (B ) = R(ς) Γ B 0 R(ς) Γ B 0 = B 2 P, (3.44) Cal )T (B PCal sin(dip) sin(dip) and substituting R(ς) T. R(ς) = R( ς). R(ς) = I results in: T cos(dip) cos(dip) (B ) = B 0 Γ T R(ς) T R(ς) Γ B 0 = B 2 P. (3.45) Cal )T (B PCal sin(dip) sin(dip) It was proved that, by incorporating the spurious rotation matrix R(ς), the corrected locus of measurements still lies on the surface of a sphere with radius B centered at the origin (Ozyagcilar 2012b). The reason is that a sphere is still a sphere under arbitrary rotation. If the constraint is applied that the inverse soft-iron matrix W 1 is symmetric, then it is impossible for any spurious rotation matrix to be incorporated in the calibration process since any rotation matrix must be anti-symmetric. A further advantage of a symmetric inverse soft-iron matrix W 1 is the relationship between the eigenvectors and eigenvalues of matrix {W 1 } T W 1 and matrix W 1. It can be proved that 46

60 if W 1 is symmetric as {W 1 } T = W 1, then the eigenvectors of matrix {W 1 } T W 1 are identical to the eigenvectors of matrix W 1 and the eigenvalues of matrix {W 1 } T W 1 are the square of the eigenvalues of matrix W 1. Eigenvectors X i and eigenvalues e i of W 1 are defined by W 1. X i = e i. X i. Therefore: {W 1 } T W 1 X i = {W 1 } T e i X i = e i W 1 X i = e i 2 X i. (3.46) Since the eigenvectors of the ellipsoid fit matrix {W 1 } T W 1 represent the principal axes of magnetometer measurement ellipsoidal locus, then constraining the inverse soft iron matrix W 1 to be symmetric shrinks the ellipsoid into a sphere along the principal axes of the ellipsoid without applying any additional spurious rotation (Ozyagcilar 2012b). The symmetric inverse soft-iron matrix W 1 can be computed from the square root of the ellipsoid fit matrix {W 1 } T W 1 as following: {W 1 } T W 1 = W 1 W 1 W 1 = [ {W 1 } T W 1 ] 1/2. (3.47) This is not always a reasonable assumption and it can be accounted for the residuals in post process. Furthermore, examination of experimental data indicated that the careful installation of magnetometers axes aligned with the body axes results in an ellipsoidal locus having major and minor axes aligned with the body axes Least-Squares Estimation The calibration algorithm develops an iterated least-squares estimator to fit the ellipsoid a b c parameters to the measured magnetic field data. Substituting {W 1 } T W 1 = b e f in to c f I Equation (3.33) results in: 47

61 a b c B Px V x B 2 = (B Px V x B Py V y B Pz V z ) b e f B Py V y, (3.48) c f I B Pz V z B = a(b Px V x ) 2 + e B Py V y 2 + I(B Pz V z ) 2 + 2b(B Px V x ) B Py V y + +2c(B Px V x )(B Pz V z ) + 2f B Py V y (B Pz V z ). (3.49) The equations of the estimator can be obtained by linearizing Equation (3.49). The estimator has perturbations of the ellipsoid parameters comprising three parameters of hard-iron offset and six components of the soft-iron matrix (Gebre-Egziabher et al. 2001). Thus, given an initial guess of the unknown parameters, the estimated perturbations are sequentially added to the initial guess and the procedure is repeated until convergence is achieved (Gebre-Egziabher et al. 2001). To linearize Equation (3.49), the perturbation of B is written as δb, given by: δb = B δv δa + B δb + B δc + B x + B δv y + B δv z + B δe V x V y V z a b c e where + B f δf + B δi, (3.50) I B 2a (B Px V x ) 2b B Py V y 2c (B Pz V z ) =, (3.51) V x 2B B 2e B Py V y 2b (B Px V x ) 2f (B Pz V z ) =, (3.52) V y 2B B 2I (B Pz V z ) 2c (B Px V x ) 2f (B Py V y ) =, (3.53) V z 2B B (B Px V x ) 2 =, (3.54) a 2B B 2(B Px V x ) B Py V y, (3.55) b = 2B 48

62 B 2(B Px V x )(B Pz V z ) =, c 2B (3.56) B (B Py V y ) 2 =, e 2B (3.57) B 2 B Py V y (B Pz V z ), (3.58) f = 2B B (B Pz V z ) 2. (3.59) I = 2B The given or known inputs to the calibration algorithm are the measured magnetometer outputs B Px, B Py, and B Pz and the magnitude of geomagnetic field vector in the geographic area where the calibration is being performed. Note that the B Px, B Py, and B Pz values have been taken in N positions even though, for the sake of simplicity, the explicit notation (index) has been dropped in the above equations. In matrix notation, (3.50) can be expressed as: δb 1 δb 2 = δb N 1 δb N B B B B B B B B B V x V y V z 1 a 1 b c 1 e f 1 I B B B B B B B B B V x V 2 y V 2 z 2 a 2 b 2 c 2 e 2 f 2 I 2 B B B B B B B B B V x V N 1 y V N 1 z N 1 a N 1 b N 1 c N 1 e N 1 f N 1 I N 1 B B B B B B B B B V x V N z N a N b N c N e N f N I N V y N 49

63 δvx δv y δv z δa δb. (3.60) δc δe δf δi Equation (3.60) is in the form δb = ζ δx, where δx, the vector of unknowns is given by: δx = [δv x δv y δv z δa δb δc δe δf δi] T. (3.61) The vector δb is the difference between the known geomagnetic field strength and its magnitude computed from the magnetic measurements. An estimate of the successive perturbations of the ellipsoid parameters V x, V y, V z, a, b, c, e, f and I is obtained by using the following iterative algorithm. Firstly, select a non-zero initial guess for V x, V y, V z, a, b, c, e, f and I. Secondly, form Equation (3.60) by the initial values. Thirdly, obtain a least square estimate for δx as follows: δx = (ζ T ζ) 1 ζ T δb. (3.62) Then, update the unknown parameters by adding the δx perturbations to the current values of parameters. Finally, return to the second step and repeat until convergence is achieved. Convergence is achieved when the estimate of V x, V y, V z, a, b, c, e, f and I do not change from one iteration to the next. By imposing the symmetric constraint stated in the last section, the inverse soft-iron matrix is obtained from the square root of the ellipsoid fit matrix. The estimated calibration parameters can then be used in Equation (3.39) to transform the measured raw data lying on the ellipsoid into the corrected field measurements lying on the sphere centered at the 50

64 origin with radius equal to the geomagnetic field in the absence of hard and soft-iron interference. The computed azimuth using these corrected measurements will be highly accurate Establishing Initial Conditions The stability of the least squares solution is sensitive to the quality of the initial conditions used to start the algorithm. The closer the initial guesses are to the actual value of the nine ellipsoidal parameters, the more stable the solution becomes. Since a judicious selection of initial conditions enhances the performance of the calibration, I will propose an algorithm to establish the initial conditions for the iterative least-squares algorithm. Equation (3.49) of the ellipsoidal locus is non-linear in nature. Nevertheless, it can be treated as a desirable linear system by breaking the parameter identification problem given by Equation (3.49) in to two steps so as to estimate a good approximation of the initial values of the parameters. The proposed two step linear solution will now be explained Step 1: Hard-Iron Offset estimation The hard-iron correction may be sufficient for the probe without strong soft-iron interference because in most cases hard iron biases will have a much larger contribution to the total magnetic corruption than soft iron distortions. A simple solution can be permitted for the case where the hard-iron offset dominates and soft-iron effects can be ignored. Therefore the soft-iron matrix is assumed to be an identity matrix and Equation (3.33) simplifies to sphere locus: B p V T B p V = B 2. (3.63) This simplification results in determining just three calibration parameters modeling the hardiron offset plus the geomagnetic field strength B. My applied Matlab code fits these four model parameters of the above mentioned sphere to the series of magnetometer measurements taken 51

65 under different probe orientations, while minimizing the fit error in a least-squares sense. The least-squares method minimizes the 2 norm of the residual error by optimizing the calibration fit and determines the sphere with radius equal to the geomagnetic field strength B centered at the hard-iron offset V. The number of measurements used to compute the calibration parameters must be greater than or equal to four since there are four unknowns to be determined (Ozyagcilar 2012b). The detail of the computation of hard-iron offsets has been published by Ozyagcilar via Freescale application notes, number AN4246. The data is now centered at the origin but still highly distorted by soft-iron effects. The computed azimuth will not be accurate after applying hard-iron corrections only. The calibrated measurements can now be passed to the second step of the algorithm for calculating the soft-iron interference Step 2: Solving Ellipsoid Fit Matrix by an Optimal Ellipsoid Fit to the Data Corrected for Hard Iron Biases The hard-iron vector V and the magnitude of geomagnetic field solved from step 1 are applied in the step 2 solution. Equation (3.33) is then written as: a b c B xcor_h B 2 = (B xcor_h B ycor_h B zcor_h ) b e f B ycor_h, (3.64) c f I B zcor_h Where B xcor_h, B ycor_h and B zcor_h are acquired by subtraction of the hard-iron vector V (obtained from step 1) from B x, B y and B z measurements respectively. a B xcor_h 2 + e B ycor_h 2 + I B zcor_h 2 + 2b B xcor_h B ycor_h + 2c B xcor_h B zcor_h + 2f B ycor_h B zcor_h B 2 = r. (3.65) 52

66 The least-squares method minimizes the 2 norm of the residual error vector r by fitting 6 components of the ellipsoid fit matrix {W 1 } T W 1 to the series of B Cor_h taken at N positions expressed as follows: B xcor 2 h B 2 1 ycorh B 2 1 zcorh B 1 xcorh B ycorh B 1 xcorh B zcorh 1 B 2 xcorh B 2 2 ycorh B 2 2 zcorh B 2 xcorh B ycorh B 2 xcorh B zcorh 2 B 2 xcorh B 2 ycorh B 2 zcorh B xcorh B ycorh B xcorh B zcorh N 1 N 1 N 1 N 1 N 1 B xcor 2 h B 2 N ycorh B 2 N zcorh B N xcorh B ycorh B N xcorh B zcorh N where in Equation (3.66) (3.66) B xcor 2 h 1 B 2 xcorh 2 B 2 xcorh N 1 B 2 xcorh N B 2 ycorh 1 B 2 ycorh 2 B 2 ycorh N 1 B 2 ycorh N B 2 zcorh 1 B 2 zcorh 2 B 2 zcorh N 1 B 2 zcorh N B xcorh B ycorh 1 B xcorh B ycorh 2 B xcorh B ycorh N 1 B xcorh B ycorh N B xcorh B zcorh 1 B xcorh B zcorh 2 B xcorh B zcorh N 1 B xcorh B zcorh N 53

67 2 B is denoted as matrix A and 2 B is denoted as vector l. The least-squares solution for the B 2 B 2 vector of unknowns is given by: a e I 2b = ( A T A) 1 A T l. (3.67) 2c 2f Afterwards, the inverse soft-iron matrix W 1 can be computed from the square root of the ellipsoid fit matrix {W 1 } T W 1. This algorithm can compute initial values of hard-iron and soft-iron distortions by magnetometer measurements in the complete absence of a-priori information about the direction and strength of the geomagnetic field. 3.3 Well path Design and Planning Well path design and planning employs several methods of computation of well trajectory parameters to create the well path. Each method is able to provide pictorial views both in the vertical and horizontal plane of the trajectory position of the drilling bit in the wellbore. Eventually, it is been able to compute the position at each survey station and therefore predict the length and direction from a survey station relative to the target position. This helps to detect the deviations with less ease and therefore initiate the necessary directional corrections or adjustment in order to re-orient the drilling bit to the right course before and during the drilling operations (Amorin and Broni-Bediako 2010). These computations are required to be done ahead of time before drilling resumes and also during drilling operations to minimize risk and the uncertainty surrounding hitting a predetermined target (Sawaryn and Thorogood 2003). Therefore, as the 54

68 well is surveyed during the various stages of drilling and construction, the position of the well path is recorded and plotted as a course line or trajectory in software plots (Lowdon and Chia 2003). The survey calculation methods of well trajectory available in the industry are the Tangential, Balanced Tangential, Average Angle, Mercury, Radius of Curvature and the Minimum Curvature methods. The main difference in all these techniques is that one group uses straight line approximations and the other assumes the wellbore is more of a curve and is approximated with curved segments. The Tangential, Balanced Tangential, Average Angle and Mercury are applicable to a wellbore trajectory which follows a straight line course, while the Radius of Curvature is strictly applicable to a wellbore trajectory that follows a curved segment. The Minimum Curvature method is applicable to any trajectory path. Bourgoyne et al. (1991) showed that the Tangential method, which is a simplistic method assuming straight-line segments with constant angles along the well trajectory, shows considerable error for the northing, easting and elevation, which makes it no longer preferred in the industry. The differences in results obtained using the Balanced Tangential, Average Angle, Mercury, Radius of Curvature and Minimum Curvature are very small, hence any of the methods could be used for calculating the well trajectory. Realistically, well paths are curved as the wellbore trajectory is built up. The method of applying a minimum curvature to the well path takes into account the graduation of the angles in three dimensions along the wellbore trajectory and, hence, is a better approximation. Minimum Curvature is the most widely preferred method in the oil industry since it is applicable to any trajectory path and thus more emphasis would be placed on this rather than the other methods (Amorin and Broni-Bediako 2010). All the Minimum Curvature methods assume that the hole is 55

69 a spherical arc with a minimum curvature or a maximum radius of curvature between stations, and the wellbore follows a smoothest possible circular arc between stations; that is the two adjacent survey points lie on a circular arc. This arc is located in a plane whose orientation is defined by known inclination and direction angles at the ends of the arc (Bourgoyne et al. 1991). The calculation process requires data input containing measured Depth, inclination angles, and corrected Bearing (Azimuths) with their corresponding descriptive data of the station ID. Moreover, spatial data of the reference station (initial or starting coordinates) and magnetic declination are required. The direction for the magnetic declination angle must be specified; if the magnetic declination is to westward, the sign is negative otherwise it is positive. Figure 3-1 shows the geometry of the Tangential method and Figure 3-2 shows the geometry of the Minimum Curvature method. Figure 3-1. Representation of the geometry of the tangential method(amorin and Broni- Bediako 2010) 56

70 Figure 3-2. Representation of the geometry of the minimum curvature method (Amorin and Broni-Bediako 2010) The Minimum Curvature method effectively fits a spherical arc between points by calculating the dog-leg (DL) curvature and scaling by a ratio factor (RF). The spatial coordinates of easting, northing, and elevation can be computed by the Minimum Curvature method as follows (Amorin and Broni-Bediako 2010): TVD = MD (cos I 1 + cos I 2 )(RF), (3.68) 2 North = MD (sin I 1 cos AZ 1 + sin I 2 cos AZ 2 )(RF), (3.69) 2 East = MD [(sin I 1 sin AZ 1 +sin I 2 sin AZ 2 )](RF), (3.70) 2 DL = cos 1 {cos(i 2 I 1 ) sin I 1 sin I 2 [1 cos(az 2 AZ 1 )]}, (3.71) RF = 2 tan DL DL 2, (3.72) 57

71 where = Change in parameter, MD = Measured depth, TVD = True vertical depth, AZ 1 = Azimuth angle at upper survey point, AZ 2 = Azimuth angle at lower survey point, I 1 = Inclination angle at upper survey point, I 2 = Inclination angle at lower survey point, DL = dog-leg curvature, RF = Ratio factor for minimum curvature. 3.4 Summary In the directional drilling operation, the computing device on the surface is programmed in accordance with the magnetic correction methods. For this research, I have developed my Matlab program either in accordance with MSA or hard- and soft-iron algorithm. The inputs to the program include the x-axis, y-axis, and z-axis components of the local magnetic and gravitational field at each survey station. Furthermore, an external estimate of the local geomagnetic field at the location of the wellbore is added to the program inputs. The magnetic disturbances solved by the program are used to correct the magnetic measurements. The corrected magnetic field measurements are then used in the well-known azimuth expressions such as (2.19) and (2.20) to derive the corrected borehole azimuth along the borehole path at the point at which the readings were taken. Finally, the position of the well path is achieved as a trajectory in Matlab software plots by the use of minimum curvature method. The following flowcharts shows the steps taken for MSA as well as hard- and soft-iron model: 58

72 Start hard- and soft-iron model Start MSA model Input magnetic and gravity measurements Input geomagnetic referencing values of field strength and declination Input geomagnetic referencing values of field strength, dip, and declination Initialize hard-iron components V x, V y, and V z through step1: Eq. (3.63) Initialize magnetic perturbations as zero No Initialize soft-iron matrix components a, b, c, e, f, and I through step2: Eq. (3.67) Estimate perturbations by Eq. (3.62) and update parameters Iteration completion? Yes Inverse soft-iron matrix is obtained from Eq. (3.47) Estimate perturbations by Eq. (3.19) and update parameters Iteration completion? Yes Correct magnetic observations by Eqs. (3.1) through (3.3) No Correct magnetic observations by Eq. (3.39) Calculate corrected azimuth from Eq. (2.22) Calculate horizontal pictorial view of the wellbore by Eqs. (3.69) and (3.70) 59

73 Chapter Four: RESULTS and ANALYSIS In this section the evaluation results of magnetic compensation models is presented and compared through real, simulated and experimental investigations. All calculations and graphs have been implemented in Matlab. 4.1 Simulation Studies for Validation of the Hard and Soft-Iron Iterative Algorithm A set of data was created to assess the performance of the aforementioned hard and soft-iron magnetometer calibration algorithm. The locus of magnetometer measurements obtained would cover the whole sphere or ellipsoidal surface if during the calibration procedure the magnetometer assembly is rotated through the entire 3D space. As it will be seen from the experimental data set shown in the next figures, this is not always possible and only a small portion of the sphere is present. However, for the simulation studies it was possible to cover the spherical surface by assuming a sensor measuring the magnetic field while rotating through a wide range of high side, inclination, and azimuth angles. In the case where there are no magnetic disturbances and no noise, equation (3.27) can calculate data points of magnetic field B P lying on a sphere with radius equal to B centered at origin, where in Equation (3-27), V x = V y = V z = 0 and W = Identity Matrix. It is assumed that the simulated wellbore drilling takes place in a location where B = 500 mgauss and DIP = 70. It is noted that since Gauss is a more common unit in directional drilling, I use mgauss rather that SI unit of Tesla. Figure 4-1 and Figure 4-2 quantify the size of the simulated magnetometer measurement locus in the absence of any interferences and noises. As shown in Figure 4-1 and Figure 4-2, a range of inclination and highside angles between 0 to 360 degrees are applied to Equation (3.27) while at 60

74 each case of inclination and highside value, the azimuth varies from 0 to 360 degree and thus a circle of magnetic points is created which totally leads to 555 data points. As shown, the locus of the magnetometer measurements under arbitrary orientation changes will lie on the surface of a sphere centered at the zero field with radius equal to 500 mgauss. 500 HighSide : 90 degree Bz 0 mgauss By mgauss Bx mgauss 500 Figure 4-1. Sphere locus with each circle of data points corresponding to magnetic field measurements made by the sensor rotation at highside 90,with a specific inclination and a cycle of azimuth values from 0, 10, 20,, 360. Disturbing this sphere locus by substituting soft-iron matrix, W, and hard-iron vector, V, given in Table 4-1 into Equation (3.27) leads to the ellipsoidal locus indicated in Figure

75 500 Inclination : 90 degree Bz 0 mgauss By mgauss Y Bx mgauss X 500 Figure 4-2. Sphere locus with each circle of data points corresponding to magnetic field measurements made by the sensor rotation at inclination 90,with a specific highside and a cycle of azimuth values from 0, 10, 20,, 360. Hard-Iron(mGauss) Table 4-1. The ellipsoid of simulated data Actual Values Soft-Iron W V x = V y = V z =

76 PRESS A KEY TO GO TO THE NEXT ITERATION Bz 0 Raw Data mgauss Initial Calibration -200 Sphere -400 Ellipsoide Iteration Iteration By 0 mgauss 0 Bx mgauss 500 Figure 4-3. A schematic illustrating the disturbed data lying on an ellipsoid with characteristics given in Table 4-1.

77 Removing the unwanted magnetic interference field in the vicinity of the magnetometers from a real data set leaves errors due to magnetometer measurement noise which is shown in Figure 4-4. Based on this, the measurement noise standard deviation σ was evaluated as 0.3 mgauss. Therefore, the simulated data have been contaminated by adding a random normally distributed noise of σ = 0.3mGauss. Probability Density Function Mean = mgauss Standard Deviation=0.3mGauss Magnetic Field Strength (mgauss) Figure 4-4. Histogram of the magnetometer output error based on real data of a case study Table 4-2 shows six cases of simulation studies to quantify the estimation accuracy as a function of initial values and amount of noise added to the data points simulated on the ellipsoid of Figure 4-3. Figure 4-5 through Figure 4-7 illustrate the first three cases evaluating the performance of the iterative least-squares estimator initialized by the two-step linear estimator. In the absence of noise (Figure 4-5) the algorithm converges exactly to the actual values. When the measurement noise is increased to 0.3 and 6 mgauss, the results shown in Figure 4-6 and Figure 4-7 are obtained respectively. The algorithm is seen to converge in all these three cases. 64

78 Table 4-2. Parameters solved for magnetometer calibration simulations #Case Hard-Iron (mgauss) Initial Values Soft-Iron W Noise (mgauss) Hard-Iron (mgauss) Estimated Values Soft-Iron W 65 I Figure 4-5 II Figure 4-6 III Figure 4-7 IV Figure 4-8 V Figure 4-9 VI Figure 4-10 V x = V y = V z = V x = V y = V z = V x = V y = V z = V x = 180 V y = 120 V z = 270 V x = 180 V y = 120 V z = 270 V x = 180 V y = 120 V z = V x = 200 V y = 100 V z = 300 V x = V y = V z = V x = V y = V z = V x = V y = V z = V x = V y = V z = V x = V y = V z = Divergence Divergence

79 In cases IV (Figure 4-8) and V (Figure 4-9), the initial conditions were chosen randomly without using the two-step linear estimator. It is seen that the algorithm diverges under these random initial guesses. In case VI (Figure 4-10), initial guesses of hard-iron parameters were picked randomly from a normal distribution with a mean equal to the actual bias and a standard deviation of 50 mgauss but soft-iron parameters were initialized very close to two-step linear estimator. It is seen that case VI will converge even with random normally distributed noise of 6 mgauss. This means that, the divergence is primarily due to the initial guesses assigned to softiron parameters being away from the actual values. In solving the hard- and soft-iron algorithm, it was necessary to ensure avoiding illconditioning by examining the condition number of the matrix ζ T ζ during iterations. For this purpose, firstly, my Matlab code reduced the matrix (ζ T ζ) to Reduced Row Echelon Form through Gauss Elimination with Pivoting (the reason is that pivoting, defined as swapping or sorting rows or columns in a matrix through Gaussian elimination, adds numerical stability to the final result) (Gilat 2008). Secondly, the condition number of the reduced matrix (ζ T ζ) was calculated. For instance, in the convergence case III (Figure 4-7) initialized with two step linear estimator, the condition number at all iterations was calculated equal to 1. Further more in the divergence case V (Figure 4-9), where initial conditions were chosen randomly without two step linear solutions, the condition number until iteration of about 500 was calculated equal to 1 and, finally, due to improper initializing, after iteration of about 500 the condition number was calculated as infinity and the solution became singular. As a result, the problem is wellconditioned and divergence is due to the improper initializing. 66

80 The above six cases investigated for smaller strips of the measurement locus than the data points of Figure 4-3 indicated that the algorithm diverged repeatedly when smaller locus was used while it converged more often when a larger strip of the measurement locus was available. The results show that the data noise tolerated can be larger when a larger measurement locus of the modeled ellipsoid is available, although the algorithm cannot tolerate unreliable initial guesses even if the data is error-free. The algorithm initialized by the two-step linear estimator also diverges under high noise levels but not as often as it did when the initial guesses are unrealistic. The difference in initial conditions, however, is not the only cause of the divergence because these results show just a limited number of simulation locus out of many. In summary, it is implied that, initializing by the two-step linear estimator provides superior performance. It can tolerate higher noise and it requires a smaller portion of the measurement locus than the case where the iterative least-squares algorithm is used alone. However, it is also concluded that for relatively low cost magnetometers with relatively large magnitude output noise, this algorithm is not suitable unless a large portion of the ellipsoid is covered. Vx(mGauss) Vy(mGauss) W(1,1) W(1,2) W(2,3) W(2,2) Vz(mGauss) Iteration W(1,3) Estimated Actual Iteration W(3,3) Iteration Figure 4-5. Case #I: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the least-squares iterations 67

81 Vx(mGauss) W(1,1) W(2,2) Vy(mGauss) Vz(mGauss) W(1,3) W(1,2) Iteration W(2,3) W(3,3) Iteration Iteration Figure 4-6. Case #II: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the Vy(mGauss) Vx(mGauss) Vz(mGauss) W(1,1) least-squares iterations Iteration W(1,2) Estimated Actual Iteration W(1,3) W(2,2) W(2,3) W(3,3) Iteration Figure 4-7. Case #III: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the least-squares iterations 68

82 Vx(mGauss) W(1,1) W(2,2) Vy(mGauss) W(1,2) W(2,3) Vz(mGauss) Iteration Iteration Iteration W(1,3) Figure 4-8. Case #IV: Divergence of hard-iron (in mgauss) and soft-iron (unit-less) Vx(mGauss) Vy(mGauss) Vz(mGauss) W(3,3) estimates for the least-squares iterations W(1,3) W(1,2) W(1,1) Iteration Iteration Iteration Figure 4-9. Case #V: Divergence of hard-iron (in mgauss) and soft-iron (unit-less) W(2,2) W(2,3) W(3,3) estimates for the least-squares iterations

83 Vx(mGauss) W(1,1) W(2,2) Vy(mGauss) W(1,2) Estimated Iteration Actual Iteration Iteration Figure Case #VI: Parameters of hard-iron (in mgauss) and soft-iron (unit-less) for the least-squares iterations Vz(mGauss) W(1,3) W(2,3) W(3,3) 4.2 Experimental Investigations Laboratory Experiment The hard and soft-iron magnetometer calibration algorithm were further validated on an experimental data set collected in the University of Calgary laboratory located in the basement of engineering building. The results were compared with the MSA magnetic compensation method while incorporating diurnal variation corrections Experimental Setup For this purpose, we ran an experiment which models the MWD tool by mounting a low cost Micro Electro-Mechanical Systems (MEMS) integrated sensor incorporating a tri-axial gyro, accelerometer, and magnetometer on a turntable to obtain magnetic and acceleration 70

84 measurements and determine the turntable s orientation by inclination and azimuth. Since the sensor is fixed on the turntable, the readings change according to the orientation of the turntable. The MEMS sensor applied to this experiment was the STEVAL-MKI062V2 inemo inertial Module V2. The STEVAL-MKI062V2 is the second generation of the inemo module family. A complete set of communication interfaces with various power supply options in a small size form factor (4 4 cm) make inemo V2 a flexible and open demonstration platform. To aid in user development and analysis, the STEVAL-MKI062V2 demonstration kit includes a PC GUI for sensor output display and a firmware library to facilitate the use of the demonstration board features. This platform( with a size of only 4 by 4 centimeters) combines five sensors including a 6-axis sensor module (LSM303DLH : 3-axis accelerometer and 3-axis magnetometer), a 2-axis roll-and-pitch gyroscope, a 1-axis yaw gyroscope, a pressure sensor, and a temperature sensor (STEVAL-MKI062V2, 2010). For this study, effects of temperature and pressure were considered negligible and the MEMS gyroscope observations were not needed. Table 4-3 summarizes some of the features of LSM303DLH. A complete list of features of the LSM303DLH is available online at, Table 4-3. Features of 3-axis accelerometer and 3-axis magnetometer MEMS based sensors Parameter 3-axis accelerometer and 3-axis magnetometer (LSM303DLH) Magnetic Range ±1.3 to ± 8.1 Gauss Linear Acceleration Range ±2 g / ±4 g / ±8 g Operational Power Supply Range 2.5 V to 3.3 V (Voltage) Operating Temperature Range -30 to +85 C Storage Temperature Range -40 to +125 C 71

85 MEMS sensors suffer from various errors that have to be calibrated and compensated to get acceptable results. For this study, the MEMS accelerometers had already been calibrated to estimate and characterize the deterministic sensor errors such as bias, scale factor, and nonorthogonality (non-deterministic sensor noises were considered negligible). Based on the accelerometer calibration report, the MEMS accelerometers were well fabricated, not far away from the ideal case, and the scale factors as well as the misalignments were all in a small range 1. The calibration of the MEMS magnetometers using the hard- and soft-iron algorithm as well as the MSA method was examined in this study Turntable Setup The experiment was done by using a single-axis turntable, which does not require special aligned mounting. The turntable, capable of horizontally rotating clock-wise and counter-clockwise and vertically rotating in major directions of 0, 45, 90, 180, 270 and 360 degrees, had a feedback control to displace the sensor to designated angular positions. The turntable, controlled using a desktop PC, provided the condition where the magnetic survey probe was placed in a calibrated test stand and then the stationary stand was rotated through a series of directions. Then, a graph can show azimuth errors defined as the difference between the nominal test stand angles and the measured angles with and without correction. The post-calibration angular position calculated analytically from experimental data is compared with turntable heading inputs to verify how accurate the proposed algorithms could mathematically compensate for magnetic interference errors. 1 Thanks to Micro Engineering Dynamics and Automation Laboratory in department of Mechanical & Manufacturing Engineering at the University of Calgary for the calibration of the MEMS accelerometers and the collection of the experimental data. 72

86 Data Collection Procedure for Magnetometer Calibration For the process of magnetic interference calibration, it was required to take stationary measurements as the sensor fixed in location is rotated at attitudes precisely controlled. The number of attitudes must be at least as large as the number of the error parameters in order to avoid singularities when the inverse of the normal matrix is computed. In an ideal laboratory calibration, the stationary magnetometer and accelerometer measurements applied to the correction algorithm were collected from the stated experimental setup at attitudes of turntable with precise inclinations of 0 and 90 degrees. In the inclination test of 0 degree, the desired attitude measurements were made at five different angular positions through clockwise rotations of 0, 90, 180, 270 and 360 degrees. Table 4-4 indicates the tests measured by the sensors under the specific conditions. All data were collected at 100 Hz sampling frequency. After the preliminary experiments, it was found out that the electro-magnetic field generated from the table motor itself caused interference. Thus an extended sensor holder was developed, placing the sensors two feet away in the normal direction of the table surface to isolate the magnetometers from the electro-magnetic field generated by the table motor, the data collecting computer and the associated hardware (See Figure 4-11). Table 4-4. Turn table setup for stationary data acquisition Stationary Measurement Stationary Measurement File no. Inclination (degree) Angular Position (degree) File no Inclination (degree) Angular Position (degree) #1 0 0 # # # # # # # # #

87 Figure Experimental setup of MEMS integrated sensors on turn table at 45 inclination Heading Formula When the coordinate system of sensor package was set up at the arrangement of the three orthogonal axes shown in Figure 2-2, the azimuth would be determined by Equation (2.20). However, identifying different axes arrangements of laboratory experiment when reading raw data files lead to different azimuth formulas as follows: 2 2 (G z B y G y B z ) G x + G y + G z Azimuth = tan 1 2. (4.1) B x G y + G 2 z G x B z G z B y G y The order of rotation matrices and the choice of clock-wise or counter-clock-wise rotation can lead to different azimuth formulas. Therefore the orientation of the axes of magnetometer and accelerometer sensors needs to be noticed as experimental conditions. By considering the axis orientation of sensors, the correct azimuth formula was derived as Equation (4.1). The inclination was calculated from Equation (2.12) or (2.13). The experiment was performed at two different nominal inclination angles of 0 and 90 degrees in order to verify whether the 74 2

88 inclination angle was correctly observed in this experiment. The experimental results show that there is approximately a +3 degrees offset at 0 degree inclination, while the offset is -3 degrees at 90 degrees inclination (see Figure 4-12). The reason is that the cosine function in the inclination formula (Equation (2.13)) is not capable of differentiating positive and negative angles. Regardless of this calculation error, the offset would be consistently 3 degrees. It can be suggested that the turntable has an offset inclination angle of 3 degrees around test stand inclination angles of 0 and 90 degrees and therefore the inclination angle was correctly observed in this experiment #6 #7 #8 #9 #10 Inclination(Degree) #1 #2 #3 #4 # Samples x 10 4 Figure Inclination set up for each test Correction of the Diurnal Variations Diurnal variations are fluctuations with a period of about one day. The term diurnal simply means daily and this variation is called diurnal since the magnetic field seems to follow a 75

89 periodic trend during the course of a day. To determine the specific period and amplitude of the diurnal effect being removed, a second magnetometer is used as a base station located at a fixed location, which will measure the magnetic field for time-based variations at specific time intervals, every second, for instance. In this experiment, the time series data was gathered through a long time period of about five days ( hours) in time intervals of one second at a reference station close to the sensors mounted on the turn table but sufficiently remote to avoid significant interference. This project aims to remove the diurnal variations from this time series data. To remove noise spikes from the signal and fill in missing sample data from the signal, a median filter is applied. This median filter replaces each element in the data with the median value over the length of the filter (I chose the length of filter equal to 100 elements in the data). The data were acquired with a sample rate of 1 Hz. As we are only interested in the hourly magnetic variations over the five days period, the secondary fluctuations only contribute noise, which can make the hourly variations difficult to discern. Thus the lab data is smoothed from a sample rate of 1 Hz to a sample of one hour by resampling the median filtered signal (see Figure 4-13). The magnetic time series containing a periodic trend during the course of a day as diurnal effect are transferred into the frequency domain and makes it possible to determine the exact frequency (around 1/24 hour = ) of the periodic diurnal effect. Therefore, a filter is applied in time domain to attenuate the frequencies in a narrow band around the cut-off frequency of diurnal effects which was computed as (1/hour) as shown in Figure 4-14 where the largest peek corresponds to the frequency of 0.41 (1/hour). 76

90 As shown in Figure 4-14, there are two smaller peeks. According to the literature, the Earth s magnetic field undergoes secular variations on time scales of about a year or more which reflect changes in the Earth s interior. These secular variations can be predicted by global geomagnetic models such as IGRF through magnetic observatories which have been around for hundreds of years. Shorter time scales mostly arising from electric currents in the ionosphere and magnetosphere or geomagnetic storms can cause daily alterations referred as diurnal effects (Buchanan et al. 2013). As a result, the two peeks smaller than the diurnal peek cannot be due to variations in the Earth s geomagnetic field. They are most likely caused by the sensor noise and other man-made magnetic interferences present in the laboratory and affecting the time series data (In the laboratory, it was impossible to isolate all the magnetic interferences affecting the time series data) Magnetic Strength in a Sample Rate of 1 HZ(original signal) Resampled Magnetic Strength in a Sample Rate of 1 Hour Magnetic Strength(mGauss) Time (hours) Figure The observations of the geomagnetic field strength follow a 24 hour periodic trend. 77

91 In the data processing, the magnetometers must be synchronized to provide proper corrections when removing the time-based variations. Otherwise noise is added to the corrected survey data. Therefore, diurnal corrections are made at time 3 pm when the turn table rotations listed in Table 4-4 were implemented. The difference of red signal from the blue signal at 3pm shown in Figure 4-15 was subtracted from the magnetic field strength value acquired from the IGRF model at University of Calgary location in the month the experiment was performed (Table 4-5). Since, in the laboratory, it was impossible to isolate all the magnetic interferences affecting the time series data gathered, the absolute values of the time series cannot be reliable and thus the diurnal correction is applied to IGRF values. Single-Sided Amplitude Spectrum in Frequency Domain Before Filtering(Sample of One Hour) After Filter Removing Diurnal Effect Frequency(1/hour) 0.041(1/hour) = 24 hour Figure Geomagnetic field intensity in the frequency domain 78

92 Magnetic Strength (mgauss) in Time Domain Before Filtering(Median Hourly Resampled) After Filter Removing Diurnal Effect Subtracing Filtered Signal from Original am 6am 12pm 6pm 12am 6am 12pm 6pm 12am 6am 12pm 6pm 12am 6am 12pm 6pm 12am 6am 12pm Time(hour) Figure Geomagnetic field intensity in the time domain Table 4-5. Diurnal correction at laboratory March 2013 University of Calgary Laboratory Latitude: N Longitude: W Altitude(meter): 1111 IGRF Magnetic Field Strength (mgauss) IGRF Dip Angle IGRF Declination Angle Magnetic Field Strength = Diurnal Corrected (mgauss) Variations = Dip Angle Calibration Coefficients The magnetic calibration was examined for the data collected at inclination 0 (the fisrt five angular positions listed in Table 4-4). Table 4-6 demonstrates the solved parameters of the hard- 79

93 and soft-iron calibration algorithm, as well as the MSA correction and compares the results with and without diurnal corrections. It is seen in the Table 4-6 that the difference between hard-iron coefficients solved with and without applying diurnal corrections is very negligible. The locus of measurements is shown in Figure Table 4-6. Parameters in the magnetometer calibration experiment Initial Values of Hard-Iron Vector (mgauss) IGRF IGRF + Diurnal Correction V x = V x = V y = V y = V z = V z = Initial values of Soft-Iron Matrix IGRF IGRF + Diurnal Correction Estimated Values of Hard-Iron Vector (mgauss) IGRF IGRF + Diurnal Correction V x = V x = V y = V y = V z = V z = IGRF Estimated Values of Soft-Iron Matrix IGRF + Diurnal Correction MSA Parameters (mgauss) IGRF IGRF + Diurnal Correction B x = B x = B y = B y = B z = B z =

94 In Figure 4-16, the raw measurement data are shown by the red color dots on the ellipsoid, after calibration the locus of measurements will lie on the sphere which has a radius equal to the magnitude of the local magnetic field vector. The solved magnetic disturbances in table 4-6 are applied to correct the magnetic experimental data. The corrected magnetic field measurements are then used in the well-known azimuth expressions such as (2.19) and (2.20) to derive the corrected azimuth. Figure 4-17 and Figure 4-18 show the experimental trace comparing the azimuths computed using the sensor measurements at inclination 0, after and before calibration, with respect to the nominal heading inputs of Table 4-4. In Figure 4-17, georeferncing was obtained by the IGRF model corrected for diurnal variations, and in Figure 4-18 georeferncing was obtained only by the IGRF model. 600 PRESS A KEY TO GO TO THE NEXT ITERATION Bz 0 mgauss By 0 mgauss Bx mgauss 500 Raw Data Initial Calibration Sphere Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Ellipsoid Figure Portion of the ellipsoid representing the locus of magnetometer measurements from laboratory experimental data 81

95 270 #3 #4 Azimuth(Degree) Azimuth(Degree) #1 #2 Raw Azimuth Only Hard-Iron Calibration+IGRF+Diurnal Correction MSA+IGRF+Diurnal Correction Samples Figure Headings calibrated by MSA versus hard and soft iron (georeferncing obtained by IGRF model corrected for diurnal effects) #2 #3 #4 #5 Raw Azimuth MSA+IGRF Hard-Iron and Soft-Iron Calibration+IGRF 0 #1 # Samples Figure Headings calibrated by MSA versus hard and soft iron (georeferncing obtained by IGRF model without diurnal corrections) 82

96 Comparative results of the correction methods illustrated in Figures 4-17 and 4-18 are summarized in Table 4-7 to quantify the results. Figures 4-17 and 4-18 as well as Table 4-7 demonstrate that the heading residuals for the trace calibrated by MSA are less than 3 degrees while hard-iron and soft-iron calibration leads to higher residuals. It is also demonstrated that applying diurnal field correction will show no noticable improvement in heading compensation. It shows a small difference between iterative algorithms compensating for both hard- and softiron effects with respect to the first step of the linear estimator correcting only for hard iron biases. The analysis performed on the limited set of the experimental data compared the postcalibrated angular position by MSA versus hard- and soft-iron while correcting for diurnal variations. This verified that the MSA algorithm provides the most accurate heading solution either with or without diurnal correction. This implies that the hard-iron correction is much more essential than the soft-iron correction, although compensating for both hard- and soft-iron coefficients provides more accurate results. Table 4-7. A comparative summary of headings calibrated by different methods with respect to the nominal heading inputs Averaged Azimuth Values (degree) #1 #2 #3 #4 #5 Nominal Test Stand Heading Raw MSA + IGRF MSA + IGRF + Diurnal Hard and Soft-Iron + IGRF Hard and soft-iron + IGRF + Diurnal Only Hard-Iron + IGRF + Diurnal

97 4.3 Simulated Wellbore A simulated well profile is presented to compare the quality of MSA as well as hard-iron and soft-iron calibration and verify the calculations. Measured depth values for 80 sample points lying on a suggested wellbore horizontal profile were defined as known values to simulate the associated wellbore trajectory. For simplicity, the mathematical model of minimum curvature mentioned in the methodology section relating east and north coordinates on the wellbore horizontal profile to wellbore headings can be substituted by the equations of Balanced Tangential which do not need a ratio factor. The spatial coordinates of easting, northing, and elevation can be computed by the Balanced Tangential method as follows (Amorin and Broni-Bediako 2010): TVD = MD (cos I 1 + cos I 2 ), (4.2) 2 North = MD (sin I 1 cos AZ 1 + sin I 2 cos AZ 2 ), (4.3) 2 East = MD [(sin I 1 sin AZ 1 +sin I 2 sin AZ 2 )]. (4.4) 2 Applying the Balanced Tangential equations of (4.3) and (4.4), and assuming the first sample point inclination and azimuth as known values of AZ 1 = 117 and I 1 = 4, it is possible to calculate inclination and azimuth at subsequent points denoted by I 2 and A 2, respectively. Equations (4.3) and (4.4) can be solved to give: sin I 2 = 2 2 = 2 North MD cos AZ 1 sin I 1 + East MD sin AZ 1 sin I 1 2 2, (4.5) MD 2 84

98 2 North MD cos AZ 1 sin I 1 2 cos AZ 2 =. (4.6) MD sin I 2 For simplicity, inclination values are assumed to be derived between zero and 90 and azimuth values between zero and 180. It was necessary to ensure that sin and cos derived values are between -1 and +1. Therefore, the first suggested values of East and North were altered by trial and error to impose sin and cos conditions (Figure 4-19). At this stage the inclination and geographic azimuth values at all sample points have been determined. In the inclination Equation (2.12), by considering G x and G y as free unknowns it is possible to calculate G z. Highside angle is computed from Equation (2.11). In the case where there are no magnetic disturbances and no noise, V x = V y = V z = 0 and W = Identity Matrix are substituted in Equation (3.27) to calculate data points of magnetic field B P along the above wellbore trajectory. These data points are lying on a sphere with radius equal to B centered at the origin. 2 0 First Suggested Trajectory Final Trajectory -50 South(-)/North(+) (meter) West(-)/East(+) (meter) Figure Simulated wellbore horizontal profile 85

99 It is assumed that the simulated wellbore drilling takes place at the University of Calgary location. The values of DIP and B in Equation (3.27) are thus provided from Table 4-8. Magnetic azimuth values applied to Equation (3.27) are computed based on the declination values given in Table 4-8. The wellbore path moves through a series of positions with inclinations ranging from near vertical in the upper section of the wellbore referred to as BUILD, reaching to approximately horizontal in the down section of wellbore referred to as LATERAL. The first 28 sample points belong to the BUILD section and the last 51 sample points belong to LATERAL section. The BUILD section of the wellbore reached a measured depth (MD) of about 570 meter with a maximum inclination of 84 degrees and 80 meters horizontal displacement. The LATERAL section of the wellbore reached along measured depth (MD) of about 1080 meter with a maximum inclination of 90 degrees and 500 meter horizontal displacement. Geomagnetic referencing values for the BUILD section of the wellbore are different from those of the LATERAL section as indicted in Table 4-8. Table 4-8. Geomagnetic referencing values applied for the simulated wellbore March University of Calgary Location Latitude: N Longitude: W BULID LATERAL 2013 Altitude (meter) IGRF Magnetic Field Strength (mgauss) IGRF Dip angle IGRF Declination angle Disturbing this sphere locus by substituting the soft-iron matrix W and hard-iron vector V values in Table 4-9 into equation (3.27) leads to the ellipsoidal locus indicated in Figure

100 and Figure 4-21, which quantify the size of the magnetometer measurement locus simulated for BUILD and LATERAL sections of the simulated wellbore respectively. Table 4-9. The ellipsoid of simulated data BUILD : Actual Values Hard-Iron (mgauss) Soft-Iron W V x = V y = 0.1 Symmetric V z = LATERAL : Actual Value Hard-Iron (mgauss) Soft-Iron W V x = V y = 0.15 Non-Symmetric V z = The simulated data has been contaminated by adding a random normally distributed noise of σ = 0.3 mgauss and σ = 6 mauss to the data. Simulated wellbore data is listed in appendix G. The solved parameters of magnetic interference correction are stated in Table PRESS A KEY TO GO TO THE NEXT ITERATION 500 Bz 0 mgauss By mgauss Bx mgauss 500 Raw Data Initial Calibration Sphere Ellipsoide Iteration 1 Iteration 7 Figure Portion of the ellipsoid representing the locus of magnetometer measurements from BUILD section of the simulated wellbore(magnetic coordinates in mgauss) 87

101 PRESS A KEY TO GO TO THE NEXT ITERATION Bz mgauss 0 Raw Data -200 Initial Calibration Sphere -400 Ellipsoide Iteration 1 Iteration By 0 mgauss Bx -500 mgauss Figure Portion of the ellipsoid representing the locus of magnetometer measurements from LATERAL section of the simulated wellbore(magnetic coordinates in mgauss

102 Table Calibration parameters solved for simulated wellbore Case# Noise mgauss Hard-Iron mgauss Estimated Values Soft-Iron W MSA Correction mgauss I II III LATERAL BUILD LATERAL BUILD LATERAL BUILD Error free V x = V y = V z = V x = V y = V z = V x = V y = V z = V x = V y = V z = V x = V y = V z = V x = V y = V z = B x = B y = B z = B x = B y = B z = B x = B y = B z = B x = B y = B z = B x = B y = B z = B x = B y = B z = Moreover, the unreliability of conventional magnetic correction (SSA) discussed earlier in chapter2 is also demonstrated in Figure 4-22 through Figure 4-25, which compare the performance of SSA with MSA and hard- and soft-iron calibration. SSA is not stable particularly in LATERAL section. The major drawback of SSA was that it loses accuracy as the survey instrument approaches a high angle of inclination, particularly towards the east/west direction. This is shown in Figure 4-23, when SSA is most unstable at inclination 90 degrees and azimuth around 90 degrees. 89

103 BUILD LATERAL Measured Depth (meter) Figure Conventional correction is unstable in LATERAL sectionsince it is near horizontal east/west 350 Raw Azimuth(degree) Calibrated 300 Azimuth(degree) by IGRF+Multi-Survey Calibrated Azimuth(degree) by IGRF+Conventional Calibrated Azimuth(degree) by IGRF+Hard and Soft Iron Calibration 250 LATERAL Inclination(degree) Figure Conventional correction instability based on inclination 90

104 As explained in the methodology, the hard- and soft-iron calibration process transfers the magnetic measurements on a sphere locus with radius equal to the reference geomagnetic field strength, while the geomagnetic field direction (dip) is not involved in the calibration algorithm. On the other hand, the MSA methodology applies to the correction process both direction and strength of the reference geomagnetic field. These facts are illustrated in Figure 4-24 and Figure 4-25 where the green line (hard- and soft-iron) is the closest trace to IFR magnetic strength and the blue line (MSA) is the closest trace to the IGRF dip angle. 610 BUILD LATERAL IGRF Mangetic Strength(mGauss) 550 Observed Mangetic Strength(mGauss) Calibrated Mangetic Strength(mGauss) by IGRF+Multi-Survey Calibrated Mangetic Strength(mGauss) by IGRF+Conventional 540 Calibrated Mangetic Strength(mGauss) by IGRF+Hard and Soft Iron Calibration Survey point no. Figure Calculated field strength by calibrated measurements The simulated well profile has been achieved through minimum curvature trajectory computations explained in the methodology section. Figure 4-26 through Figure 4-28 present pictorial view of the simulated wellbore in the horizontal plane under the conditions stated in Table 4-10 and show that the conventional method is entirely unprofitable. As it can be seen, the data also requires the magnetic declination to attain the geographic azimuth which is a requisite to the computation of the wellbore horizontal profile

105 BUILD LATERAL IGRF Dip(degree) Observed Dip(degree) Calibrated Dip(degree)by IGRF+Multi-Survey Calibrated Dip(degree)by IGRF+Conventional Calibrated Dip(degree)by IGRF+Hard and Soft Iron Calibration Survey point no. South(-)/North(+) (meter) Figure Calculated field direction by calibrated measurements Raw Data MSA+IFR Conventional+IFR Hard-Soft Iron+IFR Real Path West(-)/East(+) (meter) 600 Figure Case #I: Wellbore pictorial view of the simulated wellbore in horizontal plane (no error) 92

106 South(-)/North(+) (meter) Raw Data MSA+IFR Conventional+IFR Hard-Soft Iron+IFR West(-)/East(+) (meter) Figure Case #II: Wellbore pictorial view of the simulated wellbore in horizontal plane South(-)/North(+) (meter) (random normally distributed noise of 0.3 mgauss) Raw Data MSA+IFR Conventional+IFR Hard-Soft Iron+IFR Real Path West(-)/East(+) (meter) Figure Case #III: Wellbore pictorial view of the simulated wellbore in horizontal plane (random normally distributed noise of 6 mgauss) 93

107 Table 4-11 shows a summary of comparative wellbore trajectory results from correction methods for case III of Table Correction Method Table Comparative wellbore trajectory results of all correction methods Case #III (random normally distributed noise of 6 mgauss) East Displacement East meter East Real Path (Diff from Real Path) meter North Displacement North meter North Real Path (Diff from Real Path) meter Closure Distance from Real Path meter Raw Data MSA Hard-Soft Iron Real Path As demonstrated earlier, the hard- and soft-iron iterative algorithm is not suitable with relatively large magnitude output noise unless a large portion of the ellipsoid is covered. Actually, the data noise tolerated can be larger when a larger measurement locus of the modeled ellipsoid is available. However, Figure 4-20 and Figure 4-21 indicate that a small portion of the ellipsoid is covered. Therefore, in the presence of a random normally distributed noise of 6 mgauss, the wellbore trajectory corrected by the hard- and soft-iron method is away from the real path. On the other hand, Table 4-11 indicates that MSA corrected surveys have produced a significant improvement in surveyed position quality (by as much as 85%), over the raw data surveyed position when compared to the real path and allowed the well to achieve the target. 94

108 4.4 A Case Study Comparison of the quality of hard- and soft-iron calibration as well as MSA, which are techniques providing compensation for drillstring magnetic interference, have been demonstrated. Geomagnetic referncing is obtained by IGRF versus IFR to investigate that the benefits of techniques can be further improved when used in conjunction with IFR. A case study of a well profile that uses these techniques is presented and compared with an independent navigation grade gyroscope survey for verification of the calculations since gyros are reported to have the best accuracy for wellbore directional surveys. The most benefitial technique to drilling projects is illustrated. Real data were scrutinized for outliers in order to draw meaningful conclusions from it. Outliers was rejected in data by computing the mean and the standard deviation of magnetic strength and dip angle using all the data points and rejecting any that are over 3 standard deviations away from the mean. In this case study, the survey probe is moved through the wellbore at a series of positions with inclinations ranging from near vertical in the upper ( BUILD ) section of the wellbore reaching to approximately horizontal in the down ( LATERAL ) section of the wellbore. Geomagnetic referencing values for the BUILD section of the wellbore are different from those for the LATERAL section as indicted in Table The solutions for each case have been listed in Table Geomagnetic referencing Table Geomagnetic referencing values Field Strength (mgauss) Dip (degrees) Declination (degrees) IFR (LATERAL) IFR (BIULD) IGRF

109 The BUILD section of the wellbore reached a measured depth (MD) of about 750 meters with a maximum inclination of 75 degrees and 250 meter horizontal displacement. The LATERAL section of the wellbore reached a measured depth (MD) of about 1900 meter with a maximum inclination of 90 degrees and 1100 meter horizontal displacement. Table Calibration parameters solved for the case study BUILD IFR IGRF Hard-Iron V x = V x = Vector V y = V y = (mgauss) V z = V z = Soft-Iron Matrix B x = B x = MSA B y = B y = (mgauss) B z = B z = LATERAL IFR IGRF Hard-Iron V x = V x = Vector V y = V y = (mgauss) V z = V z = Soft-Iron Matrix MSA (mgauss) B x = B x = B y = B y = B z = B z = Moreover, the unreliability of conventional magnetic correction (SSA) discussed earlier in chapter2 is also demonstrated in Figure 4-29 through Figure 4-32, which compare the performance of SSA with MSA and hard- and soft-iron calibration. Even though IFR was used in each case, SSA is not stable particularly in LATERAL section. The major drawback of SSA was that it loses accuracy as the survey instrument approaches a high angle of inclination, particularly 96

110 towards the east/west direction. This is shown in Figure 4-32, where SSA is most unstable at inclination 90 degrees and azimuth around 270 degrees ZOOM2 ZOOM1 Inclination (degree) Raw Azimuth(degree) Calibrated Azimuth(degree) by IFR+Multi-Survey Calibrated Azimuth(degree) by IFR+Conventional Calibrated Azimuth(degree) by IFR+Hard and Soft Iron Calibration Measured Depth (meter) Figure Conventional correction is unstable in LATERAL sectionwhen it is near horizontal east/west Inclination (degree) Raw Azimuth(degree) Calibrated Azimuth(degree) by IFR+Multi-Survey Calibrated Azimuth(degree) by IFR+Conventional Calibrated Azimuth(degree) by IFR+Hard and Soft Iron Calibration 290 LATERAL Measured Depth (meter) Figure Zoom1 of Figure

111 Inclination (degree) Raw Azimuth(degree) Calibrated Azimuth(degree) by IFR+Multi-Survey Calibrated Azimuth(degree) by IFR+Conventional Calibrated Azimuth(degree) by IFR+Hard and Soft BUILD Iron Calibration Measured Depth (meter) Figure Zoom2 of Figure LATERAL Raw Azimuth(degree) Calibrated Azimuth(degree) by IFR+Multi-Survey Calibrated Azimuth(degree) by IFR+Conventional Calibrated Azimuth(degree) by IFR+Hard and Soft Iron Calibration Inclination (degree) Figure Conventional correction instability based on inclination 98

112 Figure 4-33 and Figure 4-34 indicate that the green line (hard and soft-iron) is the closest trace to IFR magnetic strength and blue line (MSA) is the closest trace to the IFR dip angle. The reason is the same as those explained for the simulated well path in section (4.3) IFR Mangetic Strength(mGauss) Observed Mangetic Strength(mGauss) Calibrated Mangetic Strength(mGauss) by IFR+Multi-Survey Calibrated Mangetic Strength(mGauss) by IFR+Conventional Calibrated Mangetic Strength(mGauss) by IFR+Hard and Soft Iron Calibration Survey point no. Figure Calculated field strength by calibrated measurements 76.5 IFR Dip(degree) Observed Dip(degree) Calibrated Dip(degree)by IFR+Multi-Survey Calibrated Dip(degree)by IFR+Conventional Calibrated Dip(degree)by IFR+Hard and Soft Iron Calibration 76 BUILD LATERAL Survey Point no. Figure Calculated field direction by calibrated measurements 99

113 The well profile has been estimated through minimum curvature trajectory computations explained in the methodology section. Table 4-14 shows a summary of comparative wellbore trajectory results from correction methods using the case study data. Method Raw Data MSA Hard- Soft Iron Table Comparative wellbore trajectory results of all correction methods Geomagnetic referencing East Displacement East meter East Gyro (Diff from Gyro) meter North Displacement North meter North Gyro (Diff from Gyro) meter Closure Distance from Gyro meter IGRF IFR IGRF IFR IGRF IFR Gyroscope Figure 4-35 shows the respective pictorial view in the horizontal plane. In Figure 4-35, deviation of the conventional corrected trajectory from gyro survey (the most accurate wellbore survey) indicates that conventional method is entirely inaccurate. Figure 4-35 and Table 4-14 demonstrate that the positining accuracy gained by multistation analysis surpasses hard and softiron compenstion resuts. Moreover, real-time geomagnetic referencing ensures that the position difference of all correction methods with respect to gyro survey is enhanced when IFR is applied. Table 4-14 indicates that MSA combined with IFR-corrected surveys have produced a significant improvement in surveyed position quality (by as much as 84%), over the raw data surveyed position when compared to a gyro survey as an independent reference, and allowed the well to achieve the target (there was no geometric geologic target defined for the case study). 100

114 This limited data set confirms but does not yet support a conclusion that magnetic surveying accuracy and quality can be improved by mapping the crustal magnetic field of the drilling area and combining with the use of multistation analysis. It is also clear that without the combination of MSA with IFR, the potential for missing the target would have been very high. 4.5 Summary The robustness of the hard- and soft-iron algorithm was validated through the simulation runs and it was discovered that the iterative least-squares estimator is sensitive to three factors comprising initial values, sampling and sensor noise. If the initial values are not close enough to the actual values, the algorithm may diverge and the amount of noise that can be tolerated is affected by the shape of the sampling locus of measurements. The experimental analysis verified that MSA model provides the most accurate magnetic compensation, either with or without diurnal correction. Both the simulated and real wellbore profile corrections indicated that MSA model has produced significant improvement in surveyed position accuracy over hard- and softiron model especially when combined with IFR-corrected surveys. 101

115 0 102 South(-)/North(+) (meter) LATERAL BUILD Raw Data I GRF(declination) MSA+IGRF Conventional+IGRF Hard-Soft Iron+IGRF Raw Data I FR(declination) MSA+IFR Conventional+IFR Hard-Soft Iron+IFR Gyro West(-)/East(+) (meter) Figure Wellbore pictorial view in horizontal plane by minimum curvature

116 Chapter Five: CONCLUSIONS and RECOMMENDATIONS for FUTURE RESEARCH 5.1 Summary and Conclusions In this study, a set of real data, simulated data and experimental data collected in the laboratory were utilized to perform a comparison study of magnetic correction methods compensating for the two dominant error sources of the drillstring-induced interference and unmodeled geomagnetic field variations. The hard- and soft-iron mathematical calibration algorithms were validated for determining permanent and induced magnetic disturbances through an iterative least-squares estimator initialized using the proposed two-step linear solution. The initialization provided superior performance compared to random initial conditions. The simulation and experimental runs validated the robustness of the estimation procedure. As reported in some previous publications, the hard- and soft-iron calibration algorithm is limited to the estimation of the hard-iron biases and combined scale factor, and some of the softiron effects by assuming the soft-iron matrix to be diagonal. However, this study makes it possible to extend the applicability of this algorithm to all soft-iron coefficients and misalignment errors by considering the soft-iron matrix to be a symmetric matrix with non-zero off-diagonal components. However, the small difference between the iterative algorithm, compensating for both hard-iron and soft-iron effects with respect to the first step of the linear solution correcting only for hard iron biases shows that soft-iron compensation can be neglected. The results were compared with SSA and MSA correction methods while incorporating real time geomagnetic referencing and diurnal variation corrections. It is demonstrated that SSA is significantly unstable at high angles of inclination, particularly towards the east/west direction; thus SSA is no longer applicable in the industry. Finally, the results support that the positining 103

117 accuracy gained by multistation analysis surpasses hard- and soft-iron compenstion resuts. That is because, the amount of noise that can be tolerated by hard- and soft-iron algorithm is affected by the shape of the sampling locus of measurements. This algorithm is not suitable for relatively large magnitude output noise unless a large portion of the ellipsoid is covered. However, it is unlikely that a single magnetic survey tool would see such a wide range in a well trajectory. Investigations in this study performed on the limited data sets show excellent agreement with what is done in the industry, which believes that the the analysis of data from multiple wellbore survey stations, or MSA, is the key to addressing drillstring interference (Buchanan et al. 2013). There are some evidences that improvements in the compensation of magnetic disturbances are limited. The reason is that a well can typically take many days or weeks to drill, and the disturbance field effects will be largely averaged over this time period. However, this is not the case for wells drilled very quickly and so magnetic surveys are conducted in a short time frame. Therefore, it is expected that applying the diurnal field correction will show very little improvement in the surveyed position of a wellbore. The experimental data provided in the laboratory incorporating diurnal variation corrections also confirms the fact that applying the diurnal field correction will yield no noticable improvement in heading compensation. The real wellbore investigated in this study was not subject to this level of service and so the contribution of the diurnal field could not be established for a real data set. Potential improvements in the accuracy of magnetic surveys have been suggested by taking advantage of IFR data, which take into account real-time localized crustal anomalies during surveys. The benefit of real-time geomagnetic referencing is two fold; firstly to provide the most accurate estimate of declination and, secondly, to provide the most accurate estimate of the strength and dip of the local magnetic field that the survey tool should have measured. This 104

118 allows the MSA algorithm to correct the survey based on the actual local magnetic field at the site and therefore further reduced positional uncertainty is achieved (Lowdon and Chia 2003). The IFR correction effect was not presented in the experimental analysis done in this study. Therefore, in the experimental investigation, the magnetic surveying quality has been corrected without the crustal field, using a standard global geomagnetic main field model such as IGRF as a reference model. However, a limited analysis of real data confirmed (but the limited data set does not yet support a conclusion) that the position accuracy of all correction methods with respect to a gyro survey can be improved by mapping the crustal magnetic field of the drilling area. Investigations of the case study suggest that mapping the crustal magnetic anomalies of the drilling area through IFR and combining with an MSA compensation model provides a significant improvement in surveyed position quality (by as much as 84%), over the raw data surveyed position when compared to a gyro survey as an independent reference, thus allowing the well to achieve the target. It is also implied that without the combination of MSA with IFR, the potential for missing the target would have been very high. The wellbore positional accuracies generally available in the modern industry are of the order of 0.5% of the wellbore horizontal displacement. This equals to 5.5 meters for the 100 lateral section of the case study wellbore trajectory with horizontal displacement of 1100 meter. In this thesis, the position accuracy of the case study wellbore trajectory compensated by utilization of MSA in conjunction with real-time geomagnetic referencing achieved the closure distance of meter (Table 4-14) which shows a %18 improvement over the 5.5 meters of the positional accuracy by MWD surveys availbale in the modern industry. On the other hand, 105

119 hard- and soft-iron calibration provides the closure distance of meters (Table 4-14) which is not acceptable in the current industry. Well positioning accuracy approach provided by a gyro, can be delivered when MSA is applied in conjunction with IFR, thus providing a practical alternative to gyro surveying generally with little or no impact on overall well position accuracy and with the practical benefit of reduced surveying cost per well. Wells are drilled today without the use of gyro survey in the survey program entirely because evaluation works such as this research have been done. Although the magnetic survey tool is still important for the oil industry, an independent navigation-grade gyro survey has a lower error rate compared to magnetic tools and is widely used as a reference to verify how accurate the MSA can compensate the magnetic interference and control drilling activities in high magnetic interference areas where one cannot rely on magnetic tools. 5.2 Recommendations for Future Research There are limitations and cautions regarding the hard and soft-iron, as well as the MSA models which are recommended for future investigations in order to more accurately compensate for the magnetic disturbances during directional drilling Cautions of Hard-Iron and Soft-iron Calibration Limitations and cautions of the hard and soft model are as follows: (i) The linearity assumption about the relation of the induced soft-iron field with the inducing local geomagnetic field is not accurate (Ozyagcilar 2012b). The complex relationship between the induced field with the inducing local geomagnetic field is illustrated by a hysteresis loop (Thorogood et al. 1990). Further investigations on modeling of the hysteresis effects are recommended for the future research. 106

120 (ii) It should be noted that magnetometer measurements used to fit the calibration parameters should be taken as the sensor is rotated in azimuth, inclination and highside. The reason is that taking scatter data at different orientation angels prevents the magnetometer noise from dominating the solution (Ozyagcilar 2012b). It is apparent that the magnetometer measurements made at the same orientation will be identical apart from sensor noise. Therefore, it is recommended to use the accelerometer sensor to select various magnetometer measurements for calibration taken at significantly different inclinations (Ozyagcilar 2012b). This is possible where the calibration process is performed under controlled conditions by placing the sensor package in a calibrated precision stand and the stand can then be oriented in a wide range of positions which are designed to give the best possible spread in attitude so that warrantee the best possible resolution of calibration factors. However, it is unlikely that a single magnetic survey tool would see such a wide range in a single run (Ozyagcilar 2012b). Therefore coefficients acquired from downhole calibration computations cannot be expected to provide equal accuracy. On the other hand, the soft-iron induced error varies with the orientation of the probe relative to the inducing local geomagnetic field and therefore downhole acquisition of soft-iron coefficients is essential. Since the soft-iron effects are negligible compared to the hard-iron effects, it is recommended that the calibration values obtained in the laboratory for significant hard-iron effects be replaced with measurements taken in the downhole environment and the negligible soft-iron effects can be disregarded in directional drilling operations Cautions of MSA Technique Since MSA corrects for drillstring interference by deriving a set of magnetometer correction coefficients common to a group of surveys, it implies that the state of magnetization remains unchanged for all surveys processed as a group (Brooks et al. 1998). However, drillstring 107

121 magnetization may have been acquired or lost slowly during the course of the drilling operation (Brooks et al. 1998). The reason is that BHA magnetic interference is caused by repeated mechanical strains applied to ferromagnetic portions of the BHA in the presence of the geomagnetic field during drillstring revolutions (Brooks 1997). Therefore, to create valid data sets for calculating accurate sensor coefficients through the MSA calibration process, it is recommended to use data from a minimum number of surveys. Furthermore, it is recommended to group together a sufficiently well-conditioned data set showing a sufficient change in toolface attitude along with a sufficient spread in azimuth or inclination (Brooks et al. 1998). In MSA method, after identifying and correcting most of systematic errors common to all surveys in the data set, the residual errors modeled as random errors or sensor noise can be estimated from sensor specifications and knowledge of the local field or it can be estimated more directly from the residual variance minimized in the calibration process of MSA. In a way that after the iteration converges to a solution, the residual value of V is used as a quality indicator and as an input quantity for the calculation of residual uncertainty (Brooks et al. 1998). The MSA numerical algorithm operates on several surveys simultaneously. The simultaneous measurements taken at several survey stations provide additional information, which can be used to perform a full calibration by solving for additional unknown calibration parameters, including magnetometer and accelerometer coefficients affecting one or more axes (Brooks et al. 1998). However, accelerometer errors are not routinely corrected since there is no significant improvement. As evidenced by position comparisons here, the most beneficial technique for correction of BHA magnetic disturbances is achieved by the application of MSA. However, as this has not been fully established or agreed amongst the directional surveying community, and due to the 108

122 very limited availability of real data sets, conclusion of this nature is not drawn here but is only implied. Availability of case studies presenting a wide range of well locations and trajectories in varying magnetic environments is desired in the future. 109

123 References Amorin R. and Broni-Bediako E., 2010, Application of Minimum Curvature Method to Well path Calculations, Journal of Applied Sciences, Engineering and Technology 2, 7. Anon A., 1999, Horizontal and multilateral wells: Increasing production and reducing overall drilling and completion costs, Journal of Petroleum Technology, 51, 7. Aster R.C., Borchers B. and Thurber C., 2003, Parameter Estimation and Inverse Problems. Bourgoyne A.T., Millheim K.K., Chenevert M.E. and Young F.S., 2005, Applied Drilling Engineering, Tenth Printing Society of Petroleum Engineers Text Series, Richardson, TX, USA. Bourgoyne A.T., Millheim K.K., Chenvert M.E. and Young F.S., 1991, Applied Drilling Engineering, SPE Textbook Series, 2, Brooks A.G., 1997, Method of Correcting Axial and Transverse Error Components in Magnetometer Reading During Wellbore Survey Operations, U.S. patent No. 5, 623, 407, April. Brooks A.G., Goodwin A., 1994, Method of Correcting Axial and Transverse Error Components in Magnetometer Reading During Wellbore Survey Operations, European patent No EP B1, Nov. Brooks A.G., Gurden P.A., Noy K.A., 1998, Practical Application of a Multiple-Survey Magnetic Correction Algorithm, paper SPE presented at SPE Annual Technical Conference, New Orleans, Sep Buchanan A., Finn C.A., Love J.J., Worthington E.W., Lawson F.,Maus S., Okewunmi S. and Poedjono B., 2013, Geomagnetic Referencing - the Real-Time Compass for Directional Drillers, oilfield review, Autumn 2013, Schlumberger. Carden R.S. and Grace R.D., 2007, Horizontal and Directional Drilling. 110

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129 116 APPENDIX A: SIMULATED WELLBORE

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