Reduction of Wellbore Positional Uncertainty During

University of Calgary

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2015-02-04

Directional Drilling

Hadavand Zahra

Hadavand Z (2015) Reduction of Wellbore Positional Uncertainty During Directional Drilling

(Unpublished masters thesis) University of Calgary Calgary AB doi1011575PRISM27569

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master thesis

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UNIVERSITY OF CALGARY

Reduction of Wellbore Positional Uncertainty During Directional Drilling

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

copy Zahra Hadavand 2015

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

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 amp 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

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

ldquoWords will never say how grateful I am to yourdquo

Table of Contents

Abstract ii Acknowledgements iii Dedication iv Table of Contentsv List of Tables vii List of Symbols and Abbreviations xi

CHAPTER ONE INTRODUCTION1 11 Problem statement3

111 Borehole Azimuth Uncertainty3 112 Geomagnetic Referencing Uncertainty 5

12 Thesis Objectives 6 13 Thesis Outline 7

CHAPTER TWO REVIEW OF DIRECTIONAL DRILLING CONCEPTS AND THEORY 8

21 Wellbore Depth and Heading 8 22 Review of Sources and Magnitude of Geomagnetic Field Variations9

221 Review of Global Magnetic Models10 222 Measuring Crustal Anomalies Using In-Field Referencing (IFR) Technique 11 223 Interpolated IFR (IIFR) 12

23 Theory of Drillstring Magnetic Error Field 13 24 Ferromagnetic Materials Hard-Iron and Soft-Iron Interference 14 25 Surveying of Boreholes 15 26 Heading Calculation 17 27 Review of the Principles of the MWD Magnetic Surveying Technology21 28 Horizontal Wells Azimuth 22 29 Previous Studies24

291 Magnetic Forward Modeling of Drillstring25 292 Standard Method 25 293 Short Collar Method or Conventional Magnetic Survey (Single Survey) 26 294 Multi-Station Analysis (MSA) 28 295 Non-Magnetic Surveys 30

210 Summary30

CHAPTER THREE METHODOLOGY 32 31 MSA Correction Model 32 32 Hard-Iron and Soft-Iron Magnetic Interference Calibration38

321 Static Hard-Iron Interference Coefficients 38 322 Soft-Iron Interference Coefficients39 323 Relating the Locus of Magnetometer Measurements to Calibration Coefficients

40 324 Calibration Model41 325 Symmetric Constrait 44 326 Least-Squares Estimation 47

327 Establishing Initial Conditions 51 3271 Step 1 Hard-Iron Offset estimation51 3272 Step 2 Solving Ellipsoid Fit Matrix by an Optimal Ellipsoid Fit to the Data

Corrected for Hard Iron Biases 52 33 Well path Design and Planning 54 34 Summary58

CHAPTER FOUR RESULTS AND ANALYSIS60 41 Simulation Studies for Validation of the Hard and Soft-Iron Iterative Algorithm60 42 Experimental Investigations 70

421 Laboratory Experiment70 4211 Experimental Setup70 4212 Turntable Setup72 4213 Data Collection Procedure for Magnetometer Calibration 73

422 Heading Formula 74 423 Correction of the Diurnal Variations 75 424 Calibration Coefficients79

43 Simulated Wellbore 84 44 A Case Study 95 45 Summary101

CHAPTER FIVE CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH103

51 Summary and Conclusions 103 52 Recommendations for Future Research106

521 Cautions of Hard-Iron and Soft-iron Calibration 106 522 Cautions of MSA Technique 107

REFERENCES 110

APPENDIX A SIMULATED WELLBORE116

List of Tables

Table 4-1 The ellipsoid of simulated data 62

Table 4-2 Parameters solved for magnetometer calibration simulations 65

Table 4-3 Features of 3-axis accelerometer and 3-axis magnetometer MEMS based sensors 71

Table 4-4 Turn table setup for stationary data acquisition 73

Table 4-5 Diurnal correction at laboratory 79

Table 4-6 Parameters in the magnetometer calibration experiment 80

Table 4-7 A comparative summary of headings calibrated by different methods with respect to the nominal heading inputs 83

Table 4-8 Geomagnetic referencing values applied for the simulated wellbore 86

Table 4-10 Calibration parameters solved for simulated wellbore 89

Table 4-11 Comparative wellbore trajectory results of all correction methods 94

Table 4-12 Geomagnetic referencing values 95

Table 4-13 Calibration parameters solved for the case study 96

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 16

Figure 2-3 Horizontal component of error vector 24

Figure 2-4 Eastwest component of error vector 24

Figure 2-5 Conventional correction by minimum distance 29

Figure 3-1 Representation of the geometry of the tangential method 56

Figure 3-2 Representation of the geometry of the minimum curvature method 57

Figure 4-1 Sphere locus with each circle of data points corresponding to magnetic field measurements made by the sensor rotation at highside 90deg 61

Figure 4-2 Sphere locus with each circle of data points corresponding to magnetic field measurements made by the sensor rotation at inclination 90deg 62

Figure 4-3 A schematic illustrating the disturbed data lying on an ellipsoid 63

Figure 4-4 Histogram of the magnetometer output error based on real data of a case study 64

Figure 4-5 Case I Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for the least-squares iterations 67

Figure 4-6 Case II Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for the least-squares iterations 68

Figure 4-7 Case III Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for the least-squares iterations 68

Figure 4-8 Case IV Divergence of hard-iron (in mGauss) and soft-iron (unit-less) estimates for the least-squares iterations 69

Figure 4-9 Case V Divergence of hard-iron (in mGauss) and soft-iron (unit-less) estimates for the least-squares iterations 69

Figure 4-10 Case VI Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for the least-squares iterations 70

Figure 4-11 Experimental setup of MEMS integrated sensors on turn table at 45deg inclination 74

Figure 4-12 Inclination set up for each test 75

Figure 4-13 The observations of the geomagnetic field strength follow a 24 hour periodic trend 77

Figure 4-14 Geomagnetic field intensity in the frequency domain 78

Figure 4-15 Geomagnetic field intensity in the time domain 79

Figure 4-16 Portion of the ellipsoid representing the locus of magnetometer measurements from laboratory experimental data 81

Figure 4-17 Headings calibrated by MSA versus hard and soft iron (georeferncing obtained by IGRF model corrected for diurnal effects) 82

Figure 4-18 Headings calibrated by MSA versus hard and soft iron (georeferncing obtained by IGRF model without diurnal corrections) 82

Figure 4-19 Simulated wellbore horizontal profile 85

Figure 4-20 Portion of the ellipsoid representing the locus of magnetometer measurements from BUILD section of the simulated wellbore 87

Figure 4-21 Portion of the ellipsoid representing the locus of magnetometer measurements from LATERAL section of the simulated wellbore 88

Figure 4-22 Conventional correction is unstable in LATERAL section 90

Figure 4-23 Conventional correction instability based on inclination 90

Figure 4-24 Calculated field strength by calibrated measurements 91

Figure 4-25 Calculated field direction by calibrated measurements 92

Figure 4-26 Case I Wellbore pictorial view of the simulated wellbore in horizontal plane (no error) 92

Figure 4-27 Case II Wellbore pictorial view of the simulated wellbore in horizontal plane (random normally distributed noise of 03 mGauss) 93

Figure 4-28 Case III Wellbore pictorial view of the simulated wellbore in horizontal plane (random normally distributed noise of 6 mGauss) 93

Figure 4-30 Zoom1 of Figure 4-29 97

Figure 4-35 Wellbore pictorial view in horizontal plane by minimum curvature 102

Symbol

1198601198851

1198601198852

BN BE BV

BV(119899) Bh(119899)

BV(ref) Bh(ref)

Bx By and Bz

BxCorr(119899) ByCorr(119899) BzCorr(119899)

Bxm(119899) Bym(119899) Bzm(119899)

List of Symbols and Abbreviations

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 Earthrsquos coordinate frame

vertical and horizontal components of magnetic field at 119899P

survey station

reference value of vertical and horizontal components of

geomagnetic field

geomagnetic components along instrument-fixed coordinate

corrected magnetic components at 119899P

th survey station in

instrument-fixed coordinate frame

measured magnetic components at 119899P

dip angle of geomagnetic vector

ldquodog-legrdquo curvature

magnitude of gravity vector

g gravity vector

Gx Gy Gz gravity components along instrument-fixed coordinate frame

HS borehole highside angle

I borehole inclination

1198681 inclination angle at upper survey point

1198682 inclination angle at lower survey point

MD measured depth

N number of surveys

RF ratio factor for minimum curvature

TVD true vertical depth

unit vectors in Earthrsquos coordinate frame UNEV

unit vectors in instrument-fixed coordinate frame UXYZ

V hard-iron vector

components of hard-iron vector along instrument-fixed Vx Vy and Vz

coordinate frame

W soft-iron matrix

∆ change in parameter

∆119860119885 borehole azimuth error

∆Bx ∆By drillstring magnetic error field in cross-axial direction

∆Bz drillstring magnetic error field in axial direction

εx εy εz small perturbations of ∆Bx ∆By ∆Bz

119985 variance

Abbreviation Description

BGGM British Global Geomagnetic Model

BHA Bottom-Hole-Assembly

HDGM High Definition Geomagnetic Model

IFR In-Field Referencing

IGRF International Geomagnetic Reference Field

IIFR Interpolated IFR

MEMS Micro Electro-Mechanical Systems

mGauss miliGauss

MSA Multi-Station Analysis

MWD Measurement While Drilling

NMDC Non-Magnetic Drill Collars

NOAA National Oceanic and Atmospheric Administration

nT nanoTesla

SSA Single Station Analysis

WBM Wellbore Mapping

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

A perpendicular pair or an orthogonal triad of accelerometers measure the Earthrsquos 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 engineerrsquos 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

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

05 of the wellbore horizontal displacement

11 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

111 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)

(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 toolrsquos 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

112 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

12 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

bull 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

bull Estimate the applicability of the magnetic compensation methods including single-

survey 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

bull 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

bull Correct the case study wellbore trajectory by applying the most accurate magnetic

compensation solution for drillstring-induced interference and combine the results with real-

time 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

13 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 hard-

iron 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

Chapter Two REVIEW of DIRECTIONAL DRILLING CONCEPTS and THEORY

21 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 ldquoheadingrdquo 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 instrumentrsquos longitudinal axis with respect to the direction of the Earthrsquos 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

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 ldquoabsolute azimuthrdquo or ldquocorrected azimuthrdquo 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)

22 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

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 Earthrsquos 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 Earthrsquos 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

ldquosecularrdquo 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 Earthrsquos surface and interacting with the main field

(Wolf and deWardt 1981)

Fields created by the magnetization of rocks and minerals in the Earthrsquos crust typically found

in deep basement formations in the vicinity of drilling (crustal anomalies) (Bourgoyne et al

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

221 Review of Global Magnetic Models

In order to keep track of the ldquosecularrdquo 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

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

222 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

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)

223 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

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)

23 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 ie

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 Earthrsquos 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 ∆B119911 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)

24 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 1000120583119879 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

estimate and subtract the hard-iron offset through correction methods of drilling assembly

corrupting magnetic field

25 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 Earthrsquos 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 toolrsquos 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 lsquodownrsquo

and with three magnetometers it is always possible to figure out which way is the magnetic

The set of Earth-fixed axes (N E V) shown in Figure 2-2 is delineated with V being in the

direction of the Earthrsquos 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 ie 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

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) ldquoHSrdquo 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

unit vector observed in the survey toolrsquos coordinate system to the Earthrsquos 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 (119866119909 119866119910 119866119911)

and (119861119909 119861119910 119861119911) 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

26 Heading Calculation

The transformation between unit vectors observed in the survey toolrsquos coordinate system (X

Y Z) and the Earthrsquos coordinate system (N E V) is performed by the vector Equation (21)

U119873119864119881 = 120546119885 119868 119867119878 U119883119884119885 (21)

where 119880119873 119880119864 and 119880119881 are unit vectors in the N E and V directions 119880119883 119880119884 and 119880119885 are unit

vectors in the X Y and Z directions respectively and 120546119885 119868 119867119878 represent the rotation

matrices according to Russell and Russell (1978)

cos 119860119885 minussin 119860119885 0 119860119885 = sin 119860119885 cos 119860119885 0 (22)

0 0 1 17

cos 119868 0 sin 119868 119868 = 0 1 0 (23)

minussin 119868 0 cos 119868

cos 119867119878 minus sin 119867119878 0 119867119878 = sin 119867119878 cos 119867119878 0 (24)

The vector operation for a transformation in the reverse direction can be written as

= 119880 119883119884119885 119867119878119879 119868 119879 120546119885T UNEV (25)

The first step is to calculate the borehole inclination angle and highside angle Operating the

vector Equation (25) on the Earthrsquos gravity vector results in Equation (26)

119866119909 cos 119867119878 sin 119867119878 0 cos 119868 0 minus sin 119868 cos 119860119885 sin 119860119885 0 0 119866119910 = minus sin 119867119878 cos 119867119878 0 0 1 0 minussin 119860119885 cos 119860119885 0 0 (26 ) 119866119911 0 0 1 sin 119868 0 cos 119868 0 0 1 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 119880119881 = 119866119909 119880119883 + 119866119910 119880119884 + 119866119911 119880119885 (27)

It is assumed that the probe is not undergoing any acceleration except for the Earthrsquos gravity

field In the absence of external forces in static state the accelerometer experiences only the

Earth gravity with components 119866119909 119866119910 119866119911 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 119866119909 119866119910 and 119866119911 are substantially identical to the respective actual Earthrsquos

gravity field (because accelerometers are not affected by magnetic interference) Equations (28)

through (210) provide gravity field components in the (X Y Z) frame

119866119909 = minusg cos 119867119878 sin 119868 (28)

119866119910 = g sin 119868 sin 119867119878 (29)

119866119911 = g cos 119868 (210)

Thus the highside angle HS can be determined from

119866119910tan 119867119878 = (211) minus119866119909

The inclination angle can be determined from

2Gx2 + Gysin 119868 (212)

cos 119868 =

Gzcos 119868 = (213) 2Gx2 + Gy2 + Gz

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

119861 = 119861119873 119880119873 + 119861119864 119880119864 + 119861119881 119880119881 = 119861119909 119880119883 + 119861119910 119880119884 + 119861119911 119880119885 (214)

where 119861119873 119861119864 and 119861119881 are the geomagnetic field components in (N E V) frame Operating the

vector Equation (21) on the magnetic field vector results in Equation (215)

119861119873 B cos(DIP)119861119864 = 0 119861119881 B sin(DIP)

cos 119860119885 minus sin 119860119885 0 cos 119868 0 sin 119868 cos 119867119878 minussin 119867119878 0 119861119909= sin 119860119885 cos 119860119885 0 0 1 0 sin 119867119878 cos 119867119878 0 119861119910 (215)

0 0 1 minussin 119868 0 cos 119868 0 0 1 119861119911

2 12where strength of geomagnetic field B is obtained as 1198611198732 + 119861119881 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 (215) yields magnetic field components in the (N E V) frame as follows

119861119873 = cos 119860119885 cos 119868 119861119909 cos 119867119878 minus 119861119910 sin 119867119878 + 119861119911 sin 119868

minus sin 119860119885 119861119909 sin 119867119878 + 119861119910 cos 119867119878 (216)

119861119864 = sin 119860119885 cos 119868 119861119909 cos 119867119878 minus 119861119910 sin 119867119878 + 119861119911 sin 119868

+ cos 119860119885 119861119909 sin 119867119878 + 119861119910 cos 119867119878 (217)

119861119881 = minussin 119868 119861119909 cos 119867119878 minus 119861119910 sin 119867119878 + 119861119911 cos 119868 (218)

The calculated azimuth at the instrument location is the azimuth with respect to the Earthrsquos

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 119861119864 is zero and then the

azimuth is derived from Equation (217) by

sin 119860119885 minus 119861119909 sin 119867119878 + 119861119910 cos 119867119878 (219)

cos 119860119885 =

cos 119868 119861119909 cos 119867119878 minus 119861119910 sin 119867119878 + 119861119911 sin 119868

The azimuth angle is derived as a function of the inclination angle the highside angle and the

magnetic field components 119861119909 119861119910 and 119861119911 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

convert from three orthogonal accelerations and three orthogonal magnetic field measurements

to the wellbore azimuth

119861119909 119866119910 minus 119861119910 119866119909 1198661199092 + 1198661199102 + 1198661199112sin 119860119885 (220)

cos 119860119885 = 119866119911 119861119909 119866119909 +119861119910 119866119910 + 119861119911 1198661199092 + 1198661199102

If the X-Y plane of the body coordinate system is level (ie 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)

By119860119885 = minustanminus1 (221) Bx

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-

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

27 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

(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 eg 10 m) while the BHA is completely stationary (Thorogood

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)

28 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

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

Earthrsquos 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 (222) shows that the azimuth error equals the

ratio of the east-west component of the drillstring error vector and the Earthrsquos horizontal field as

shown in Figure 2-3 and Figure 2-4

∆B119911 sin 119868 sin 119860119885∆119860119885 = (222)

B cos(DIP)

∆119860119885 = Borehole Azimuth error ∆Bz= drillstring magnetic error field in axial direction

119868 = Borehole inclination AZ= Borehole azimuth

DIP= dip angle of geomagnetic vector |B| = Strength of geomagnetic field

∆Bz sin 119868 = Horizontal component of the drillstring error vector

∆Bz sin 119868 sin 119860119885 = EastWest component of the drillstring error vector

BN = 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

Figure 2-4 Eastwest component of error vector (modified from Grindrod and Wolff 1983)

29 Previous Studies

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

291 Magnetic Forward Modeling of Drillstring

The magnitude of error field ∆119861119911 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 assemblyrsquos 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

292 Standard Method

Russel amp Roesler (1985) and Grindord amp 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

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

293 Short Collar Method or Conventional Magnetic Survey (Single Survey)

This method is called ldquoshort collarrdquo 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 ldquoshort collarrdquo 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 119861119909 and 119861119910 uncorrupted ie 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 119861119909 and 119861119910 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

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 ∆119861119911 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 Bz then Bz is

calculated from equation (223) and substituted in to equation (219) to yield the absolute

azimuth angle (Russell 1987)

Bz = B2 minus 1198611199092 minus 119861119910

If the vertical component of the true geomagnetic field BV is known then Bz can be

calculated from equation (224)

119861119911 = (119861119881 119892 minus 119861119909 119866119909minus119861119910 119866119910)119866119911 (224)

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

mentioned here that there are other types of SSA computational procedures published by other

researchers which seek to determine both axial and transverse

294 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 2-5

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 toolrsquos 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

it still results in greater accuracy of azimuth even in the critical attitudes near horizontal east-

west (Brooks 1997)

295 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

210 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

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 soft-

iron magnetic calibration is examined and investigated for the directional drilling application

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 well-

known azimuth expressions such as (219) and (220) 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

31 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 ∆119861119909 ∆119861119910 and ∆119861119911 are

affecting the measured components 119861119909119898(120003) 119861119910119898(120003) and 119861119911119898(120003) at 120003P

th survey station in the (X

Y Z) frame respectively the corrected values are calculated by

119861119909119862119900119903119903(120003) = 119861119909119898(120003) minus ∆119861119909 (31)

119861119910119862119900119903119903(120003) = 119861119910119898(120003) minus ∆119861119910 (32)

119861119911119862119900119903119903(120003) = 119861119911119898(120003) minus ∆119861119911 (33)

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

119861119881(119903119890119891) 119861ℎ(119903119890119891) respectively Moreover the vertical component of the local magnetic field at the

120003P

th survey station denoted as 119861119881(120003) is computed by the vector dot product

119861 g119861119881 = (34)

By substituting Equations (27) (214) for the 120003P

th survey station the computed value of local

field is obtained from

119861119909119862119900119903119903(120003) 119866119909(120003) + 119861119910119862119900119903119903(119899) 119866119910(120003) + 119861119911119862119900119903119903(120003) 119866119911(120003)119861119881(120003) = (35) 2 05

119866119909(120003)2 + 119866119910(120003)

2 + 119866119911(120003)

2 05119861ℎ(120003) = 119861119909119862119900119903119903(120003)

2 + 119861119910119862119900119903119903(120003)2 + 119861119911119862119900119903119903(120003)

2 minus 119861119881(120003) (36)

Values of the computed magnetic field 119861119881(120003) and 119861ℎ(120003) for a survey group at a range of 120003 =

1 hellip N will typically exhibit some scatter with respect to the reference value of 119861119881(119903119890119891) and

119861ℎ(119903119890119891) therefore reflecting the varying direction of drillstring magnetization error (Brooks

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)

119873 2 21119985 =(119873minus1)

119861ℎ(120003) minus 119861ℎ(119903119890119891) + 119861119881(120003)minus119861119881(119903119890119891) (37) 120003=1

The unknown biases are solved for by minimizing this scatter through minimizing the

variance 119985 expressed in equation (37) This can be accomplished by differentiating equation

(37) with respect to the small unknown biases and setting the results to zero

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 Newtonrsquos 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 US pat Nos 5623407 to Brooks and are also explained as

following

MSA Computation

From equation (37) where the small perturbations of ∆119861119909 ∆119861119910 and ∆119861119911 can be denoted as

120576119909 120576119910 and 120576119911 differentiations give

120597120597119985 119865 120576119909 120576119910 120576119911 = =

120597120597120576119909

119873

120597120597119861ℎ(120003) 120597120597119861ℎ(119903119890119891)= 119861ℎ(120003) minus 119861ℎ(119903119890119891) minus 120597120597120576119909 120597120597120576119909

120003=1

120597120597119861119881(120003) 120597120597119861119881(119903119890119891)+ 119861119881(120003) minus 119861119881(119903119890119891) minus = 0 (38) 120597120597120576119909 120597120597120576119909

120597120597119985 119866 120576119909 120576119910 120576119911 = =

120597120597120576119910

119873

120597120597119861ℎ(120003) 120597120597119861ℎ(119903119890119891)= 119861ℎ(120003) minus 119861ℎ(119903119890119891) minus 120597120597120576119910 120597120597120576119910

120003=1

120597120597119861119881(120003) 120597120597119861119881(119903119890119891)minus = 0 (39) + 119861119881(120003) minus 119861119881(119903119890119891) 120597120597120576119910 120597120597120576119910

120597120597119985 119867 120576119909 120576119910 120576119911 = =

120597120597120576119911

119873

120597120597119861ℎ(120003) 120597120597119861ℎ(119903119890119891)= 119861ℎ(120003) minus 119861ℎ(119903119890119891) minus 120597120597120576119911 120597120597120576119911

120003=1

120597120597119861119881(120003) 120597120597119861119881(119903119890119891)minus = 0 (310) + 119861119881(120003) minus 119861119881(119903119890119891) 120597120597120576119911 120597120597120576119911

The differentiations 119865 119866 and 119867 are nonlinear with respect to 120576119909 120576119910 and 120576119911 An approximate

solution can therefore be found by linearizing equations (38) through (310) by an iterative

technique such as Newtonrsquos method The linearized form of 119865 119866 and 119867 denoted as 119891 119892 and ℎ

119891 = 1198861 (120576119909 minus 120576119909 prime) + 1198871 120576119910 minus 120576119910

prime + 1198881 (120576119911 minus 120576119911 prime) + 119865 120576119909 prime 120576119910

prime 120576119911 prime = 0 (311)

prime 120576119911 prime = 0 (312)

ℎ = 1198863 (120576119909 minus 120576119909 prime) + 1198873 120576119910 minus 120576119910

prime 120576119911 prime = 0 (313)

120597120597119865 120576119909 prime 120576119910

prime 120576119911 prime 120597120597119865 120576119909 prime 120576119910

prime 120576119911 prime1198861 = 1198871 = 1198881 = (314) 120597120597120576119909 120597120597120576119910 120597120597120576119911

120597120597119866 120576119909 prime 120576119910

prime 120576119911 prime 1198862 = 1198872 = 1198882 = (315)

120597120597120576119909 120597120597120576119910 120597120597120576119911

120597120597119867 120576119909 prime 120576119910

prime 120576119911 prime 1198863 = 1198873 = 1198883 = (316)

120597120597120576119909 120597120597120576119910 120597120597120576119911

The primed error terms 120576119909 prime 120576119910

prime and 120576119911 prime represent the previous estimates of these values The

linearized equations (311) through (313) can be solved for the unknowns 120576119909 120576119910 and 120576119911 by

iterative technique of Newtonrsquos method in which successive approximations to 120576119909 120576119910 and 120576119911 are

found by (Brooks et al 1998)

120576119909 minus 120576119909 prime

120576119910 minus 120576119910 prime

120576119911 minus 120576119911 prime

minus1

⎡1205971205972119985 120576119909

prime 120576119910 prime 120576119911 prime 1205971205972119985 120576119909

prime 120576119910 prime 120576119911 prime⎤

⎢ 1205971205971205761199092 120597120597120576119909120597120597120576119910 120597120597120576119909120597120597120576119911 ⎥ ⎛ 120597120597120576119909 ⎞⎢1205971205972119985 120576119909

prime 120576119910 prime 120576119911 prime ⎥ ⎜120597120597119985 120576119909

prime 120576119910 prime 120576119911 prime ⎟

= minus ⎢ ⎥ ⎜ ⎟ (317) ⎢ 120597120597120576119910120597120597120576119909 1205971205971205761199102 120597120597120576119910120597120597120576119911 ⎥ ⎜ 120597120597120576119910 ⎟

⎜ ⎟⎢1205971205972119985 120576119909 prime 120576119910

prime 120576119911 prime ⎥ 120597120597119985 120576119909 prime 120576119910

prime 120576119911 prime⎢ ⎥ ⎣ 120597120597120576119911120597120597120576119909 120597120597120576119911120597120597120576119910 1205971205971205761199112 ⎦ ⎝ 120597120597120576119911 ⎠

120576119909 120576119910120576119911

prime120576119909prime = 120576119910 prime120576119911

minus1 prime prime prime prime prime prime prime ⎡1205971205972119985 120576119909 120576119910 120576119911prime 1205971205972119985 120576119909 120576119910 120576119911prime 1205971205972119985 120576119909 120576119910 120576119911prime 120597120597119985 120576119909prime 120576119910 120576119911prime⎤ ⎢ 1205971205971205761199092 120597120597120576119909120597120597120576119910 120597120597120576119909120597120597120576119911 ⎥ ⎛ 120597120597120576119909 ⎞⎢ prime prime prime prime prime prime ⎥ prime prime1205971205972119985 120576119909 120576119910 120576119911prime 1205971205972119985 120576119909 120576119910 120576119911prime 1205971205972119985 120576119909 120576119910 120576119911prime ⎜120597120597119985 120576119909 120576119910 120576119911prime ⎟ minus ⎢ ⎥ ⎜ ⎟ (318) ⎢ 120597120597120576119910120597120597120576119909 1205971205971205761199102 120597120597120576119910120597120597120576119911 ⎥ ⎜ 120597120597120576119910 ⎟

⎜ ⎟⎢ prime prime prime prime prime prime ⎥ prime prime1205971205972119985 120576119909 120576119910 120576119911prime 1205971205972119985 120576119909 120576119910 120576119911prime 1205971205972119985 120576119909 120576119910 120576119911prime 120597120597119985 120576119909 120576119910 120576119911prime⎢ ⎥ ⎣ 120597120597120576119911120597120597120576119909 120597120597120576119911120597120597120576119910 1205971205971205761199112 ⎦ ⎝ 120597120597120576119911 ⎠

prime prime 120576119909 120576119909prime 1198861 1198871 1198881 minus1 119865 120576119909 120576119910 120576119911prime

prime prime prime120576119910 = 120576119910 minus 1198862 1198872 1198882 119866 120576119909 120576119910 120576119911prime (319) prime120576119911 120576119911 1198863 1198873 1198883 119867 120576119909prime 120576119910prime 120576119911prime

The updated bias estimates of 120576119909 120576119910 and 120576119911 obtained from equation (319) is replaced with

previous estimates of 120576119909 120576119910 and 120576119911 in equations (31) through (33) to refine the values of

119861119909119862119900119903119903(120003) 119861119910119862119900119903119903(120003) and 119861119911119862119900119903119903(120003) 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 120576119909 120576119910 and 120576119911 falling below a predefined minimum limit or a specified number of iterations

having been performed (Brooks et al 1998)

The derivatives 1198861 through 1198883 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 1198861 is derived by

119873

1205971205972119861ℎ(120003)1198861 = 119873 minus 119861ℎ(119903119890119891) (320) 1205971205971205761199092

120003=1

In the case where the central values are the mean values of the computed local magnetic field

which are not constant the coefficient 1198861 is derived more complicated as

119873119873 ⎡ 1205971205972119861ℎ(120003) ⎤ ⎢ 2 1 2 1 ⎛ 1205971205971205761199092 ⎞⎥120597120597119861ℎ(120003) 120597120597119861119881(120003) 120003=1 1198861 = 119873 minus ⎢ ⎜119861ℎ(120003) ⎟⎥ (321) ⎢ 120597120597120576119909 119873

+ 120597120597120576119909 119873

+ ⎜ 119873 ⎟⎥

⎢ ⎥120003=1 ⎣ ⎝ ⎠⎦

2⎡ 2 ⎤

1205971205972119861ℎ(120003) 1 119866119909(120003) 119861119909119862119900119903119903(120003) + 119861119881(120003)

minus119866|g119909|

(120003)

⎢ ⎥⎛ ⎞= ⎢1 minus | minus ⎥ (322) 1205971205971205761199092 119861ℎ(120003) |g minus119861ℎ(120003)⎢ ⎥

⎣ ⎝ ⎠ ⎦

120597120597119861ℎ(120003) 119861119909119862119900119903119903(120003) + 119861119881(120003)

minus119866|g119909|

(120003)

= (323) 120597120597120576119909 minus119861ℎ(120003)

120597120597119861119881(120003) minus119866119909(120003)= (324) 120597120597120576119909 |g|

Similar calculations can derive the remaining coefficients 1198871 through 1198883 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 (219) and (220) to

derive the corrected borehole azimuth (Brooks 1997)

32 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 httpwwwfreescalecom 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

321 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

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 V119909 V119910 and V119911 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)

322 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 times 3 matrix 119882119878119900119891119905 The components of 119882119878119900119891119905

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 119882119878119900119891119905

represents the effective coefficient relating the induced field generated in the x-direction in

response to an inducing field in the y-direction Thus 119882119878119900119891119905 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 times 3 gain matrix 119882119866119886119894119899 which is also referred to as scale factor error Another 3 times 3 matrix

119882119873119900119899119874119903119905ℎ119900119892 referred to as misalignment error is used to model

(i) The rotation of magnetometer relative to the instrument-fixed coordinate system (X Y

(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)

119882119873119900119899119874119903119905ℎ119900119892 can be modelled as the following symmetric matrix (Gebre-Egziabher et al 2001)

1 minus120576119911 120576119910

119882119873119900119899119874119903119905ℎ119900119892 = 120576119911 1 minus120576119909 (325) minus120576119910 120576119909 1

The three independent parameters 120576119909 120576119910 and 120576119911defining the matrix 119882119873119900119899119874119903119905ℎ119900119892represent

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 soft-

iron matrix 119882 defined as

119882 = 119882119873119900119899119874119903119905ℎ119900119892 119882119866119886119894119899 119882119878119900119891119905 (326)

The process of calibrating a triad of magnetometers involves estimating the hard-iron vector

V and the soft-iron matrix W defined above

323 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

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

324 Calibration Model

Equation (215) 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 119882 into the inverse form of

equation (215) yields the magnetometer measurement affected by both hard-iron and soft-iron

distortions as illustrated in equation (327) where 119861119875 denotes the distorted magnetometer

measured at a survey point

119861119875119909119861119875 = 119861119875119910 =

119861119875119911

119882 cos 119867119878 minus sin 119867119878

sin 119867119878 cos 119867119878

cos 119868 0

0 1 minus sin 119868

0 cos 119860119885 minussin 119860119885

sin 119860119885 cos 119860119885

cos(DIP)0 +

0 0 1 sin 119868 0 cos 119868 0 0 1 sin(DIP)

119881119909119881119910

119881119911 (327)

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)

119882minus1 119861119901 minus V119879119882minus1 119861119901 minus V = 119861119901 minus V

119879119882minus1119879119882minus1 119861119901 minus V = B2 (328)

substituting from equation (327) and denoting

cos 119867119878 sin 119867119878 0 cos 119868 0 minus sin 119868 cos 119860119885 sin 119860119885 0 minus sin 119867119878

0 cos 119867119878

0 sin 119868

0 cos 119868

minussin 119860119885 0

cos 119860119885 0

= Γ (329)

results in

119882minus1 119861119901 minus V = Γ B cos(DIP)

0 sin(DIP)

Therefore it is proved that

119882minus1 119861119901 minus V119879119882minus1 119861119901 minus V = Γ B

cos(DIP)0

sin(DIP)

119879

Γ B cos(DIP)

0 sin(DIP)

= B2 (331)

In general the locus of the vector 119861119901 lying on the surface of an ellipsoid with center

coordinate of the vector V is expressed as

119861119901 minus V119879

A 119861119901 minus V = 119888119900119899119904119905 (332)

Where matrix A must be symmetric Equation (331) and (332) are similar since it can be

easily proved that the matrix 119882minus1119879119882minus1 is symmetric as 119860119879 = [119882minus1119879119882minus1]119879 =

119882minus1119879119882minus1 = A Thus in a strong hard and soft-iron environment the locus of raw

magnetometer measurements forms the surface of an ellipsoid defined by

119861119901 minus V119879119882minus1119879119882minus1 119861119901 minus V = B2 (333)

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 119882minus1119879119882minus1 which is transposed square of

the inverse soft-iron matrix 119882minus1

In the absence of soft-iron and misalignment errors the diagonal 3 times 3 scale factor

matrix 119882119866119886119894119899 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 axesrsquo magnitudes determined by the scale factor errors 119904119891119909 119904119891119910 and 119904119891119911 along

the instrument-fixed coordinate frame This can be expressed mathematically as follows

(1 + 119904119891119909) 0 0 119882119866119886119894119899 = 0 (1 + 119904119891119910) 0 (334)

0 0 (1 + 119904119891119911)

1⎡ 0 0 ⎤1 + 119904119891119909 ⎢ ⎥

minus1 ⎢ 1 ⎥119882119866119886119894119899 = ⎢ 0 0 ⎥ (335) 1 + 119904119891119910 ⎢ ⎥1⎢ ⎥0 0⎣ 1 + 119904119891119911⎦

119861119901 minus V119879

119882119866119886119894119899minus1119879 119882119866119886119894119899minus1 119861119901 minus V = 1198612 (336)

2⎡ 0 0 ⎤ ⎢ 1 +

1 119904119891119909

2⎢ 1 ⎥119861119901 minus V

119879

⎢ 0 0 ⎥ 119861119901 minus V = 1198612 (337) ⎢ 1 + 119904119891119910

⎢ ⎥0 0 ⎣ 1 +

1 119904119891119911

Mathematically the locus of measurements is described by the following equation

2 2 2 119861119875119909 minus 119881119909 119861119875119910 minus 119881119910 + 119861119875119911 minus 119881119911+ = 1198612 (338)

1 + 119904119891119909

1 + 119904119891119910

1 + 119904119891119911

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)

325 Symmetric Constrait

The linear ellipsoid model of 119861119901 minus V119879119882minus1119879119882minus1 119861119901 minus V = B2 is solved for the

transposed square of the inverse soft-iron matrix 119882minus1119879119882minus1and the hard-iron vector V by

optimum fitting of the ellipsoid to the measured data points The ellipsoid fit provides the matrix

119882minus1119879119882minus1 whereas the calculation of the calibrated magnetic field measurement 119861119875119862119886119897

according to equation (339) requires the inverse soft-iron matrix 119882minus1

119861119875119909 minus 119881119909

119861119875119862119886119897 = 119882minus1 119861119875119910 minus 119881119910 (339) 119861119875119911 minus 119881119911

Although it is trivial to compute the ellipsoid fit matrix 119882minus1119879119882minus1 from the inverse soft-

iron matrix 119882minus1 there is no unique solution to the inverse problem of computing 119882minus1 from the

matrix 119882minus1119879119882minus1 The simplest solution is to impose a symmetric constraint onto the inverse

soft-iron matrix 119882minus1 As proved in equation (332) the matrix 119882minus1119879119882minus1 is symmetric with

only six independent coefficients while the soft-iron matrix 119882 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 119882minus1 also be

symmetric with six degrees of freedom The reading 119861119901 is calibrated by an estimated hard-iron

offset 119881119862119886119897 and an estimated soft-iron matrix 119882119862119886119897 then the resulting corrected magnetic field

measurement 119861119875119862119886119897 is given by

cos(DIP)119861119875119862119886119897 = 119882C119886119897minus1 119861119901 minus 119881119862119886119897 = 119882119862119886119897minus1 119882 Γ B 0 + 119881 minus 119881119862119886119897 (340)

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 (340) it is necessary that 119882119862119886119897minus1119882 = I and 119881119862119886119897 = 119881 thus resulting in

) = B2 (341) (119861119875119862119886119897)119879(119861119875119862119886119897

Mathematically an arbitrary rotation can be incorporated in the estimated inverse soft-iron

minus1 matrix 119882119862119886119897 as 119882119862119886119897minus1119882 = 119877(120589) where 119877(120589) is the arbitrary rotation matrix by angle 120589

Considering that the transpose of any rotation matrix is identical to rotation by the inverse angle

leads to 119877(120589)119879 = 119877(minus120589) Substituting 119881119862119886119897 = 119881 in equation (340) yields

cos(DIP)119861119875119862119886119897 = 119882119862119886119897minus1 119861119901 minus 119881119862119886119897 = 119882119862119886119897minus1 119882 Γ B 0 (342)

sin(DIP)

cos(DIP) 119879 cos(DIP)) = 119882119862119886119897minus1 119882 Γ B 0 119882119862119886119897minus1 119882 Γ B 0 (343) (119861119875119862119886119897)

119879(119861119875119862119886119897 sin(DIP) sin(DIP)

then substituting 119882119862119886119897minus1 119882 = 119877(120589) yields

cos(DIP) 119879 cos(DIP)) = 119877(120589) Γ B 0 119877(120589) Γ B 0 = B2 (344) (119861119875119862119886119897)

119879(119861119875119862119886119897 sin(DIP) sin(DIP)

and substituting 119877(120589)119879 119877(120589) = 119877(minus120589) 119877(120589) = 119868 results in

cos(DIP) 119879 cos(DIP)) = B 0 Γ119879 119877(120589)119879 119877(120589) Γ B 0 = B2 (345) (119861119875119862119886119897)

119879(119861119875119862119886119897 sin(DIP) sin(DIP)

It was proved that by incorporating the spurious rotation matrix 119877(120589) 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 119882minus1 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 119882minus1 is the relationship between

the eigenvectors and eigenvalues of matrix 119882minus1119879119882minus1 and matrix 119882minus1 It can be proved that

if 119882minus1 is symmetric as 119882minus1119879 = 119882minus1 then the eigenvectors of matrix 119882minus1119879119882minus1 are

identical to the eigenvectors of matrix 119882minus1 and the eigenvalues of matrix 119882minus1119879119882minus1 are the

square of the eigenvalues of matrix 119882minus1 Eigenvectors 119883119894 and eigenvalues 119890119894 of 119882minus1 are

defined by 119882minus1 119883119894 = 119890119894 119883119894 Therefore

119882minus1119879119882minus1 119883119894 = 119882minus1119879119890119894 119883119894 = 119890119894119882minus1119883119894 = 1198901198942119883119894 (346)

Since the eigenvectors of the ellipsoid fit matrix 119882minus1119879119882minus1 represent the principal axes of

magnetometer measurement ellipsoidal locus then constraining the inverse soft iron matrix 119882minus1

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 119882minus1 can be computed from the square root of the

ellipsoid fit matrix 119882minus1119879119882minus1 as following

119882minus1119879119882minus1 = 119882minus1119882minus1 119882minus1 = [ 119882minus1119879119882minus1]12 (347)

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

magnetometersrsquo axes aligned with the body axes results in an ellipsoidal locus having major and

minor axes aligned with the body axes

326 Least-Squares Estimation

The calibration algorithm develops an iterated least-squares estimator to fit the ellipsoid

119886 119887 119888 parameters to the measured magnetic field data Substituting 119882minus1119879119882minus1 = 119887 119890 119891 in to

119888 119891 119868

Equation (333) results in

119886 119887 119888 119861119875119909 minus 119881119909

B2 = (119861119875119909 minus 119881119909 119861119875119910 minus 119881119910 119861119875119911 minus 119881119911) 119887 119890 119891 119861119875119910 minus 119881119910 (348) 119888 119891 119868 119861119875119911 minus 119881119911

|119861| = 119886(119861119875119909 minus 119881119909)2 + 119890 119861119875119910 minus 119881119910

2 + 119868(119861119875119911 minus 119881119911)2 + 2119887(119861119875119909 minus 119881119909) 119861119875119910 minus 119881119910 +

(349) +2119888(119861119875119909 minus 119881119909)(119861119875119911 minus 119881119911) + 2119891 119861119875119910 minus 119881119910 (119861119875119911 minus 119881119911)

The equations of the estimator can be obtained by linearizing Equation (349) 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 (349) the perturbation of |119861| is written as 120575119861 given by

120597120597119861 120575a +

120597120597119861 120575c +

120597120597119861

120597120597b 120575b +

120597120597119861120575119861 = 120575119881119909 +

120597120597119861 120575119881119910 +

120597120597119861 120575119881119911 +

120597120597119861 120575e

120597120597119881119909 120597120597119881119910 120597120597119881119911 120597120597a 120597120597c 120597120597e

+ 120597120597119861

120597120597f 120575f +

120597120597119861

120597120597I 120575I (350)

120597120597119861 minus2119886 (119861119875119909 minus 119881119909) minus 2b 119861119875119910 minus 119881119910 minus 2c (119861119875119911 minus 119881119911) = (351)

120597120597119881119909 2119861

120597120597119861 minus2119890 119861119875119910 minus 119881119910 minus 2b (119861119875119909 minus 119881119909) minus 2f (119861119875119911 minus 119881119911) = (352)

120597120597119881119910 2119861

120597120597119861 minus2119868 (119861119875119911 minus 119881119911) minus 2c (119861119875119909 minus 119881119909) minus 2f (119861119875119910 minus 119881119910) = (353)

120597120597119881119911 2119861

120597120597119861 (119861119875119909 minus 119881119909)2

= (354) 120597120597a 2119861

120597120597119861 2(119861119875119909 minus 119881119909) 119861119875119910 minus 119881119910 (355)

120597120597b =

2119861

120597120597119861 2(119861119875119909 minus 119881119909)(119861119875119911 minus 119881119911) = (356)

120597120597c 2119861

120597120597119861 (119861119875119910 minus 119881119910)2

= (357) 120597120597e 2119861

120597120597119861 2 119861119875119910 minus 119881119910 (119861119875119911 minus 119881119911) (358)

120597120597f =

2119861

120597120597119861 (119861119875119911 minus 119881119911)2

(359) 120597120597I

=2119861

The given or known inputs to the calibration algorithm are the measured magnetometer

outputs 119861119875119909 119861119875119910 and 119861119875119911 and the magnitude of geomagnetic field vector in the geographic

area where the calibration is being performed Note that the 119861119875119909 119861119875119910 and 119861119875119911 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 (350) can be expressed as

⎡ 1205751198611 ⎤1205751198612⎢ ⎥⋮ =⎢ ⎥ ⎢120575119861119873minus1⎥ ⎣ 120575119861119873 ⎦

120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861⎡ ⎤ ⎢ 120597120597119881119909 120597120597119881119910 120597120597119881119911 1 120597120597a 1 120597120597b

120597120597c 1 120597120597e 120597120597f 1 120597120597I ⎥1 1 1 1 1

⎢ ⎥ 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861⎢ ⎥ ⎢ 120597120597119881119909 2 120597120597119881119910 2

120597120597119881119911 2 120597120597a 2 120597120597b 2 120597120597c 2 120597120597e 2 120597120597f 2 120597120597I 2 ⎥ ⎢ ⎥⋮ ⋱ ⋮⎢ ⎥ 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861⎢ ⎥ ⎢ 120597120597119881119909 119873minus1 120597120597119881119910 119873minus1

120597120597119881119911 119873minus1 120597120597a 119873minus1 120597120597b 119873minus1 120597120597c 119873minus1 120597120597e 119873minus1 120597120597f 119873minus1 120597120597I 119873minus1 ⎥ ⎢ ⎥ 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861 120597120597119861⎢ ⎥ ⎣ 120597120597119881119909 119873 120597120597119881119910 119873

120597120597119881119911 119873 120597120597a 119873 120597120597b 119873 120597120597c 119873 120597120597e 119873 120597120597f 119873 120597120597I 119873 ⎦

⎡120575119881119909⎤120575119881119910⎢ ⎥120575119881119911⎢ ⎥ ⎢ 120575a ⎥

times ⎢ 120575b ⎥ (360) ⎢ 120575c ⎥ ⎢ 120575e ⎥ ⎢ 120575f ⎥ ⎣ 120575I ⎦

Equation (360) is in the form 120575119861 = the vector of unknowns is given by where 120575119883120577 120575119883

120575 = [120575119881119909 120575119881119910 120575119881119911 120575a 120575b 120575c 120575e 120575f 120575I]119879 (361)

The vector 120575119861 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 119881119909 119881119910 119881119911 a b c 119890 119891 and I is obtained by

using the following iterative algorithm

Firstly select a non-zero initial guess for 119881119909 119881119910 119881119911 a b c 119890 119891 and I Secondly form

Equation (360) by the initial values Thirdly obtain a least square estimate for 120575 as follows

120575 = (120577119879120577)minus1120577119879 120575119861 (362)

Then update the unknown parameters by adding the 120575 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 119881119909 119881119910 119881119911 a b c 119890 119891 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 (339) to transform the measured raw data

lying on the ellipsoid into the corrected field measurements lying on the sphere centered at the

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

327 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 (349) 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 (349) 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

3271 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 (333) simplifies to sphere locus

119861119901 minus V119879119861119901 minus V = B2 (363)

This simplification results in determining just three calibration parameters modeling the hard-

iron 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

under different probe orientations while minimizing the fit error in a least-squares sense The

least-squares method minimizes the 2ndashnorm 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

3272 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 (333) is then written as

119886 119887 119888 119861119909119862119900119903_ℎ

B2 = (119861119909119862119900119903_ℎ 119861119910119862119900119903_ℎ 119861119911119862119900119903_ℎ) 119887 119890 119891 119861119910119862119900119903_ℎ (364) 119888 119891 119868 119861119911119862119900119903_ℎ

Where 119861119909119862119900119903_ℎ 119861119910119862119900119903_ℎ and 119861119911119862119900119903_ℎ are acquired by subtraction of the hard-iron vector V

(obtained from step 1) from 119861119909 119861119910 and 119861119911 measurements respectively

119886 119861119909119862119900119903_ℎ2 + 119890 119861119910119862119900119903_ℎ

2 + 119868 119861119911119862119900119903_ℎ2 + 2119887 119861119909119862119900119903_ℎ 119861119910119862119900119903_ℎ + 2119888 119861119909119862119900119903_ℎ 119861119911119862119900119903_ℎ

+ 2119891 119861119910119862119900119903_ℎ 119861119911119862119900119903_ℎ minus B2 = 119903 (365)

The least-squares method minimizes the 2ndashnorm of the residual error vector 119903 by fitting 6

components of the ellipsoid fit matrix 119882minus1119879119882minus1to the series of 119861119862119900119903_ℎ taken at N positions

expressed as follows

⎡ 119861119909119862119900119903ℎ21

119861119910119862119900119903ℎ21

119861119911119862119900119903ℎ21

119861119909119862119900119903ℎ times 119861119910119862119900119903ℎ 1

119861119909119862119900119903ℎ times 119861119911119862119900119903ℎ 1

⎢ 119861119909119862119900119903ℎ22

119861119910119862119900119903ℎ22

119861119911119862119900119903ℎ22

119861119909119862119900119903ℎ times 119861119910119862119900119903ℎ 2

119861119909119862119900119903ℎ times 119861119911119862119900119903ℎ 2⎢ ⋮ ⋱ ⋮⎢

⎢ 119861119909119862119900119903ℎ 119861119910119862119900119903ℎ

119861119911119862119900119903ℎ 119861119909119862119900119903ℎ

times 119861119910119862119900119903ℎ 119861119909119862119900119903ℎ

times 119861119911119862119900119903ℎ⎢ 2119873minus1

2119873minus1

2119873minus1 119873minus1 119873minus1

⎣ 119861119909119862119900119903ℎ2119873

119861119910119862119900119903ℎ2119873

119861119911119862119900119903ℎ2119873

119861119909119862119900119903ℎ times 119861119910119862119900119903ℎ 119873

119861119909119862119900119903ℎ times 119861119911119862119900119903ℎ 119873

where in Equation (366)

⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 119861119909119862119900119903ℎ

119861119909119862119900119903ℎ 22⋮

119861119909119862119900119903ℎ 2119873minus1

119861119909119862119900119903ℎ 2119873

119861119910119862119900119903ℎ 21

119861119910119862119900119903ℎ 22

119861119910119862119900119903ℎ 2119873minus1

119861119910119862119900119903ℎ 2119873

119861119911119862119900119903ℎ 21

119861119911119862119900119903ℎ 22

119861119911119862119900119903ℎ 2119873minus1

119861119911119862119900119903ℎ 2119873

119861119909119862119900119903ℎ times 119861119910119862119900119903ℎ 1

119861119909119862119900119903ℎ times 119861119910119862119900119903ℎ 2

⋱ 119861119909119862119900119903ℎ

times 119861119910119862119900119903ℎ 119873minus1

119861119909119862119900119903ℎ times 119861119910119862119900119903ℎ 119873

119861119909119862119900119903ℎ times 119861119911119862119900119903ℎ 1

119861119909119862119900119903ℎ times 119861119911119862119900119903ℎ 2

⋮ 119861119909119862119900119903ℎ

times 119861119911119862119900119903ℎ 119873minus1

119861119909119862119900119903ℎ times 119861119911119862119900119903ℎ 119873

⎤ is denoted as matrix A and ⎢

⎥ is denoted as vector l The least-squares solution for the ⎢ ⎥ ⎢B2⎥ ⎣B2⎦

vector of unknowns is given by

119886⎡ 119890 ⎤ ⎢ 119868 ⎥ ⎢2119887⎥ = ( A119879A)minus1A119879119897 (367) ⎢ ⎥ ⎢2119888⎥ ⎣2119891⎦

Afterwards the inverse soft-iron matrix 119882minus1 can be computed from the square root of the

ellipsoid fit matrix 119882minus1119879119882minus1 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

33 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

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

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

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)

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 ldquodog-legrdquo (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

∆119879119881119863 = ∆119872119863

(cos 1198681 + cos 1198682)(RF) (368) 2

∆119873119900119903119905ℎ = ∆119872119863

(sin 1198681 times cos 1198601198851 + sin 1198682 times cos 1198601198852)(RF) (369) 2

∆119864119886119904119905 = ∆119872119863

[(sin 1198681 times sin 1198601198851 +sin 1198682 times sin 1198601198852)](RF) (370) 2

DL = cosminus1 cos(1198682 minus 1198681) minus sin 1198681 times sin 1198682 times [1 minus cos(1198601198852 minus 1198601198851)] (371)

119877119865 = DL

times tan 119863119871

∆= Change in parameter MD = Measured depth TVD = True vertical depth

1198601198851 = Azimuth angle at upper survey point

1198601198852 = Azimuth angle at lower survey point

1198681 = Inclination angle at upper survey point

1198682 = Inclination angle at lower survey point

DL = ldquodog-legrdquo curvature

119877119865 = Ratio factor for minimum curvature

34 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 (219) and (220) 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

Start hard- and soft-iron model Start MSA model

Input magnetic and gravity measurements

Input geomagnetic referencing values Input geomagnetic referencing values of field strength and declination of field strength dip and declination

Initialize magnetic perturbations as zero

Estimate perturbations by Eq (319) and update

parameters

Iteration completion

Correct magnetic observations by Eqs (31) through (33)

Calculate corrected azimuth from Eq (222)

Calculate horizontal pictorial view of the wellbore by Eqs

(369) and (370)

Initialize hard-iron components 119881119909 119881119910 and 119881119911 through step1 Eq

Initialize soft-iron matrix components a b c e f and I

through step2 Eq (367)

parameters

Correct magnetic observations by Eq (339)

Inverse soft-iron matrix is obtained from Eq (347)

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

41 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 (327) can

calculate data points of magnetic field 119861119875 lying on a sphere with radius equal to B centered at

origin where in Equation (3-27) 119881119909 = 119881119910 = 119881119911 = 0 and 119882 = Identity Matrix It is assumed that

the simulated wellbore drilling takes place in a location where B = 500 119898119866119886119906119904119904 and DIP =

70deg It is noted that since Gauss is a more common unit in directional drilling I use 119898Gauss

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 (327) while at

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 119898119866119886119906119904119904

500 -500

500 HighSide 90 degree

Bx mGauss

mGaussBy

mGauss Bz

Figure 4-1 Sphere locus with each circle of data points corresponding to magnetic field

measurements made by the sensor rotation at highside 90degwith a specific inclination and a

cycle of azimuth values from 120782deg 120783120782deg 120784120782deg hellip 120785120788120782deg

Disturbing this sphere locus by substituting soft-iron matrix W and hard-iron vector V

given in Table 4-1 into Equation (327) leads to the ellipsoidal locus indicated in Figure 4-3

0Bz mGauss

-500 500

Inclination 90 degree

mGauss mGauss

-500 -500 Figure 4-2 Sphere locus with each circle of data points corresponding to magnetic field

measurements made by the sensor rotation at inclination 90degwith a specific highside and a

Table 4-1 The ellipsoid of simulated data

Actual Values

Hard-Iron(119898Gauss) Soft-Iron 119882

119881119909 = 200 08633 minus00561 minus00236 119881119910 = 100 minus00561 08938 00379

minus00236 00379 08349119881119911 = minus300

PRESS A KEY TO GO TO THE NEXT ITERATION

Raw Data Bz 0 mGauss Initial Calibration

-200 Sphere Ellipsoide

-400 Iteration 1 Iteration 5 -600

500 500

0By 0mGauss Bx -500 mGauss -500

Figure 4-3 A schematic illustrating the disturbed data lying on an ellipsoid with characteristics given in Table 4-1

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 120590 was evaluated as 03 119898119866119886119906119904119904

Therefore the simulated data have been contaminated by adding a random normally distributed

noise of 120590 = 03119898119866119886119906119904119904

abilit

Mean = 5779 mGauss Standard Deviation=03mGauss

5765 577 5775 578 5785 579 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 03 and 6 119898Gauss 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

Table 4-2 Parameters solved for magnetometer calibration simulations

Case Hard-Iron (119898Gauss)

Initial Values

Soft-Iron W Noise (119898Gauss) Hard-Iron

(119898Gauss)

Estimated Values

Soft-Iron W

I Figure

Vx = 2005884 Vy = 986962 Vz = minus3004351

08630 minus00559 minus00234 minus00559 08935 00381 minus00234 00381 08352

0 119881119909 = 200 119881119910 = 100 119881119911 = minus300

II Figure

Vx = 2005993 Vy = 987076 Vz = minus3004401

03 119881119909 = 2000130 119881119910 = 1000145 119881119911 = minus3000068

III Figure

Vx = 2008056

Vy = 989251 Vz = minus3005314

119881119909 =2002640515488068

119881119910 =1002955703924412

119881119911 =-3001296263083428

IV Figure

Vx = 180 Vy = 120

Vz = minus270 minus11 001 003 001 12 minus003 003 minus003 11

0 119881119909 = 1996864 119881119910 = 997188 119881119911 = minus2996086

Divergence

V Figure

Vx = 180 Vy = 120

03 119881119909 = 1996244 119881119910 = 996399 119881119911 = minus2995569

Divergence

VI Figure 4-10

Vx = 180 Vy = 120

Vz = minus270 086 minus005 minus002 minus005 089 003 minus002 003 083

119881119909 =2002640515488051

119881119910 =1002955703924423

119881119911 =-3001296263083437

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 119898Gauss 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

119898Gauss This means that the divergence is primarily due to the initial guesses assigned to soft-

iron parameters being away from the actual values

In solving the hard- and soft-iron algorithm it was necessary to ensure avoiding ill-

conditioning by examining the condition number of the matrix ζTζ during iterations For this

purpose firstly my Matlab code reduced the matrix (ζTζ) to ldquoReduced Row Echelon Formrdquo

through ldquoGauss Elimination with Pivotingrdquo (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 (ζTζ) 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 well-

conditioned and divergence is due to the improper initializing

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

2005 08938

200 1 2 3 4

1 2 3 4

1 2 3 4 Iteration

1 2 3 4 0863

1 2 3 4 -00236

-00235

Iteration 1 2 3 4

Iteration Estimated Actual

08632 08936

00381 -00559 100

0038 -0056 99

00379 -00561

1 2 3 4 1 2 3 4 1 2 3 4

Figure 4-5 Case I Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for the

least-squares iterations

12 201 087

1 2 3 4 1 2 3 4

2005 200

086 06 1995

1 2 3 4

0045 101 -005

1 2 3 4

004 100 0035 -006 99

003 98 1 2 3 4 1 2 3 4

-299 084

1 2 3 4

1 2 3 4 1 2 3 4

-002 -300 083

-301 -003

Iteration Iteration Iteration

Figure 4-6 Case II Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for the

220 12 09

1088 086 084

200 08 06

082 180 0 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

120 008 -002 -004 -006 -008

006 004 100 002

080 -01 0 1 2 3 4 5 6 0 1 2 3 4 5 6 1 2 3 4 5 6

0 1 2 3 4 5-320 Iteration

1 2 3 4 5

-006 -004 -002

Iteration 1 2 3 4 5 6

081 082 083 084 085

Figure 4-7 Case III Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for the

180 190 200 210

-600 -400 -200

0 50 100 0 50 100 0 50 100

100 110

0 50 100 0 50 100 0 50 100

-300 W(1

3) 400-200

200-400 -600 0-310

0 50 100 0 50 100 0 50 100 Iteration Iteration Iteration

Figure 4-8 Case IV Divergence of hard-iron (in mGauss) and soft-iron (unit-less)

estimates for the least-squares iterations

210 200 0

200 20

-200 190 -400

180 -600 0

0 50 100 0 50 100 0 50 100

100 110

0 50 100 0 50 100 0 50 100

200 600

-280 0

3) 400-200 -290

-400 200 -600 0

Figure 4-9 Case V Divergence of hard-iron (in mGauss) and soft-iron (unit-less)

12 092

09 1 088 086

06 084 0 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

115 005 0

004 110 -002

003 105 -004 002 -006 100 001 -008

0 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

0 1 2 3 4 5 6

-004 -002

0 002 004

Figure 4-10 Case VI Parameters of hard-iron (in mGauss) and soft-iron (unit-less) for

the least-squares iterations

42 Experimental Investigations

421 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

4211 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

measurements and determine the turntablersquos 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 (4times 4 119888119898) 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

httpwwwstcomwebenresourcetechnicaldocumentdatasheetCD00260288pdf

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 plusmn13 to plusmn 81 Gauss Linear Acceleration Range plusmn2 g plusmn4 g plusmn8 g

Operational Power Supply Range 25 V to 33 V (Voltage) Operating Temperature Range -30 to +85 degC Storage Temperature Range -40 to +125 degC

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 non-

orthogonality (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 range1

The calibration of the MEMS magnetometers using the hard- and soft-iron algorithm as well

as the MSA method was examined in this study

4212 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-clock-

wise 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

Thanks to Micro Engineering Dynamics and Automation Laboratory in department of Mechanical amp Manufacturing Engineering at the University of Calgary for the calibration of the MEMS accelerometers and the collection of the experimental data

4213 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

1 0 0 6 90 -2 0 90 7 90 -3 0 180 8 90 -4 0 270 9 90 -5 0 360 10 90 -

Figure 4-11 Experimental setup of MEMS integrated sensors on turn table at

45deg inclination

422 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 (220)

However identifying different axes arrangements of laboratory experiment when reading raw

data files lead to different azimuth formulas as follows

2⎡ (GzBy minus GyBz) Gx2 + Gy2 + Gz ⎤ Azimuth = tanminus1 ⎢ ⎥ (41)

⎢Bx Gy2 + Gz2 minusGx BzGz minus ByGy ⎥ ⎣ ⎦

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 (41)

The inclination was calculated from Equation (212) or (213) The experiment was performed

at two different nominal inclination angles of 0 and 90 degrees in order to verify whether the

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 (213)) 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

1 2 3 4 5

0 05 1 15 2 25 3 Samples x 104

Figure 4-12 Inclination set up for each test

423 Correction of the Diurnal Variations

Diurnal variations are fluctuations with a period of about one day The term diurnal simply

means lsquodailyrsquo and this variation is called diurnal since the magnetic field seems to follow a

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 (11798 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

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

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 124ℎ119900119906119903 = 04166666666) 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 0041 (1hour) as shown in Figure 4-14

where the largest peek corresponds to the frequency of 041 (1hour)

As shown in Figure 4-14 there are two smaller peeks According to the literature the Earthrsquos

magnetic field undergoes secular variations on time scales of about a year or more which reflect

changes in the Earthrsquos 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 Earthrsquos 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

0 20 40 60 80 100 Time (hours)

Figure 4-13 The observations of the geomagnetic field strength follow a 24 hour periodic

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

00 005 01 015 02 025 03 035 04 045 05 Frequency(1hour)

Before Filtering(Sample of One Hour) After Filter Removing Diurnal Effect

0041(1hour) = 24 hour

Figure 4-14 Geomagnetic field intensity in the frequency domain

Before Filtering(Median Hourly Resampled) After Filter Removing Diurnal Effect Subtracing Filtered Signal from Original

12am 6am 12pm 6pm 12am 6am 12pm 6pm 12am 6am 12pm 6pm 12am 6am 12pm 6pm 12am 6am 12pm Time(hour)

Figure 4-15 Geomagnetic field intensity in the time domain

Table 4-5 Diurnal correction at laboratory

University of Calgary Laboratory Latitude 5108deg N

Longitude minus11413deg 119882 Altitude(meter) 1111

March 2013

IGRF Magnetic Field Strength (119898Gauss) 566726 IGRF Dip Angle 7349deg

IGRF Declination Angle 1483deg

Diurnal Corrected Variations

Magnetic Field Strength (119898Gauss)

566726-34375= 563288

Dip Angle 7349deg minus 00915deg = 73398deg

424 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-

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 4-16

Table 4-6 Parameters in the magnetometer calibration experiment

Initial Values of Hard-Iron Vector (119898Gauss)

IGRF IGRF + Diurnal Correction

119881119909 = minus2453563123417183 119881119909 = minus2453563123417181 119881119910 = 5778527890716003 119881119910 = 57785278907160040

119881119911 = minus69721746424075958 119881119911 = minus69721746424075960

Initial values of Soft-Iron Matrix IGRF IGRF + Diurnal Correction

06715 minus00553 minus00169 06675 minus00549 minus00168 minus00553 07022 00252 minus00549 06979 00251 minus00169 00252 06612 minus00168 00251 06572

Estimated Values of Hard-Iron Vector (119898Gauss) IGRF IGRF + Diurnal Correction

119881119909 = minus251492417079711 119881119909 = minus251492417079716 119881119910 = 5729617551799971 119881119910 = 5729617551799970 119881119911 = minus79854119493792 119881119911 = minus79854119493795

Estimated Values of Soft-Iron Matrix

07523 minus00821 minus00103

IGRF minus00821 08135 00239

minus00103 00239 06206

IGRF + Diurnal Correction

IGRF MSA Parameters (119898Gauss)

IGRF + Diurnal Correction ∆119861119909 = minus2371932 ∆119861119909 = minus2372463 ∆119861119910 = 538988 ∆119861119910 = 542609

∆119861119911 = minus4671157∆119861119911 = minus4652156

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 (219) and (220) 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 0deg 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

-500 0

500 -600

Raw Data Initial Calibration Sphere Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Ellipsoid

3 4 21

610Bz mGauss

mGaussBy

Bx mGauss

Figure 4-16 Portion of the ellipsoid representing the locus of magnetometer measurements

from laboratory experimental data

ree) 180

Raw Azimuth Only Hard-Iron Calibration+IGRF+Diurnal Correction MSA+IGRF+Diurnal Correction

0 2000 4000 6000 8000 10000 12000 14000 16000 Samples

Figure 4-17 Headings calibrated by MSA versus hard and soft iron (georeferncing

obtained by IGRF model corrected for diurnal effects)

Raw Azimuth MSA+IGRF Hard-Iron and Soft-Iron Calibration+IGRF

2000 4000 6000 8000 10000 12000 14000 16000 Samples

obtained by IGRF model without diurnal corrections)

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 soft-

iron 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 post-

calibrated 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 0deg 90deg 180deg 270deg 360deg

Raw -278032deg 145415deg 2606828deg 2877881deg -278609deg

MSA + IGRF -11433deg 889201deg 1824533deg 2709831deg -11238deg

MSA + IGRF + Diurnal -10673deg 888897deg 1823915deg 2710439deg -10476deg

Hard and Soft-Iron + IGRF 49429deg 888421deg 1732424deg 2810430deg 50596deg Hard and soft-Iron + IGRF +

Diurnal 49429deg 888421deg 1732424deg 2810430deg 50596deg

Only Hard-Iron + IGRF + Diurnal 81334deg 862955deg 1732443deg 2788222deg 82538deg

43 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)

∆119879119881119863 = ∆119872119863

(cos 1198681 + cos 1198682) (42) 2

∆119873119900119903119905ℎ = ∆119872119863

(sin 1198681 times cos 1198601198851 + sin 1198682 times cos 1198601198852) (43) 2

∆119864119886119904t = ∆119872119863

[(sin 1198681 times sin 1198601198851 +sin 1198682 times sin 1198601198852)] (44) 2

Applying the Balanced Tangential equations of (43) and (44) and assuming the first sample

point inclination and azimuth as known values of 1198601198851 = 117deg and 1198681 = 4deg it is possible to

calculate inclination and azimuth at subsequent points denoted by 1198682 and 1198602 respectively

Equations (43) and (44) can be solved to give

sin 1198682 =

= 2 times ∆119873119900119903119905ℎ minus

∆119872119863 times cos 1198601198851 times sin 1198681 + ∆119864119886119904119905 minus ∆119872119863 times sin 1198601198851 times sin 1198681

2 2 (45) ∆1198721198632

2 times ∆119873119900119903119905ℎ minus ∆119872119863 times cos

2 1198601198851 times sin 1198681

cos 1198601198852 = (46) ∆119872119863 times sin 1198682

For simplicity inclination values are assumed to be derived between zero and 90deg and

azimuth values between zero and 180deg It was necessary to ensure that sin and cos derived values

are between -1 and +1 Therefore the first suggested values of ∆119864119886119904119905 and ∆119873119900119903119905ℎ 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

(212) by considering 119866119909 and 119866119910 as free unknowns it is possible to calculate 119866119911 Highside angle

is computed from Equation (211)

In the case where there are no magnetic disturbances and no noise 119881119909 = 119881119910 = 119881119911 =

0 and 119882 = ldquoIdentity Matrixrdquo are substituted in Equation (327) to calculate data points of

magnetic field 119861119875 along the above wellbore trajectory These data points are lying on a sphere

with radius equal to B centered at the origin

First Suggested Trajectory Final Trajectory

0 100 200 300 400 500 600 West(-)East(+) (meter)

Figure 4-19 Simulated wellbore horizontal profile

It is assumed that the simulated wellbore drilling takes place at the University of Calgary

location The values of DIP and B in Equation (327) are thus provided from Table 4-8

Magnetic azimuth values applied to Equation (327) 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 ldquoBUILDrdquo reaching to approximately

horizontal in the down section of wellbore referred to as ldquoLATERALrdquo The first 28 sample

points belong to the ldquoBUILDrdquo section and the last 51 sample points belong to ldquoLATERALrdquo

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

University of Calgary Location Latitude 5108deg 119873

Longitude minus11413deg 119882 BULID LATERAL

2013 Altitude (meter) 1111 900 IGRF Magnetic Field Strength (119898Gauss) 566726 566782

IGRF Dip angle 7349deg 7349deg IGRF Declination angle 1483deg 1481deg

Disturbing this sphere locus by substituting the soft-iron matrix W and hard-iron vector V

values in Table 4-9 into equation (327) leads to the ellipsoidal locus indicated in Figure 4-20

and Figure 4-21 which quantify the size of the magnetometer measurement locus simulated for

BUILD and LATERAL sections of the simulated wellbore respectively

BUILD Actual Values Hard-Iron (119898Gauss) Soft-Iron 119882 119881119909 = minus015 10014 minus00005 minus00004 119881119910 = minus01 Symmetric minus00005 10014 minus00007 119881119911 = 13 minus00004 minus00007 10014

LATERAL Actual Value Hard-Iron (119898Gauss) Soft-Iron 119882

119881119909 = 035 10010 00010 00023 119881119910 = 015 Non-Symmetric minus00010 10009 00020 119881119911 = 9 minus00023 minus00015 10699

The simulated data has been contaminated by adding a random normally distributed noise

of 120590 = 03 119898119866119886119906119904119904 and 120590 = 6 119898119886119906119904119904 to the data Simulated wellbore data is listed in

appendix G The solved parameters of magnetic interference correction are stated in Table 4-10

Bz 0mGauss

mGauss

Raw Data Initial Calibration Sphere Ellipsoide

-500 Iteration 1 Iteration 7 500

mGauss -500 -500

from BUILD section of the simulated wellbore(magnetic coordinates in 119950gauss)

200 Bz

0mGauss Raw Data

-200 Initial Calibration

-400 Sphere Ellipsoide Iteration 1 Iteration 6

-500 0By

mGauss -500

mGauss Bx

Figure 4-21 Portion of the ellipsoid representing the locus of magnetometer measurements from LATERAL section of the

simulated wellbore(magnetic coordinates in 119950Gauss

Table 4-10 Calibration parameters solved for simulated wellbore

Case Noise 119898Gauss Hard-Iron

119898Gauss

Estimated Values

Soft-Iron W MSA Correction 119898Gauss

Error free

119881119909 = minus01500 119881119910 = minus01000 119881119911 = 13000

10014 minus00005 minus00004 minus00005 10014 minus00007 minus00004 minus00007 10014

∆119861119909 = minus05100 ∆119861119910 = minus02613 ∆119861119911 = 19029

L 119881119909 = 03500 119881119910 = 01500 119881119911 = 90000

10010 0000 00001 00000 10009 00003 00001 00003 10699

∆119861119909 = 03290 ∆119861119910 = 02436 ∆119861119911 = 87431

119881119909 = minus05549 119881119910 = minus03796 119881119911 = 23382

10008 minus00006 00001 minus00006 10010 minus00003 00001 minus00003 09994

∆119861119909 = minus05429 ∆119861119910 = minus02771 ∆119861119911 = 19121

L 119881119909 = 03035 119881119910 = 01707 119881119911 = 94012

10010 00000 00002 00000 10008 00018 00002 00018 10753

∆119861119909 = 03095 ∆119861119910 = 02407 ∆119861119911 = 87477

119881119909 = minus65583 119881119910 = minus55737 119881119911 = 162082

09925 minus00015 00080 minus00015 09962 00068 00080 00068 09730

∆119861119909 = minus1163895 ∆119861119910 = minus057047 ∆119861119911 = 2057959

L 119881119909 = minus03479 119881119910 =07536 119881119911 = 158978

10013 00007 00047 00007 10017 00249 00047 00249 10268

∆119861119909 = minus0064359 ∆119861119910 = 0182197 ∆119861119911 = 8610343

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 eastwest direction

This is shown in Figure 4-23 when SSA is most unstable at inclination 90 degrees and azimuth

around 90 degrees

BUILD LATERAL

300 400 500 600 700 800 900 1000 1100 Measured Depth (meter)

Figure 4-22 Conventional correction is unstable in LATERAL sectionsince it is near

horizontal eastwest

875 88 885 89 895 90 905 91 915

350 Raw Azimuth(degree) Calibrated Azimuth(degree) by IGRF+Multi-Survey Calibrated Azimuth(degree) by IGRF+Conventional Calibrated Azimuth(degree) by IGRF+Hard and Soft Iron Calibration

LATERAL

Inclination(degree)

Figure 4-23 Conventional correction instability based on inclination

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

Survey point no

Figure 4-24 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

IGRF Mangetic Strength(mGauss) Observed Mangetic Strength(mGauss) Calibrated Mangetic Strength(mGauss) by IGRF+Multi-Survey Calibrated Mangetic Strength(mGauss) by IGRF+Conventional Calibrated Mangetic Strength(mGauss) by IGRF+Hard and Soft Iron Calibration

BUILD LATERAL

24 26 28 30 32 34

20 30 40 50 60 70

Survey point no

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

BUILD LATERAL

Figure 4-25 Calculated field direction by calibrated measurements

Raw Data MSA+IFR Conventional+IFR Hard-Soft Iron+IFR Real Path

0 100 200 300 400 500 West(-)East(+) (meter)

Figure 4-26 Case I Wellbore pictorial view of the simulated wellbore in horizontal plane

(no error)

0 100 200 300 400 500 600-400

Raw Data MSA+IFR Conventional+IFR Hard-Soft Iron+IFR

West(-)East(+) (meter)

Figure 4-27 Case II Wellbore pictorial view of the simulated wellbore in horizontal plane

(random normally distributed noise of 03 mGauss)

0 100 200 300 400 500 600-300

Figure 4-28 Case III Wellbore pictorial view of the simulated wellbore in horizontal

plane (random normally distributed noise of 6 mGauss)

Table 4-11 shows a summary of comparative wellbore trajectory results from correction

methods for case III of Table 4-10

Table 4-11 Comparative wellbore trajectory results of all correction methods

Case III (random normally distributed noise of 6 119898Gauss) ∆ East ⋁ ∆North ⋁

Correction Method

East Displacement ∆East

Real Path (Diff from Real Path)

North Displacement ∆North

Closure Distance from Real Path

meter meter meter

Raw Data 586845 7695 201612 33856 347195

MSA 577522 1628 240140 4672 49475 Hard-Soft

Iron 569709 9441 260698 2523 269385

Real Path 579150 0 235468 0 0

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

119898Gauss 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

44 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 (ldquoBUILDrdquo) section of the wellbore reaching

to approximately horizontal in the down (ldquoLATERALrdquo) 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 4-12 The solutions for each case have been listed in

Table 4-13

Table 4-12 Geomagnetic referencing values

Geomagnetic referencing

Field Strength

(119898Gauss)

Dip (degrees)

Declination (degrees)

IFR (LATERAL) 57755 7558 1387 IFR (BIULD) 5775 7556 1376

IGRF 577 7552 1382

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 4-13 Calibration parameters solved for the case study

BUILD IFR IGRF

Hard-Iron Vx = 0396198 Vx = 0396198 Vector Vy = 0107228 Vy = 0107228

(119898Gauss) Vz = 0171844 Vz = 0171844 09986 56362 times 10minus4 44251 times 10minus4 09977 56313 times 10minus4 44213 times 10minus4Soft-Iron

56362 times 10minus4 09986 74496 times 10minus4 56313 times 10minus4 09977 74432 times 10minus4Matrix 44251 times 10minus4 74496 times 10minus4 09986 44213 times 10minus4 74432 times 10minus4 09977

∆119861119909 = minus0153885 ∆119861119909 = minus020028MSA ∆119861119910 = minus0121778 ∆119861119910 = minus007931(119898Gauss)

∆119861119911 = 1294122 ∆119861119911 = 1952245 LATERAL

IFR IGRF Hard-Iron Vx = 0056286 Vx = 0056286

Vector Vy = minus0047289 Vy = minus0047289 (119898Gauss) Vz = 9281106 Vz = 9281106

09990 50788 times 10minus5 00021 09980 50739 times 10minus5 00021 Soft-Iron 50788 times 10minus5 09991 minus00015 50739 times 10minus5 09981 minus00015 Matrix 00021 minus00015 09347 00021 minus00015 09338

∆119861119909 = 0349761 ∆119861119909 = 0489535MSA ∆119861119910 = minus006455 ∆119861119910 = minus0096051(119898Gauss)

∆119861119911 = 3917254 ∆119861119911 = 5600618

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

towards the eastwest direction This is shown in Figure 4-32 where SSA is most unstable at

inclination 90 degrees and azimuth around 270 degrees

400 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

0 200 400 600 800 1000 1200 1400 1600 1800 2000 Measured Depth (meter)

Figure 4-29 Conventional correction is unstable in LATERAL sectionwhen it is near

horizontal eastwest

840 860 880 900 920 940 Measured Depth (meter)

LATERAL

Figure 4-30 Zoom1 of Figure 4-29

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

84 85 86 87 88 89 90 91

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

LATERAL

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 (43)

40 50 60 70 80 90 100

Survey point no

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

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

LATERAL BUILD

45 50 55 60 65 70 75 80 85 Survey Point no

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

∆ East ∆ North

Method Geomagnetic referencing

East Displacement∆ East

⋁ Gyro (Diff from Gyro)

North Displacement∆ North

Closure Distance

from Gyro meter

meter meter Raw Data

IGRF 1351953 4743 76771 27688 28091 IFR 1351878 4668 76057 28402 28783

MSA IGRF 1350472 3262 120726 16266 16590 IFR 1351043 3833 106837 2378 45102

Hard-Soft Iron

IGRF 1350663 3453 125550 21090 21371

IFR 1350630 3420 124837 20377 20662

Gyroscope 134721 - 104460 - -

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 soft-

iron 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

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)

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

45 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 soft-

iron model especially when combined with IFR-corrected surveys

-1320 -1200 -1080 -960 -840 -720 -600 -480 -360 -240 -120

Raw DataIGRF(declination)

MSA+IGRF Conventional+IGRF Hard-Soft Iron+IGRF Raw DataIFR(declination)

MSA+IFR Conventional+IFR Hard-Soft Iron+IFR Gyro

BUILD LATERAL

West(-)East(+) (meter) Figure 4-35 Wellbore pictorial view in horizontal plane by minimum curvature

Chapter Five CONCLUSIONS and RECOMMENDATIONS for FUTURE RESEARCH

51 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 un-

modeled 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 soft-

iron 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 eastwest direction

thus SSA is no longer applicable in the industry Finally the results support that the positining

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

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

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

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 05 of the wellbore horizontal displacement This equals to 55 meters 05 times 1100 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 45102 meter (Table 4-14) which shows a 18 improvement over the 55 meters of

the positional accuracy by MWD surveys availbale in the modern industry On the other hand

hard- and soft-iron calibration provides the closure distance of 20377 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

52 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

521 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

(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

522 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

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 119985 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

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

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APPENDIX A SIMULATED WELLBORE





13 Thesis Outline













29 Previous Studies


292 Standard Method




210 Summary












3272 Step 2 Solving Ellipsoid Fit Matrix by an Optimal Ellipsoid Fit to the Data Corrected for Hard Iron Biases


34 Summary








422 Heading Formula




44 A Case Study

45 Summary





