anticollision method of active magnetic guidance ranging...

Research ArticleAnticollision Method of Active Magnetic GuidanceRanging for Cluster Wells

Xinyu Dou 123 Huaqing Liang 12 and Yang Liu12

1State Key Laboratory of Petroleum Resources and Prospecting China University of Petroleum-Beijing Beijing 102249 China2College of Geophysics and Information Engineering China University of Petroleum-Beijing Beijing 102249 China3Intelligence and Information Engineering College Tangshan University Tangshan 063000 China

Correspondence should be addressed to Huaqing Liang hqliangcupeducn

Received 6 February 2018 Revised 24 March 2018 Accepted 11 April 2018 Published 15 May 2018

Academic Editor Milica Rancic

Copyright copy 2018 Xinyu Dou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Conventional adjacent wells range scanning calculations cannot meet the accuracy demands of the anticollision measurement ofborehole distances any longer Current techniques commonly employ electromagnetic detection tools while drilling this requiresputting equipment down adjacent wells to avoid collision risks which adds more workload and costs and sometimes even affectsthe normal production of the producing wells Measuring and tracking adjacent borehole distances while drilling is an essentialprocess that guides the drill bit and effectively avoids collisions with existing wells This paper proposes an active anticollisionmethod of rotating magnetic ranging based on double symmetrical magnetic sources with opposite magnetic moments First theproposed method uses magnetic sources in the drilling well that are built into the probe tube to generate a magnetic field then theferromagnetic casing of the existingwell would bemagnetized by the abovementionedmagnetic field finally themagnetization fieldof the ferromagnetic casing is measured by a triaxial magnetometer built into the probe tube to determine the spacing and positionof the existing well Simultaneously the calculation models of magnetic flux density around the casing of the existing well andmagnetic sources are established the calculation formulae of the relative distance and position of two adjacent wells are deducedand a new variable interval section segmentation is proposed based on the Cosine theoremThe simulation results demonstrate thatthe spacing and position of the existing well are determined based on the magnetic sourcesrsquo spacing inside the probe the magneticmoment of the magnetic sources the relative permeability of the casing the diameter of the casing and the inclination between thedrilling well and the existing wellThe validity and accuracy of the active magnetizationmodel are confirmed providing theoreticalsupport for the further development of electromagnetic anticollision devices

1 Introduction

Complex wells can effectively improve single well produc-tivity and the ultimate recovery of complex oil and gasfields magnetic guidance technology is the core technologyof drilling engineering in complex structure wells [1] Theworking principle of the anticollision scanning method [2]is fitting and approximating the real drilling trajectory withinclinometer data then the relative position relationshipbetween the drilling well and the existing well can becalculated by the anticollision scanning algorithm Whenthe relative borehole distance is less than the given safetydistance the drilling direction can be adjusted to avoid acollision accident The accuracy of the anticollision scanning

method relies on inclinometer data of borehole trajectoriesfor the drilling well and the existing well borehole trajectoryfitting approximationmethod and the anticollision scanningalgorithm These data and algorithms do tend to have errorsin practical applications the cumulative errors increase withincreased drilling depth causing the accuracy of the calcula-tion results to be lower Therefore this method fails to effec-tively avoid wellbore collisions To better solve this problemit is necessary to monitor the distance between adjacent wellsand maintain it within a reasonable range in real time Thenewly emerged electromagnetic guidance tools can guaranteesafe drilling and the successful avoidance of collisions

Electromagnetic guidance tools which include the mag-netic guidance tool (MGT) [3] the RotatingMagnet Ranging

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 7583425 11 pageshttpsdoiorg10115520187583425

2 Mathematical Problems in Engineering

Service (RMRS) system [4] the Wellspot Tool (WT) [5] andthe Single Wire Guidance (SWG) [6] tool can detect thedistance between the drilling well and the existing well toavoid collision [1ndash6] MGT and RMRS are applied in adjacenttwin parallel Steam Assisted Gravity Drainage (SAGD) wellsthat guide the drilling of an injection well to ensure thatit is parallel to the production well [2 3 7ndash9] However itrequires improvement if it is to be applied to the anticollisionof cluster wells There are two primary disadvantages ofthe MGT and RMRS methods First they are difficult tooperate as a probe tube must be placed in the existing wellto guide the detection of drill bit Second the monitoringworkload of the anticollision system is quite heavy WT isnot suitable for the anticollision of cluster wells since thedetection process requires pulling out the drilling tools TheSWG tool overcomes the disadvantages of the MGT andRMRS [1 4] namely that neither reduces the workloads norrequires pulling out the drilling tool to guide the detectiondevice However there are three primary disadvantages ofthe SWG tool First a cable must be placed down the offsetwell which affects the normal production process of theexisting well Additionally a large amount of equipmentmust be utilized in order to run this system Lastly thetechnology of the SWG tool is monopolized by foreigncompanies and the service cost is expensive so domesticcompanies rarely use SWG to reduce the cost of a singlewell

With cluster wells being used in more and more appli-cations in land and sea environments the borehole distancehas been reduced from 2m times 2m to 15m times 17m [2 4] andinfilling adjustment wells are usually added to the existingwell pattern The traditional anticollision scanning methodis insufficient for cluster wellsrsquo anticollision Therefore theactive magnetic guidance ranging method is proposed toavoid borehole collision This method is based on detectingmagnetic induced intensity of the casing in the existing wellit detects the borehole distance during the drilling process inreal time without interrupting drilling or ever requiring anyother equipment to be placed in the existingwellThemethodbecomes practically significant in providing a theoreticalbasis for the research and development of active anticollisiondevices for cluster wells

2 Principle of Active MagneticGuidance Ranging

The signal sources or the probe tube must be mounted inthe existing well to measure the distance between drillingwell and offset wells for MGT RMRS SWG and otherelectromagnetic guidance tools [10ndash13] To solve the aboveproblems the active magnetic anticollision ranging methodof clusterwells is proposed based on the fact that themagneticpoles of double magnetic sources are parallel to each otherwith oppositemagneticmoments (expressed as red and greenlines in Figure 1) It does not interrupt the drilling of theproduction well nor does it require any other equipment tobe placed in the offset well yet it detects the borehole distancein real time during the drilling process This principle isshown in Figure 1

X

Geographic north

Geographic eastY

Z

Casing Pipe

Probe Tube

O

Magnetic source A

Magnetic source BSensor

Figure 1 Principle of active magnetic guidance ranging for clusterwells

Therefore the total magnetic induction intensity asdetected by a triaxial magnetometer is 0 when there is nocasing around the probe tube or when the casing is faraway from the probe tube [14] When there is casing aroundthe probe tube the triaxial magnetometer data representthe sum of the geomagnetic field the magnetic field oftwo magnetic sources and the magnetic flux density ofthe magnetizing casing which generates a magnetic fieldalong its axial direction (expressed as blue lines) As themagnetic fields of magnetic sources vary with the rotationof the drill rod the alternating magnetic field frequency ofthe casing varies along with the rotation rate of magneticsources

Based on the above described principle the relativedistance and orientation between the drilling well and theexisting well are calculated using the detection data of atriaxial magnetometer and the well trajectory of the drillingwell is adjusted appropriately to prevent collisions with theexisting well

3 Calculation Method ofAdjacent Well Distance

The calculation method of adjacent well distance is theanticollision core technology of magnetic guidance Underthe coordinate system shown in Figure 1 the casing stretchesvertically downward along the 119885-axis and the 119883-axis pointsto geographic north The outer diameter and thickness of the

Mathematical Problems in Engineering 3

X

Z

Y

I

M

O

P

r

K

R

Figure 2 Schematic diagram of magnetic dipole

casing are 119862 and 120575 respectively The inclination and azimuthof the casing are 120572 and 120593 respectively31 Distribution of Magnetic Field around Magnetic SourcesThe dimensions of the components are strictly limited by thegeometrical size of the probe tube in the drilling well Thetwo symmetrical magnetic sourcesrsquo sizes must be far less thanthe size of drill collar and the mud flow channel should bereserved The magnetic source is a cylinder with a lengthof 100sim200mm and a diameter of 20sim50mm according tocommon drill collar parameters To ensure an anticollisionsystem for cluster well applications the borehole distanceshould be at least 1m and normallymore than 3m otherwisethe probability of collision is increased The size of themagnetic source is much smaller than the adjacent boreholedistances thus the magnetic source can be regarded as amagnetic dipole [1 15 16]

The spherical coordinate system is established as shownin Figure 2 The magnetic dipole modeled with a circularcurrent loop is placed in the 119883119874119884 plane The sphericalcoordinate system can be established by taking the center ofthe magnetic dipole as the origin point and the direction ofthe magnetic moment of the magnetic dipole as the 119885-axisThe spherical coordinates of point 119875 that is any point inspace can be expressed as (119903 120579 120593) The vector distance frompoint 119875 to point 119874 is r the intersection angle between 119875119874and the 119885-axis is 120579 and the angle between the 119883-axis and119870119874 which is the projection of 119875119874 on the XY-plane is 120593Expression (1) gives the relations between the spherical andCartesian coordinates [17]

119903 = 119909sin 120579 cos120593 = 119910

sin 120579 sin120593 = 119911cos 120579

120579 = arctan(radic1199092 + 1199102119911 ) 120593 = arctan(119910119909) (1)

Assuming that point 119872 is any point on the circle itscorresponding coordinate is (119903 1205872 1205931) and the magneticflux density caused by the current unit 119868119889l in the space point119875 can be calculated by the Biot-Savart Law as follows

119889B = 1205834120587 119868119889l times a1198863 (2)

where a 119889l and 120583 are the distance vector of point119872 to point119875 the tangent vector of point 119872 and the space magneticpermeability respectively In the spherical coordinate systemthose parameters have the following forms

a = (119903 sin 120579 cos120593 minus 119877 cos1205931) 997888119894+ (119903 sin 120579 sin120593 minus 119877 sin1205931) 997888119895 + (119903 cos120593)997888119896 119889l = (119889119909 119889119910 119889119911) = (minus119877 sin120593119889120593 119877 cos120593119889120593 0)

11198863 = 1(radic1199032 + 1198772)3 [1 minus 2119903119877 sin 1205791199032 + 1198772 cos (120593 minus 1205931)]minus32 119898 = 119868119878 = 1205871198681198772

(3)

In previous expressions with 119877 the radius of the loop isdenoted and 119898 is the magnetic dipole moment By perform-ing the integral calculation and substituting Expressions (3)in formula (2) when far from the loop the above formula canbe expressed as

119861119909 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 2120579 cos1205931199032 + 1198772

119861119910 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 2120579 sin1205931199032 + 1198772

119861119911 = 1205832120587 119898(radic1199032 + 1198772)3 (1 minus 31199032sin21205792 (1199032 + 1198772)) (4)

The detecting data of the magnetic anticollision systemwith a rotating probe tube which is driven by the drill stringare collected for the cluster well while drilling Put simply themagnetic dipole rotates around the 119884-axis and the distance 119903between the casing and the magnetic dipole can be regardedas a constant 120593 = 0 0 le 120579 le 2120587

119861119909 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 21205791199032 + 1198772

119861119910 = 0119861119911 = 1205832120587 119898(radic1199032 + 1198772)3 (1 minus 31199032sin21205792 (1199032 + 1198772))

(5)

If the distance from any point 119875 on the casing to themagnetic source is much larger than the radius of themagnetic source (119903 ≫ 119877) then

119861 = radic1198612119909 + 1198612119910 + 1198612119911 = 12058311989821205871199033radic1 minus 34 sin2120579 (6)


times104

2

25

3

35

4

B (n

T)

0 215 (rad)

05

Figure 3 Magnetic flux density simulation

By assuming 120583 = 4120587times 10minus7Hm119898 = 200Asdotm2 119903 = 1m120579 isin [0 2120587] the curve of magnetic flux density 119861 is shown inFigure 3

During the rotation of the probe from Figure 3 theaxial direction of the magnetic source points directly tothe casing whenever the maximum magnetic flux density ofcasing reaches one of the corresponding coordinates namely0 05120587 120587 15120587 and 2120587 The direction of the casing canbe determined by the maximum magnetic flux density Thedistance from the probe tube to the casing can be calculatedby using the relationship between the relative distance and themagnetic flux density of the magnetizing casing Thereforethe maximum and minimum values on the curve indicatethat the axis of the magnetic source is directly pointing to thecasing at these points At the moment the curve reaches itspeak or valley the azimuth of the casing in the adjacent wellcan be determined bymeasuring themagnetic azimuth of theprobe tube with the built-in triaxial magnetometer

32Magnetizing Field Calculation of Casing Due to the com-plexity of the actual formation the following five assumptionsare proposed to simplify the calculation [18ndash20]

(1) The formation is evenly distributed and isotropic(2) The casing length is infinite(3)The radius of the casing is much less than the distance

between the drilling well and the existing well(4) The casing is isotropic(5) There are no ferromagnetic minerals with high per-

meability in the formationUnder these conditions the magnetic flux density of the

casing in the existing well can be calculated by formula (4)

321 Magnetic Flux Density Calculation of the MagnetizingCasing for the Upper Magnetic Source Taking the uppermagnetic source center of the probe tube as the origin 119862the 119884-axis and 119885-axis correspond to the axial directions ofthe probe tube and the upper magnetic source respectivelyThe coordinate system is established in Figure 4The distance119874119864 from the probe tube center point 119874 to the casing is dthe intersection angle between the axes of the casing andthe probe tube is 120572 the distance between the two magnetic

O

Y

X

ZP

A

E

B

C

D

d

h

h

r

N

M

r1

r2

2

1

Figure 4 Calculation model of magnetic intensity in the magne-tized casing

sources is 2ℎ the angle between 119875119862 and the upper magneticsource axis is 1205791 the distance from point 119875 to point 119862 is 1199031and point 119875 is any point on the casing The permeability ofthe casing is 1205831 = 1205830(1 + 120594119898) the magnetic susceptibility ofthe casing is 120594119898 and the vacuum permeability is 1205830

The magnetic flux density distribution around the uppermagnetic source can be deduced by formula (4) as follows

119861119862119909 = 31205838120587 11989811990331 sin 21205791 cos120593119861119862119910 = 31205838120587 11989811990331 sin 21205791 sin120593119861119862119911 = 1205832120587 11989811990331 (1 minus 32 sin21205791)

(7)

where 120583 is the permeability of the space surrounding themagnetic source

From the previous analysis the magnetic flux density ofthemagnetizing casing reachesmaximumwhen themagneticsource axis points to the casing Therefore the magneticflux density distribution around the magnetic source can bededuced as follows

119861119862119909 = 0119861119862119910 = 3120583119898812058711990331 sin 21205791119861119862119911 = 120583119898212058711990331 (1 minus 32 sin21205791)

(8)


For the upper magnetic source the magnetic flux densityof the magnetizing casing can be expressed as follows

1198611015840119862119909 = 01198611015840119862119910 = 31205831119898812058711990331 sin 212057911198611015840119862119911 = 1205831119898212058711990331 (1 minus 32 sin21205791)

(9)

322 Magnetic Flux Density Calculation of the MagnetizingCasing for the Lower Magnetic Source Similar to formula(7) for the lower magnetic source built into the probe tubethe magnetic flux density of the magnetizing casing can beexpressed as follows

1198611015840119863119909 = 01198611015840119863119910 = minus31205831119898812058711990332 sin 212057921198611015840119863119911 = minus 1205831119898212058711990332 (1 minus 32 sin21205792)

(10)

323 Total Magnetic Flux Density Calculation of the CasingThe formulae of the magnetic flux density of the magnetizingcasing caused by the two different magnetic sources arederived below

The magnetic flux density caused by the upper magneticsource can be calculated using the following


(11)

Similarly the magnetic flux density caused by the othermagnetic source can be expressed as

119861119875119863119909 = minus312058318120587 11989811990332 sin 21205792 cos120593119861119875119863119910 = minus312058318120587 11989811990332 sin 21205792 sin120593119861119875119863119911 = minus 12058312120587 11989811990332 (1 minus 32 sin21205792)

(12)

where the intersection angle between 119875119863 and the 119885-axisis 1205792 and the distance between point 119875 and point 119863 is 1199032The geometric relationship in Figure 4 can be expressed asfollows

1199031 = radic(119910 minus ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)21205791 = arctan

radic(119910 minus ℎ)2 + 1199092radic1198892 minus 1199092 + 119910 tan120572 1199032 = radic(119910 + ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)21205792 = arctan

radic(119910 + ℎ)2 + 1199092radic1198892 minus 1199092 + 119910 tan120572 120593 = arctan

119910119909

(13)

The total magnetic flux density of point 119875 caused by thetwo magnetic sources can be expressed as

119861119875119909 = 312058311198981199094120587radic1199092 + 1199102[[[[[[

radic(119910 minus ℎ)2 + 1199092 (radic1198892 minus 1199092 + 119910 tan120572)(radic(119910 minus ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)2)5 minus

radic(119910 + ℎ)2 + 1199092 (radic1198892 minus 1199092 + 119910 tan120572)(radic(119910 + ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)2)5

]]]]]]

119861119875119910 = 312058311198981199104120587radic1199092 + 1199102[[[[[[



]]]]]]

119861119875119911 = 12058311198982120587[[[[[[1 minus (32) (((119910 minus ℎ)2 + 1199092) ((119910 minus ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)2))

(radic(119910 minus ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)2)3


minus 1 minus (32) (((119910 + ℎ)2 + 1199092) ((119910 + ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)2))(radic(119910 + ℎ)2 + 1199092 + (radic1198892 minus 1199092 + 119910 tan120572)2)3]]]]]]

(14)

4 Calculation of the Distance betweenAdjacent Wells

The triaxial magnetometer is centered between the twomagnetic sources built into the probe tubeThe totalmagneticflux density detected by the triaxial magnetometer is the sumof the magnetic fields generated by all points of the casingthe twomagnetic sources and the constant geomagnetic fieldThe triaxial magnetometer detects the magnetic flux densitywhich is generated by the two magnetic sources and can beexpressed as follows

119861119862119863119909 = 119861119862119909 + 119861119863119909 = 0119861119862119863119910 = 119861119862119910 + 119861119863119910 = 0119861119862119863119911 = 119861119862119911 + 119861119863119911 = 0(15)

The geomagnetic field is usually acquired from interna-tional geomagnetic reference field (IGRF) data [8] and it canbe regarded as a constant over a short period of time

The element 119875 of the segmentation casing is far less thanthe distance between the probe tube and the casing so it canbe regarded as a magnetic dipole (idealized as a point 119875)

The magnetic flux density distribution around the magneticdipole 119875 is derived according to above formulae as follows

119861119909 = 3120583119898101584081205871199033 sin 2120579 cos120593119861119910 = 3120583119898101584081205871199033 sin 2120579 sin120593119861119911 = 120583119898101584021205871199033 (1 minus 32 sin2120579)

(16)

where

120579 = arctanradic1199092 + 1199102radic1198892 minus 1199092 + 119910 tan120572

119903 = radic1199092 + 1199102 + (radic1198892 minus 1199092 + 119910 tan120572)2120593 = arctan

119910119909 (17)

When (17) is substituted into formula (16) the followingformulae are deduced

119861119909 = 31205831198981015840119909 (radic1198892 minus 1199092 + 119910 tan120572)4120587(radic1199092 + 1199102 + (radic1198892 minus 1199092 + 119910 tan120572)2)5


119861119911 = 1205831198981015840 (1 minus (32) ((1199092 + 1199102) (1199092 + 1199102 + (radic1198892 minus 1199092 + 119910 tan120572)2)))2120587(radic1199092 + 1199102 + (radic1198892 minus 1199092 + 119910 tan120572)2)3

(18)

where the magnetic moment of the magnetic dipole is 1198981015840 =119881120594119898119867 The volume of casing element 119875 is 119881 The magneticsusceptibility of the casing is 120594119898 and casing diameter is C

1198981015840 = 119881120594119898radic1198612119875119909 + 1198612119875119910 + 11986121198751199111205831 (19)

During the calculation process the initial magnetic fluxdensity of point 119875 can be calculated by using formula(14) Subsequently the magnetic moment of the magneticdipole can be calculated using formula (19) Ultimatelythe triaxial magnetometer detects the magnetic flux den-sity of point 119875 which can be calculated using formulae(18)


In summary the triaxial magnetometer detects the mag-netic flux density of the magnetizing casing which is com-posed of a large number of magnetic dipole units when allthe other parameters except the variable quantity 119910 are allknown quantities Therefore 119889 can be calculated by reversededuction on the condition that the magnetic flux density isknown through measurements using the following

119861119874119909 = 119873sum119894=1

119861119875119909119894119861119874119910 = 119873sum

119894=1

119861119875119910119894119861119874119911 = 119873sum

119894=1

119861119875119911119894(20)

5 New Segmentation Strategies

A new variable section segmentation strategy is proposed inFigure 5 It is assumed that the distance between a magneticdipole unit of the casing model and the measuring point is119889119899 In the new section segmentation the parameter 119897119899 thelength of the magnetic dipole unit is determined by 119889119899 Theintersection angle between 1198891 and casing axis is 120573 (120573 = 120572 +1205872) The relationship between 119897119899 and 119889119899 can be expressed as[21ndash25]

120582 = 119889119899119897119899 (21)

where 120582 = 10 is a ratio coefficient [25] It is ensured that thesegmented unit is small enough to be regarded as a magneticdipole without error

If there is a measuring point around the casing model wecan determine its horizontal point on the casing Then themodel is divided from this point the aim of the segmentationstrategy is obtaining the formula of one unit volume 119881119899 andit can be expressed as

119881119899 = 120587119897119899120575 (119862 minus 120575) (22)

To determine 119881119899 we need to obtain a formula for thevariable 119897119899

First 1198971 is calculated by 1198971 = 1198891120582 (where 1198891 = 119889)Based on the Cosine theorem we can obtain the followingexpression

(1198971 + 11989722 )2 + 11988921 minus (1198971 + 1198972) 1198891 cos120573 = 11988922 (23)

Combined with formulae (21) and (23) the relationshipbetween 1198971 and 1198972 can be deduced using the following

1198972 = 1205822 + 14 minus 120582 cos1205731205822 minus 14 1198971 (24)

Measure point

B

A

l3

l2

l1 d1

d2

d3

Figure 5 The variable section segmentation strategy for the casing

Second two representations are shown below

119861 = 119897119899 + 119897119899minus12 119860 = 119899minus1sum119894=2

119897119894 + 119897119894minus12 (25)

We can also deduce two expressions based on the Cosinetheorem

(119861 + 119860)2 + 11988921 minus 21198891 (119861 + 119860) cos120573 = 11988921198991198602 + 11988921 minus 21198891119860 cos120573 = 1198892119899minus1 (26)

Finally by combining these expressionswith formula (21)the expression of 119897119899 can be deduced using the following

119897119899 = 119860 + (1205822 + 14) 119897119899minus1 minus 1198891 cos1205731205822 minus 14 (27)

From the expression of 119897119899 we can see that the volumesof the divided units in the new segmentation method aredifferent The unit nearer to the measuring point is smallerand the unit farther from the measuring point is largerExpression (27) is substituted into expression (21) and thenit can be used to calculate the magnetic moment

6 Experimental Results

During the simulation it is assumed that 1205830 = 4120587times10minus7Hm119898 = 10Asdotm2 ℎ = 1m 120572 = 0∘ 119909 = 0 and the axesof the probe tube and the casing are on the same planeAs the probe tube works it turns with a rotation speed oftens to hundreds of revolutions per minute The drill bit canonly advance several millimeters to tens of millimeters per


times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m

Figure 6 Magnetic flux density simulation for casing

minute Compared with the rotation rate the drilling footageper minute is much smaller Therefore it can be assumedthat the probe tube stays approximately at the same pointrelative to adjacent casing when it makes a single turn Therelative distance between probe tube and casing is set to be1m 2m 3m or 4m and the signal measured by the triaxialmagnetometer is simulated by utilizing formula (14) Figure 6indicates that the triaxial magnetometer detects the strongestmagnetic flux density when the axis of the magnetic sourcepoints directly to the casing The magnetic flux density ofthe casingrsquos magnetic field is of a symmetrical distributionaround 119910 = 0 the peak values of magnetic flux density aremainly concentrated in the vicinity of 119910 = plusmn1m and themagnetic flux density decreases with increasing 119910 Moreoverthe magnetic flux density is weaker with increasing distancebetween the adjacent wells

7 Discussion

Based on the calculations of the active anticollision methodthe influence factors of the measurement results mainlyinclude the distance between the two magnetic sources (2ℎ)in the probe tube the magnetic moment of the magneticsources (119898) the relative permeability of the casing (1205831) thediameter of the casing (119862) and the intersection angle (120572)

between the drilling well and the existing well It is assumedthat the formation is uniformly distributed and isotropicthen in the absence of any influence from nonferromagneticminerals in the surrounding stratum the various influencefactors of the measurement results can be analyzed

71 Influence of Magnetic Source on Measurement ResultsThe magnetic source is the key component of the magneticanticollision ranging system for adjacent wells its parametersdirectly affect the range and accuracy of the measured dis-tancesThemain parameters that influence the measurementaccuracy include the distance (2ℎ) between the twomagneticsources and the magnetic moment (119898) of the magneticsources By choosing 1205830 = 4120587 times 10minus7Hm 1205831 = 1000 120572 = 0∘119862 = 127mm 119898 = 10Asdotm2 and different values of 2ℎthe measurement results shown in Figure 7 indicate that thedistance between the casing and the probe tube vary in therange of 04sim2m

Figure 7 illustrates that magnetic flux density of theprobe tube detection increases with increasing 2ℎ when themagnetic moment is invariant However its variability issmall and may be ignored under the condition of 2ℎ gt12m Therefore considering the length of the probe tubeitself the maximum magnetic flux density of the probe tubeis observed under the condition of 2ℎ = 12m Influence of


times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)

Figure 7 Influence of the distance between double magneticsources on measurement results

times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)

Figure 8 Influence of magnetic moment on measurement results

the different values of the magnetic moment of the magneticsources on the magnetic flux density measurement resultsis shown in Figure 8 for 2ℎ = 12m Figure 8 indicatesthat the magnetic flux density detected by the probe tubeincreases with increasing magnetic moment when 2ℎ isinvariantTherefore we should choosemagnetic sources withlarge magnetic moments when designing the probe tubersquosstructure

72 Influence of the Casing on Measurement Results Thestrata surrounding the probe tube are less affected by themagnetic flux density generated by the magnetic sourcesbut the adjacent wellrsquos casing with high permeability isgreatly affected by the magnetic flux density generated by themagnetic sources Supposing 2ℎ= 12m120572 = 0∘119862 = 127mm119898 = 10Asdotm2 and different values of 1205831 the measurementresults are produced as shown in Figure 9 in which the

times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)

Figure 9 Influence of the relative magnetic conductivity of thecasing on measurement results

times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)

Figure 10 Influence of the diameter of casing on measurementresults

distance between the casing and the probe tube ranges within04sim2m

Figure 9 illustrates that themagnetic flux density detectedby the probe tube increases with the increasing the relativepermeability of the casing when the parameters of themagnetic source are invariantThe relative permeability of thecasing is determined by the casingrsquos properties Therefore ifthe relative permeability of the adjacent casing is measuredbefore calculating the distance between the adjacent wellsthen a more accurate relative permeability can be obtainedto ensure the accuracy of the magnetic anticollision rangingsystem

When the relative permeability of the casing is 1205831 = 1000different values are used for the diameter of the casing (C) andthe distance (d) between the casing and the probe tube rangeswithin 04sim2m the measurement results of the simulationare shown in Figure 10


times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)

Figure 11 Influence of the intersection angle 120572 on measurementresults

Figure 10 indicates that themagnetic flux density detectedby the probe tube increases as the diameter of the casingincreases when the relative permeability of the casing isinvariant and the risk of colliding with the larger diametercasing is increased under the same distance between adjacentwells

73 Influence of the Intersection Angle 120572 on MeasurementResults In the cluster wells there is an intersection angle (120572)between the drilling well and the target well Assuming that2ℎ = 12m 119898 = 10Asdotm2 1205831 = 1000 and 119862 = 1270mmwhen 120572 changes from 0∘ to 90∘ the distance between thecasing and the probe tube ranges within 04sim2m within themeasurement results simulation which produces the resultsshown in Figure 11

Figure 11 indicates that the change in the magnetic fluxdensity detected by the probe tube is reduced to half of theoriginal value when the intersection angle 120572 changes from0∘ to 50∘ In addition there is no correlation between themagnetic flux density and 120572 when 120572 gt 50∘ According to theanalysis this is because when the angle between the drillingwell and the adjacent well is too large the magnetic fieldgenerated by the magnetic source farther from the casingwithin the probe is too weak at the casing It is equivalentto the scenario in which only one magnetic source in theprobe tube generates the magnetization of the casing andthe distribution of the flux density changes resulting in thedistortion of the curve Therefore it is necessary to avoid theapplication of this method under conditions with 120572 gt 50∘8 Conclusions

Themagnetic anticollision system for cluster well is a kind ofmagnetic guidance tool that can determine the measurementdistance while drilling in real time The principle of themethod is based on detecting the magnetic flux density ofthe magnetizing casing in which the two magnetic sourcesare parallel to each other with opposite magnetic fields and

installed at two ends of the built-in tube probe First with therotation of the probe tube the triaxial magnetometer detectsthe oscillation of the signal from the casing Then the signalis transmitted to the ground by themud pulse Ultimately thedistances and azimuth of adjacent wells are calculated by dataanalysis software

The radii of the magnetic sources are far less than theborehole distance so the magnetic sources can be regardedas magnetic dipoles The magnetic flux density distributionformulae of the magnetic dipole are deduced by the magneticflux density calculation method for a magnetic dipole Thesimulation results indicate that a triaxial magnetometer candetect the maximum orminimum value of the magnetic fieldintensity when the axis of the magnetic source points directlyto the casing

The magnetic flux density detected by the probe tubeincreases with increasing distance between the double mag-netic sources when its distances exceeds 12m there is nosignificant change in the magnetic flux density detected Themagnetic flux density of the probe is positively related tothe magnetic moment of the magnetic sources the relativepermeability of the casing and the diameter of the adjacentwell casing pipe so that it can reach farther measuringdistances and higher ranging accuracies by increasing theseparametersrsquo values Accurate measurement results can beobtained by the method of adjacent well location under thecondition of 120572 lt 50∘ This method can provide theoreticalsupport and an experimental basis for the development of theactive magnetic guidance ranging for cluster wells

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the financial supportof National Science and Technology Major Project ldquoNewTechnologies of Designing and Controlling for the Com-plex Structure Wells and the Cluster Wells (Grant no2017ZX05009-003)rdquo

References

[1] D L Gao and B B Diao ldquoDevelopment of the magneticguidance drilling technique in complex well engineeringrdquoPetroleum drilling techniques vol 44 no 5 pp 1ndash9 2016

[2] D Gao B Diao Z Wu and Y Zhu ldquoResearch into magneticguidance technology for directional drilling in SAGD horizon-tal wellsrdquo Petroleum Science vol 10 no 4 pp 500ndash506 2013

[3] A-M Al-Bahlani and T Babadagli ldquoSAGD laboratory experi-mental and numerical simulation studies A review of currentstatus and future issuesrdquo Journal of Petroleum Science andEngineering vol 68 no 3-4 pp 135ndash150 2009


[4] D A Wood ldquoDrilling and borehole techniques relevant tonatural gas exploration and development A collection ofpublished research (2009-2015)rdquo Journal of Natural Gas Scienceand Engineering vol 26 pp 396ndash408 2015

[5] B Diao and D Gao ldquoA magnet ranging calculation method forsteerable drilling in build-up sections of twin parallel horizontalwellsrdquo Journal of Natural Gas Science and Engineering vol 27pp 1702ndash1709 2015

[6] B Diao and D Gao ldquoStudy on a ranging system based on dualsolenoid assemblies for determining the relative position of twoadjacent wellsrdquo CMES Computer Modeling in Engineering ampSciences vol 90 no 1 pp 77ndash90 2013

[7] B B Diao D L Gao and Z YWu ldquoMagnet ranging calculationmethod of twin parallel horizontal wells steerable drillingrdquoJournal of China university of petroleum vol 35 no 6 pp 71ndash75 2011

[8] C C Finlay S Maus C D Beggan et al ldquoInternationalGeomagnetic Reference Field The eleventh generationrdquo Geo-physical Journal International vol 183 no 3 pp 1216ndash1230 2010

[9] Z Guan Y Liu and Y Shi ldquoProblems and developing directionof anti-collision technology in the dense well pattern areardquo inProceedings of the Procedia engineering vol 7 pp 304ndash311 2010

[10] T Liu and B Wang ldquoStudy of magnetic ranging technologyin horizontal directional drillingrdquo Sensors and Actuators APhysical vol 171 no 2 pp 186ndash190 2011

[11] G Liu Q Yang Z Dong B He and Z Geng ldquoA drill bit vibra-tion anti-collision monitoring system and field experimentrdquoNatural Gas Industry vol 33 no 6 pp 66ndash70 2013

[12] M Marchetti V Sapia and A Settimi ldquoMagnetic anomalies ofsteel drums A review of the literature and research results of theINGVrdquo Annals of Geophysics vol 56 no 1 2013

[13] S D Billings C Pasion S Walker and L Beran ldquoMagneticmodels of unexploded ordnancerdquo IEEE Transactions on Geo-science and Remote Sensing vol 44 no 8 pp 2115ndash2123 2006

[14] Z Wu D Gao and B Diao ldquoAn investigation of electromag-netic anti-collision real-time measurement for drilling clusterwellsrdquo Journal of Natural Gas Science and Engineering vol 23pp 346ndash355 2015

[15] W Yang C Hu M Li M Q-H Meng and S Song ldquoAnew tracking system for three magnetic objectivesrdquo IEEETransactions on Magnetics vol 46 no 12 pp 4023ndash4029 2010

[16] Z Y Zhang C H Xiao W Chen and G Zhou ldquoExperimentresearch of magnetic dipole model applicability for a magneticobjectrdquo Journal of basic Science and engineering vol 18 no 5pp 862ndash868 2010

[17] J Simpson J Lane C Immer and R Youngquist ldquoSimpleanalytic expressions for the magnetic field of a circular currentlooprdquo NASA technical document collection document pp 1ndash32001

[18] V Sanchez L Yaoguo M N Nabighian and D L WrightldquoNumerical modeling of higher order magnetic moments inUXO discriminationrdquo IEEE Transactions on Geoscience andRemote Sensing vol 46 no 9 pp 2568ndash2583 2008

[19] B Tu Li DS and EH Lin ldquoAnalysis of drilling parallelhorizontal twin wells rotating magnetic beacons magnetic fieldstrength size in SAGDrdquo in Proceedings of the PIERS Proceedingsvol 7 pp 5ndash8 2010

[20] D Wang and D Gao ldquoStudy of magnetic vector guide systemin tubular magnet source spacerdquo Shiyou XuebaoActa PetroleiSinica vol 29 no 4 pp 608ndash611 2008

[21] S Feng D Liu X Cheng H Fang and C Li ldquoA new segmenta-tion strategy for processing magnetic anomaly detection dataof shallow depth ferromagnetic pipelinerdquo Journal of AppliedGeophysics vol 139 pp 65ndash72 2017

[22] Z Guo D Liu and Y Luo ldquoError analysis of magnetic fieldcalculation using magnetic dipole based on circular currentrdquoInternational Journal of Applied Mathematics and Statistics vol51 no 24 pp 121ndash130 2013

[23] Z-Y Guo D-J Liu Q Pan and Y-Y Zhang ldquoForward model-ing of total magnetic anomaly over a pseudo-2D undergroundferromagnetic pipelinerdquo Journal of Applied Geophysics vol 113pp 14ndash30 2015

[24] Z Guo D Liu Q Pan Y Zhang Y Li and Z Wang ldquoVerticalmagnetic field and its analytic signal applicability in oil fieldunderground pipeline detectionrdquo Journal of Geophysics andEngineering vol 12 no 3 article no 340 pp 340ndash350 2015

[25] Q Pan D-J Liu Z-Y Guo H-F Fang andM-Q Feng ldquoMag-netic anomaly inversion using magnetic dipole reconstructionbased on the pipeline section segmentation methodrdquo Journal ofGeophysics and Engineering vol 13 no 3 pp 242ndash258 2016

Hindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

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Decision SciencesAdvances in


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Submit your manuscripts atwwwhindawicom


Service (RMRS) system [4] the Wellspot Tool (WT) [5] andthe Single Wire Guidance (SWG) [6] tool can detect thedistance between the drilling well and the existing well toavoid collision [1ndash6] MGT and RMRS are applied in adjacenttwin parallel Steam Assisted Gravity Drainage (SAGD) wellsthat guide the drilling of an injection well to ensure thatit is parallel to the production well [2 3 7ndash9] However itrequires improvement if it is to be applied to the anticollisionof cluster wells There are two primary disadvantages ofthe MGT and RMRS methods First they are difficult tooperate as a probe tube must be placed in the existing wellto guide the detection of drill bit Second the monitoringworkload of the anticollision system is quite heavy WT isnot suitable for the anticollision of cluster wells since thedetection process requires pulling out the drilling tools TheSWG tool overcomes the disadvantages of the MGT andRMRS [1 4] namely that neither reduces the workloads norrequires pulling out the drilling tool to guide the detectiondevice However there are three primary disadvantages ofthe SWG tool First a cable must be placed down the offsetwell which affects the normal production process of theexisting well Additionally a large amount of equipmentmust be utilized in order to run this system Lastly thetechnology of the SWG tool is monopolized by foreigncompanies and the service cost is expensive so domesticcompanies rarely use SWG to reduce the cost of a singlewell

With cluster wells being used in more and more appli-cations in land and sea environments the borehole distancehas been reduced from 2m times 2m to 15m times 17m [2 4] andinfilling adjustment wells are usually added to the existingwell pattern The traditional anticollision scanning methodis insufficient for cluster wellsrsquo anticollision Therefore theactive magnetic guidance ranging method is proposed toavoid borehole collision This method is based on detectingmagnetic induced intensity of the casing in the existing wellit detects the borehole distance during the drilling process inreal time without interrupting drilling or ever requiring anyother equipment to be placed in the existingwellThemethodbecomes practically significant in providing a theoreticalbasis for the research and development of active anticollisiondevices for cluster wells

2 Principle of Active MagneticGuidance Ranging

The signal sources or the probe tube must be mounted inthe existing well to measure the distance between drillingwell and offset wells for MGT RMRS SWG and otherelectromagnetic guidance tools [10ndash13] To solve the aboveproblems the active magnetic anticollision ranging methodof clusterwells is proposed based on the fact that themagneticpoles of double magnetic sources are parallel to each otherwith oppositemagneticmoments (expressed as red and greenlines in Figure 1) It does not interrupt the drilling of theproduction well nor does it require any other equipment tobe placed in the offset well yet it detects the borehole distancein real time during the drilling process This principle isshown in Figure 1

X

Geographic north

Geographic eastY

Z

Casing Pipe

Probe Tube

O

Magnetic source A

Magnetic source BSensor

Figure 1 Principle of active magnetic guidance ranging for clusterwells

Therefore the total magnetic induction intensity asdetected by a triaxial magnetometer is 0 when there is nocasing around the probe tube or when the casing is faraway from the probe tube [14] When there is casing aroundthe probe tube the triaxial magnetometer data representthe sum of the geomagnetic field the magnetic field oftwo magnetic sources and the magnetic flux density ofthe magnetizing casing which generates a magnetic fieldalong its axial direction (expressed as blue lines) As themagnetic fields of magnetic sources vary with the rotationof the drill rod the alternating magnetic field frequency ofthe casing varies along with the rotation rate of magneticsources

Based on the above described principle the relativedistance and orientation between the drilling well and theexisting well are calculated using the detection data of atriaxial magnetometer and the well trajectory of the drillingwell is adjusted appropriately to prevent collisions with theexisting well

3 Calculation Method ofAdjacent Well Distance

The calculation method of adjacent well distance is theanticollision core technology of magnetic guidance Underthe coordinate system shown in Figure 1 the casing stretchesvertically downward along the 119885-axis and the 119883-axis pointsto geographic north The outer diameter and thickness of the


X

Z

Y

I

M

O

P

r

K

R




119903 = 119909sin 120579 cos120593 = 119910

sin 120579 sin120593 = 119911cos 120579



119889B = 1205834120587 119868119889l times a1198863 (2)




(3)


119861119909 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 2120579 cos1205931199032 + 1198772

119861119910 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 2120579 sin1205931199032 + 1198772

119861119911 = 1205832120587 119898(radic1199032 + 1198772)3 (1 minus 31199032sin21205792 (1199032 + 1198772)) (4)


119861119909 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 21205791199032 + 1198772

119861119910 = 0119861119911 = 1205832120587 119898(radic1199032 + 1198772)3 (1 minus 31199032sin21205792 (1199032 + 1198772))

(5)




times104

2

25

3

35

4

B (n

T)

0 215 (rad)

05










O

Y

X

ZP

A

E

B

C

D

d

h

h

r

N

M

r1

r2

2

1





(7)



119861119862119909 = 0119861119862119910 = 3120583119898812058711990331 sin 21205791119861119862119911 = 120583119898212058711990331 (1 minus 32 sin21205791)

(8)



1198611015840119862119909 = 01198611015840119862119910 = 31205831119898812058711990331 sin 212057911198611015840119862119911 = 1205831119898212058711990331 (1 minus 32 sin21205791)

(9)



(10)




(11)



(12)





119910119909

(13)


119861119875119909 = 312058311198981199094120587radic1199092 + 1199102[[[[[[



]]]]]]

119861119875119910 = 312058311198981199104120587radic1199092 + 1199102[[[[[[



]]]]]]





(14)



119861119862119863119909 = 119861119862119909 + 119861119863119909 = 0119861119862119863119910 = 119861119862119910 + 119861119863119910 = 0119861119862119863119911 = 119861119862119911 + 119861119863119911 = 0(15)





(16)

where



119910119909 (17)





(18)


1198981015840 = 119881120594119898radic1198612119875119909 + 1198612119875119910 + 11986121198751199111205831 (19)




119861119874119909 = 119873sum119894=1

119861119875119909119894119861119874119910 = 119873sum

119894=1

119861119875119910119894119861119874119911 = 119873sum

119894=1

119861119875119911119894(20)



120582 = 119889119899119897119899 (21)



119881119899 = 120587119897119899120575 (119862 minus 120575) (22)



(1198971 + 11989722 )2 + 11988921 minus (1198971 + 1198972) 1198891 cos120573 = 11988922 (23)


1198972 = 1205822 + 14 minus 120582 cos1205731205822 minus 14 1198971 (24)

Measure point

B

A

l3

l2

l1 d1

d2

d3



119861 = 119897119899 + 119897119899minus12 119860 = 119899minus1sum119894=2

119897119894 + 119897119894minus12 (25)









times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m



7 Discussion






times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









X

Z

Y

I

M

O

P

r

K

R




119903 = 119909sin 120579 cos120593 = 119910

sin 120579 sin120593 = 119911cos 120579



119889B = 1205834120587 119868119889l times a1198863 (2)




(3)


119861119909 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 2120579 cos1205931199032 + 1198772

119861119910 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 2120579 sin1205931199032 + 1198772

119861119911 = 1205832120587 119898(radic1199032 + 1198772)3 (1 minus 31199032sin21205792 (1199032 + 1198772)) (4)


119861119909 = 31205838120587 119898(radic1199032 + 1198772)3 1199032 sin 21205791199032 + 1198772

119861119910 = 0119861119911 = 1205832120587 119898(radic1199032 + 1198772)3 (1 minus 31199032sin21205792 (1199032 + 1198772))

(5)




times104

2

25

3

35

4

B (n

T)

0 215 (rad)

05










O

Y

X

ZP

A

E

B

C

D

d

h

h

r

N

M

r1

r2

2

1





(7)



119861119862119909 = 0119861119862119910 = 3120583119898812058711990331 sin 21205791119861119862119911 = 120583119898212058711990331 (1 minus 32 sin21205791)

(8)



1198611015840119862119909 = 01198611015840119862119910 = 31205831119898812058711990331 sin 212057911198611015840119862119911 = 1205831119898212058711990331 (1 minus 32 sin21205791)

(9)



(10)




(11)



(12)





119910119909

(13)


119861119875119909 = 312058311198981199094120587radic1199092 + 1199102[[[[[[



]]]]]]

119861119875119910 = 312058311198981199104120587radic1199092 + 1199102[[[[[[



]]]]]]





(14)



119861119862119863119909 = 119861119862119909 + 119861119863119909 = 0119861119862119863119910 = 119861119862119910 + 119861119863119910 = 0119861119862119863119911 = 119861119862119911 + 119861119863119911 = 0(15)





(16)

where



119910119909 (17)





(18)


1198981015840 = 119881120594119898radic1198612119875119909 + 1198612119875119910 + 11986121198751199111205831 (19)




119861119874119909 = 119873sum119894=1

119861119875119909119894119861119874119910 = 119873sum

119894=1

119861119875119910119894119861119874119911 = 119873sum

119894=1

119861119875119911119894(20)



120582 = 119889119899119897119899 (21)



119881119899 = 120587119897119899120575 (119862 minus 120575) (22)



(1198971 + 11989722 )2 + 11988921 minus (1198971 + 1198972) 1198891 cos120573 = 11988922 (23)


1198972 = 1205822 + 14 minus 120582 cos1205731205822 minus 14 1198971 (24)

Measure point

B

A

l3

l2

l1 d1

d2

d3



119861 = 119897119899 + 119897119899minus12 119860 = 119899minus1sum119894=2

119897119894 + 119897119894minus12 (25)









times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m



7 Discussion






times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









times104

2

25

3

35

4

B (n

T)

0 215 (rad)

05










O

Y

X

ZP

A

E

B

C

D

d

h

h

r

N

M

r1

r2

2

1





(7)



119861119862119909 = 0119861119862119910 = 3120583119898812058711990331 sin 21205791119861119862119911 = 120583119898212058711990331 (1 minus 32 sin21205791)

(8)



1198611015840119862119909 = 01198611015840119862119910 = 31205831119898812058711990331 sin 212057911198611015840119862119911 = 1205831119898212058711990331 (1 minus 32 sin21205791)

(9)



(10)




(11)



(12)





119910119909

(13)


119861119875119909 = 312058311198981199094120587radic1199092 + 1199102[[[[[[



]]]]]]

119861119875119910 = 312058311198981199104120587radic1199092 + 1199102[[[[[[



]]]]]]





(14)



119861119862119863119909 = 119861119862119909 + 119861119863119909 = 0119861119862119863119910 = 119861119862119910 + 119861119863119910 = 0119861119862119863119911 = 119861119862119911 + 119861119863119911 = 0(15)





(16)

where



119910119909 (17)





(18)


1198981015840 = 119881120594119898radic1198612119875119909 + 1198612119875119910 + 11986121198751199111205831 (19)




119861119874119909 = 119873sum119894=1

119861119875119909119894119861119874119910 = 119873sum

119894=1

119861119875119910119894119861119874119911 = 119873sum

119894=1

119861119875119911119894(20)



120582 = 119889119899119897119899 (21)



119881119899 = 120587119897119899120575 (119862 minus 120575) (22)



(1198971 + 11989722 )2 + 11988921 minus (1198971 + 1198972) 1198891 cos120573 = 11988922 (23)


1198972 = 1205822 + 14 minus 120582 cos1205731205822 minus 14 1198971 (24)

Measure point

B

A

l3

l2

l1 d1

d2

d3



119861 = 119897119899 + 119897119899minus12 119860 = 119899minus1sum119894=2

119897119894 + 119897119894minus12 (25)









times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m



7 Discussion






times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018










1198611015840119862119909 = 01198611015840119862119910 = 31205831119898812058711990331 sin 212057911198611015840119862119911 = 1205831119898212058711990331 (1 minus 32 sin21205791)

(9)



(10)




(11)



(12)





119910119909

(13)


119861119875119909 = 312058311198981199094120587radic1199092 + 1199102[[[[[[



]]]]]]

119861119875119910 = 312058311198981199104120587radic1199092 + 1199102[[[[[[



]]]]]]





(14)



119861119862119863119909 = 119861119862119909 + 119861119863119909 = 0119861119862119863119910 = 119861119862119910 + 119861119863119910 = 0119861119862119863119911 = 119861119862119911 + 119861119863119911 = 0(15)





(16)

where



119910119909 (17)





(18)


1198981015840 = 119881120594119898radic1198612119875119909 + 1198612119875119910 + 11986121198751199111205831 (19)




119861119874119909 = 119873sum119894=1

119861119875119909119894119861119874119910 = 119873sum

119894=1

119861119875119910119894119861119874119911 = 119873sum

119894=1

119861119875119911119894(20)



120582 = 119889119899119897119899 (21)



119881119899 = 120587119897119899120575 (119862 minus 120575) (22)



(1198971 + 11989722 )2 + 11988921 minus (1198971 + 1198972) 1198891 cos120573 = 11988922 (23)


1198972 = 1205822 + 14 minus 120582 cos1205731205822 minus 14 1198971 (24)

Measure point

B

A

l3

l2

l1 d1

d2

d3



119861 = 119897119899 + 119897119899minus12 119860 = 119899minus1sum119894=2

119897119894 + 119897119894minus12 (25)









times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m



7 Discussion






times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018










(14)



119861119862119863119909 = 119861119862119909 + 119861119863119909 = 0119861119862119863119910 = 119861119862119910 + 119861119863119910 = 0119861119862119863119911 = 119861119862119911 + 119861119863119911 = 0(15)





(16)

where



119910119909 (17)





(18)


1198981015840 = 119881120594119898radic1198612119875119909 + 1198612119875119910 + 11986121198751199111205831 (19)




119861119874119909 = 119873sum119894=1

119861119875119909119894119861119874119910 = 119873sum

119894=1

119861119875119910119894119861119874119911 = 119873sum

119894=1

119861119875119911119894(20)



120582 = 119889119899119897119899 (21)



119881119899 = 120587119897119899120575 (119862 minus 120575) (22)



(1198971 + 11989722 )2 + 11988921 minus (1198971 + 1198972) 1198891 cos120573 = 11988922 (23)


1198972 = 1205822 + 14 minus 120582 cos1205731205822 minus 14 1198971 (24)

Measure point

B

A

l3

l2

l1 d1

d2

d3



119861 = 119897119899 + 119897119899minus12 119860 = 119899minus1sum119894=2

119897119894 + 119897119894minus12 (25)









times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m



7 Discussion






times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018










119861119874119909 = 119873sum119894=1

119861119875119909119894119861119874119910 = 119873sum

119894=1

119861119875119910119894119861119874119911 = 119873sum

119894=1

119861119875119911119894(20)



120582 = 119889119899119897119899 (21)



119881119899 = 120587119897119899120575 (119862 minus 120575) (22)



(1198971 + 11989722 )2 + 11988921 minus (1198971 + 1198972) 1198891 cos120573 = 11988922 (23)


1198972 = 1205822 + 14 minus 120582 cos1205731205822 minus 14 1198971 (24)

Measure point

B

A

l3

l2

l1 d1

d2

d3



119861 = 119897119899 + 119897119899minus12 119860 = 119899minus1sum119894=2

119897119894 + 119897119894minus12 (25)









times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m



7 Discussion






times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









times107

d = 1 m

0 5 10minus5minus10

y (m)

0

05

1

15

2

25

B(n

T)times106

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 2 m

times105

0

05

1

15

2

25

B(n

T)

0 5 10minus5minus10

y (m)

d = 4 m

times105

0 5 10minus5minus10

y (m)

0

2

4

6

B(n

T)

d = 3 m



7 Discussion






times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









times106

2h = 08 m2h = 12 m2h = 10 m

1 206 120804 1814 16

Distance (m)

04

06

08

1

12

14

16

18

B(n

T)


times106

m = 10 middotＧ2

m = 20 middotＧ2

m = 30middotＧ2

05

1

15

2

25

B(n

T)

1 206 12 14 16 180804

Distance (m)




times106

1 = 2000

1 = 1000

1 = 500

1 206 12 14 16 180804

Distance (m)

0

05

1

15

2

25

3

35

4

B(n

T)


times106

C = 24448 mmC = 17780 mmC = 12700 mm

0

1

2

3

4

5

6

7

8B

(nT)

1 206 12 14 16 180804

Distance (m)






times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









times106

d = 10 md = 05 m

0

05

1

15

2

25

3

35

B(n

T)

706030 40 50 80 9010 200

(∘)









Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018






































Journal of












Journal of







Volume 2018






Volume 2018








anticollision method of active magnetic guidance ranging...

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