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Math. Model.Nat. Phenom. Vol. 10,No. ?,2015,pp. ??-?? Magnetic Flux Leakage Method: Large-Scale Approximation 5 1 0 2 AnastasiyaV.Pimenovaa,1, DenisS. Goldobina,b,c, Jeremy Levesleyb, n Andrey O. Ivantsova, Peter Elkingtond, MarkBacciarellid a J 3 a InstituteofContinuousMediaMechanics,UB RAS, Perm 614013,Russia 1 b Department ofMathematics,UniversityofLeicester, LeicesterLE1 7RH, UK ] c DepartmentofTheoreticalPhysics,Perm StateUniversity,Perm 614990,Russia n a d Weatherford, East Leake, LoughboroughLE126JX, UK - a t a d Abstract. Weconsidertheapplicationofthemagneticfluxleakage(MFL)methodtothedetection . s ofdefects inferromagnetic(steel)tubulars. Theproblemsetupcorrespondsto thecases wherethe c i distance from the casing and the point where the magnetic field is measured is small compared s y to the curvature radius of the undamaged casing and the scale of inhomogeneity of the magnetic h fieldinthedefect-freecase. Mathematicallythiscorrespondstotheplanarferromagneticlayerina p [ uniform magneticfield oriented along this layer. Defects in thelayer surface result in a strong de- 1 formationofthemagneticfield, whichprovidesopportunitiesforthereconstructionofthesurface v profile from measurements of the magneticfield. We deal with large-scale defects whose depth is 9 7 small compared to their longitudinal sizes—these being typical of corrosive damage. Within the 0 framework of large-scale approximation, analytical relations between the casing thickness profile 3 and themeasured magneticfield can bederived. 0 . 1 0 Keywords: magneticflux leakage, corrosivedefects, large-scaleapproximation 5 AMSsubject classification: 78A30,78M34,78A55 1 : v i X r a 1. Introduction The magnetic flux leakage (MFL) method is a powerful tool for non-distructive inspection of the integrityofferromagneticcasings[1, 2,3, 4, 5,6, 7]or,moregenerally,fordeterminingtheshape of ferromagnetic objects. The basic idea of the method is to reconstruct the shape features of the 1Correspondingauthor.E-mail:[email protected] 1 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory 2 1 3 Figure 1: Mirror-symmetric ferromagnetic layer of non-uniform thickness in the magnetic field H and thecoordinateframe inspected ferromagnetic object from thedeformation of themagneticfield. The conventionalway of applying the MFL method is (i) recognition of typical patterns in the magnetic field measure- ment data, (ii) identification of typical defects of the casing corresponding to these patterns, and (iii) evaluation of geometrical parameters of the defects identified from the patterns. Such an approach faces obvious problems when one deals with complex-shape defects such as metal losses due to corrosion. Although the problem of the reconstruction of an arbitrary object shape from measurements of the magnetic field (all three components of the H~-field should be measured) on some surface in space near the object is mathematically well-posed and resolvable (numerically), in practice one encounters issues which make this approach generally inapplicable (which is the reasonforusingtheconventionalpracticeoutlinedabove). However,specificallyinthecasewhere complex-shape defects are most important—in the case of corrosion damage—there is an oppor- tunity for reconstruction of the arbitrary casing thickness profile based on an analytical approach. Thisanalyticalapproachispossibleowingtothemodelreduction—thelarge-scaleapproximation, whichassumesthedefectdepthandthecasingthicknesstobesmallcomparedtothedefectwidth, as it is typical for corrosion metal losses. The corrosion loss of metal can be non-large-scale near a weld, where corrosion rapidly advances along the contact interface. Otherwise, the large-scale approximationisreasonable. In this paper we consider the application of the MFL method to the detection of defects in ferromagnetic(e.g. steel) tubularsincludingwellborecasings. Ourtreatmentis focused on theac- curate reconstruction of casing thickness profiles, in contrast to the conventional approach which is therecognitionof magneticfield patterns correspondingto catalogued typical defects by means of neural networks or similar data analysis tools. The problem setup we use corresponds to measurements made with modern devices designed for the MFL inspection of wellbore casings (e.g., see [5]). In this setup the distance from the casing and the point where the magnetic field is measured is small compared to the curvature radius of the undamaged casing and the scale of in- homogeneityofthemagneticfield in thedefect-free case. Mathematically,this correspondsto the planar ferromagnetic layer in a uniform magnetic field oriented along this layer. Defects of the layersurfaceresultinastrongdeformationofthemagneticfield,whichprovidesopportunitiesfor thereconstructionofthesurface profilefromthemeasurementsofthemagneticfield. 2 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory 2. Analytical theory: Large-scale approximation We restrict the analytical treatment to the case of large-scale defects, i.e. defect length and width is large relative to the depth or the casing (metal layer) thickness. For this case we show that the magnetic field is sensitive to the casing thickness profile, but not to the inner and outer surface profiles independently. Hence, as a starting point, one can address the problem of a symmetric layer, onesurfaceofwhich isamirrorimageoftheother. 2.1. Mirror-symmetric layer Mathematical description oftheproblem Let us consider the ferromagnetic layer confined between two surfaces z = ζ(x,y) and z = −ζ(x,y), where the (x,y)-plane is the middle plane of the layer and the z-axis is orthogonal to it. The uniform external magnetic field H is applied along the x-axis. The system is sketched in Figure1. We adoptthefollowingassumptionsfortheproblem: 1. Thelayergeometryand fields possessthesymmetryproperty (z → −z); 2. The linear magnetisation law for both the ferromagnetic material and the material around it isgivenby: B~ = µ H~ , j = 1,2; j j j 3. The magnetic permeability of the surrounding material is small compared to that of the µ ferromagnet, 1 ≫ 1; µ 2 4. Surface defects are large-scale, which means the typical longitudinal size of defects L ≫ ζ and, therefore, |∇ ζ| ≪ 1. 2 (In the following subsections we will extend our consideration beyond restrictions (1) and (2).) Henceforth, the subscripts of fields and parameters indicate the corresponding domain (1: ferromagnet, 2: upper outer area, 3: lower outer area); for the gradient and Laplace operators, ∇ and 2 ∆ , theindex“2”indicatesthetwo-dimensionalversionsofthemcalculated with respect to xand 2 y coordinates only. The ranges of parameter values of practical interest are presented in Table 1 and areconsistentwiththeassumptionsmade. According toMaxwell’sequations,wehavethefollowingequationsystem ∇×H~ = 0, j ( ∇·B~j = 0, withboundaryconditions H~ = H~ , B = B 1τ 2τ 1n 2n 3 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory Table1: Reference valuesofreal systemparameters ratioofmagneticpermeabilities µ /µ : 100−1000 1 2 longitudinalscaleofdefects L: (10−20)ζ locationofmagneticsensors z : (2−8)ζ for the normal to the surface (subscript “n”) and tangential (“τ”) components of magnetic field, respectively. When the curl of a vector field is zero in a simply-connected domain, one can introduce the scalar potential for this field within this domain. We introduce the scalar potential Φ, H~ = −∇Φ, whichobeystheequation ∆Φ = 0, (2.1) j whiletheboundaryconditionsread Φ = Φ , (2.2) 1 2 ∂Φ ∂Φ µ 1 = µ 2 . (2.3) 1 2 ∂n ∂n Since the magnetic permeability of the ferromagnet is considerably larger than that of the surrounding material, to the leading order of approximation the flux of the magnetic field does notgo outsidetheboundariesoftheferromagnet,i.e. ζ ∂Φ ∇ − 1 dz = 0. (2.4) 2 ∂x (cid:18)Z−ζ(cid:18) (cid:19) (cid:19) In this case the normal derivativesof Φ on the boundary are equal to zero, and Eq.(2.3) takes the form ∂Φ 1 = 0. (2.5) ∂n For infinitely large scale inhomogeneitiesthe magneticH-field within the layer is the same as for the defect-free planar layer, H~ = H~. Hence, we can look for the correction to the uniform 1 field H~. Onecan writedowntheTaylorseriesforΦ withrespect to z 1 z2 z4 (0) (2) (4) Φ (x,y,z) = −Hx+Φ (x,y)+Φ (x,y) +Φ (x,y) +... 1 1 1 2! 1 4! (only even powers of z are present due to the symmetry z → −z). Substituting this series into Eq.(2.1)and renamingΦ(0)(x,y)as F(x,y),onefinds 1 z2 z4 Φ (x,y,z) = −Hx+F(x,y)−∆ F(x,y) +∆2F(x,y) −... . (2.6) 1 2 2! 2 4! 4 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory Notice,herethez2n-termisoftheorderofmagnitudeofF(ζ/L)2nandthusonlyfirstseveralterms can be important for the large-scale case. Since F vanishes for infinitely large scale of defects, it shouldbesmallcompared totheleadingterm ofΦ forfinitelarge scaleLbycontinuity,i.e. 1 ∇ F ≪ H . 2 (cid:12) (cid:12) On thesurfacez = ζ(x,y),Eq.(2.2)y(cid:12)ields (cid:12) 1 ζ4 Φ (z = ζ) = Φ = −Hx+F − ∆ Fζ2+O F . (2.7) 2 1 2 2 L4 (cid:18) (cid:19) 2.2. Two-dimensional case Let us consider the two-dimensional problem of a ferromagnetic layer uniform in the y-direction. Onecan seethat forthetwo-dimensionalcasetheintegralinEq.(2.4)can befound as ζ ∂Φ − 1 dz = const = 2ζ H, (2.8) 0 ∂x Z−ζ(cid:18) (cid:19) where ζ is the z-coordinate of the undamaged surface. This significantly simplifies the task and 0 makes it possible to solve the problem analytically. Substituting expression (2.6) into the latter equation,onecan seethat ∂F ζ ζ3 = H(1− )+O H . (2.9) ∂x ζ L2 0 (cid:18) (cid:19) Φ can beexpandedintoaseries nearthesurfaceζ; 2 ∂Φ 1 ∂2Φ Φ (ζ) = Φ (h)+ 2 (ζ −h)+ 2 (ζ −h)2 +... . (2.10) 2 2 ∂z 2 ∂z2 (cid:12)z=h (cid:12)z=h (cid:12) (cid:12) (cid:12) (cid:12) To calculate(∂2Φ2/∂z2)onecan empl(cid:12)oyEq.(2.1), (cid:12) ∂2Φ ∂2Φ 2 = − 2 . (2.11) ∂z2 ∂x2 (cid:12)z=h (cid:12)z=h (cid:12) (cid:12) (cid:12) (cid:12) Hence, Eq.(2.10)can berewrittenin theform (cid:12) (cid:12) 1∂2F ∂Φ 1 ∂2Φ (h−ζ)3 Φ (h) = −Hx+F − ζ2+ 2 (h−ζ)− 2 (h−ζ)2 +O F . 2 2 ∂x2 ∂z 2 ∂x2 L3 (cid:12)z=h (cid:12)z=h (cid:18) (cid:19) (cid:12) (cid:12) (2.12) (cid:12) (cid:12) Substituting Eq.(2.9) and differentiating(cid:12) the last equation with(cid:12)respect to x, one can evaluate the x-componentofthemagneticH-field measuredat theheighthabovethelayer; ζ ∂ ∂Φ ζ2 H | = H 0 + 2 (ζ −h) +O H . (2.13) x z=h ζ ∂x ∂z L2 (cid:18) (cid:12)z=h (cid:19) (cid:18) (cid:19) (cid:12) (cid:12) (cid:12) 5 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory Let us seek a series expansion for ζ = ζ + ζ + ... , where each term of the series is small 1 2 compared tothepreviousone. To theleadingorder ofaccuracy thelast equationyields H ζ = ζ . (2.14) 1 0 H | x z=h Substituting(2.14)intoEq.(2.13)and consideringH~ = −∇Φ, onecan obtain Hζ2 ∂ H ζ = − 0 −1 H . (2.15) 2 H2| ∂x H | z x z=h (cid:18) x z=h (cid:19) (cid:16) (cid:17) Finally,tothetermsoforder(Hζ2/L2) 0 H ζ ∂ H ζ ≈ ζ 1− 0 H −1 . (2.16) 0 z H | H | ∂x H | x z=h (cid:20) x z=h (cid:18) (cid:18) x z=h (cid:19)(cid:19)(cid:21) 2.3. Asymmetric ferromagnetic layer Inthissubsectionwearguethatforlarge-scaledefectsthecaseofanasymmetriclayerisequivalent to the case of mirror-symmetric layer. Let us consider the layer confined between surfaces z = 2 ζ (x,y) and z = −ζ (x,y) with ζ 6= ζ . It is convenient to introduce the middle surface z = 2 3 3 2 3 m ζ (x,y). m ζ −ζ 2 3 ζ = , m 2 and usethecoordinateframe x = x, z = z −ζ (x). m Then ∂ ∂e ∂ζm ∂ e ∂ ∂ = − and = . ∂x ∂z ∂x ∂z ∂z ∂z In thenewcoordinatesEq.(2.1)takes theform e e e e ∂2ζ ∂ζ ∂ ∂ζ 2 ∂2Φ m m m i ∆Φ = ∆Φ − +2 + . (2.17) i i ∂x2 ∂x ∂x ∂x ∂z2 (cid:18) (cid:19) (cid:18) (cid:19) Forlarge-scaledefectstheadeditionaltermsofEq.(2.17)affectsEqs.(2.4)and(2.10)onlyinhigh- e e e e e orderterms. Hence, Eq.(2.13)is notaffected by asymmetry. 2.4. Nonlinear magnetisation law Ferromagnetic materials are characterized by hysteresis and non-linearity of the magnetisation. Hysteresis is to be avoided in inspection applications as it leads to a loss of uniqueness in the solution of the profile-reconstruction problem. For this reason, strong constant magnets are used in practice, and the system operates under conditions close to magnetic saturation (see Figure 2). Closetosaturationhysteresisbecomesinsignificantwhereasthenon-linearitybecomespronounced 6 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory Figure2: SchematicdependenceofthemagneticfieldB onthemagneticH-fieldinferromagnetic material(black solidcurves)anditslinearizationin theworkingparameter range(red line) and influences applications of the MFL method [7]. Nonetheless, non-linear-magnetisation prob- lemscanbetreatedanalyticallyforlarge-scaledefectsaswell,becausethemagneticH-fieldwithin the ferromagnetic layer deviates slightly from H~. For brevity, we consider the case of a mirror- symmetriclayerinthissection. As ∇ × H~ = 0, we still can use substitution H~ = −∇Φ. For B~ = µ(H)H~, the equation ∇·B~ = 0 takes theform 1 ∂µ H2 ∇·H~ + H~ ·∇ = 0 . µH ∂H 2 Substitutingtheseries z2 z4 (0) (2) (4) Φ (x,z) = −Hx+Φ (x)+Φ (x) +Φ (x) +... 1 1 1 2! 1 4! intothelatterequationand collectingz-free terms, onefinds ∂2Φ(0) 1 ∂µ ∂ H2 1 +Φ(2) = − 1H +... . x2 1 µ H ∂H ∂x 2 1 WithH2 = (H−(∂/∂x)Φ(0)+...)2 = H2−2H(∂/∂x)Φ(0)+...;totheleadingorderofaccuracy, 1 1 onecan obtain ∂2Φ(0) Φ(2) = −(1+β) 1 +... , (2.18) 1 ∂x2 H ∂µ β ≡ − 1 . (2.19) µ (H) ∂H 1 (cid:12)H=H (cid:12) (cid:12) One can show that 0 < β < 1. Indeed, let us wr(cid:12)ite B = α(H + H0) for H next to H (see Figure2). Thenβ = H /(H+H )withpositiveH ;therefore,β ∈ (0,1). Onthehysteresisloop, 0 0 0 whichis outofthescopeofourstudy,β can bebeyondthisrange. 7 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory Eq.(2.2)holdsvalidfornonlinearmagnetisationand yieldsamodifiedversionofEq.(2.6): 1 ∂2Φ(0) Φ (z = ζ) = Φ = −Hx+Φ(0) − (1+β) 1 ζ2 +... . (2.20) 2 1 1 2 ∂x2 Since 0 < β < 1 for real ferromagnetic materials, the change of a coefficient ahead of the last term does not change its order. As was demonstrated for the case of a mirror-symmetric layer, any terms of this order do not affect the solution to the concerned accuracy. Therefore the nonlinearityofmagnetisationdoes notaffect theleadingorderof theequationswesuggestforthe profilereconstructionprocedureforlarge-scale defects. 2.5. Three-dimensional ferromagnetic layer For the case of the three-dimensional layer Eq.(2.4) can be solved only up to the gradient of an arbitrary harmonic function of x and y, which is not very helpful for our purposes; the problem requires asomewhatdifferent approach compared tothetwo-dimensionalcase. Let us denote the magnetic potential at the height ζ as Φ(0). One can write down the Taylor 0 2 series forΦ attheheightζ; 2 ∂Φ 1 ∂2Φ Φ | = Φ(0) + 2 (ζ −ζ )+ 2 (ζ −ζ )2 +... . (2.21) 2 z=ζ 2 ∂z 0 2 ∂z2 0 (cid:12)z=ζ0 (cid:12)z=ζ0 (cid:12) (cid:12) Taking into account the boundary(cid:12)condition Φ = Φ , one(cid:12)can equate Eq.(2.6) with the latter (cid:12) 1 2 (cid:12) equationand evaluatetheleading-ordertermofthemagneticpotentialwithinthelayer: ∂Φ ζ2 Φ (x,y,z) = Φ | + 2 (ζ −ζ )+O H . (2.22) 1 2 z=ζ0 ∂z 0 L (cid:12)z=ζ0 (cid:16) (cid:17) (cid:12) Substitutingthisseries intoEq.(2.4), onefinds(cid:12) (cid:12) ∇ ·(ζH~ +ζ(ζ −ζ )∇ H +ζ∇ ζH ) = 0, (2.23) 2 0 2 z 2 z where H~ = −∇Φ | is the magnetic field at the height ζ and H is its z-component. To 2 z=ζ0 0 z solve this equation numerically it is convenient to use the exponential representation of ζ = ζ e−σ(0)−σ(1)−.... To thefirst two orders Eq.(2.23)takes theform 0 −ζ∇ σ(0) ·H~ −ζ∇ σ(1) ·H~ +ζ∇ ·H~ +∇ [ζ(ζ −ζ )∇ H +ζ∇ ζH ] = 0. (2.24) 2 2 2 2 0 2 z 2 z Hence, ∇ σ(0) ·H~ = ∇ ·H~, (2.25) 2 2 ∇ σ(1) ·H~ = ζ (1−3e−σ(0))∇ σ(0) ·∇ H −(1−e−σ(0))∆ H 2 0 2 2 z 2 z (2.26) (cid:2) +2e−σ(0)(∇ σ(0))2H −e−σ(0)∆ σ(0)H . 2 z 2 z Notice,σ(0) isnotnecessarily small. (cid:3) WithEqs.(2.25)and(2.26)onecancalculatethelayerthicknessprofile2ζ = 2ζ exp(−σ(0)− 0 σ(1)−...)fromthemagneticfieldH~ (orsomeofitscomponents)measuredatnon-largeelevation abovethelayer. 8 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory 0.11 realprofile x) reconstructed ( a t e z e, fil 0.10 o r p r e y a l 0.09 0.0 0.2 0.4 0.6 0.8 1.0 (a) x e DNS ofil0.002 DNS r p al Fourier e r Fourier m 0.000 o r f n o ti a vi-0.002 e d 0.0 0.2 0.4 0.6 0.8 1.0 (b) x Figure 3: (a): The reconstructed profile (2.16) of a ferromagnetic layer (red dashed line) compared to the original profile (black solid line) for the parameter values ζ = 0.1, a = 0.01, 0 k = 2π, µ /µ = 100. (b): Inaccuracy (the deviation from the original profile) of the profiles 1 2 reconstructed from the solution with direct numerical simulation (DNS) and the linear-in-defect solutioninFourierspace. 3. Application of the analytical technique 3.1. Validation of applicability of the analytical technique with results of numerical simulation for two-dimensional case Inordertovalidatetheapplicabilityoftheanalyticalresultsderived,wehaveconsideredthemodel caseofaferromagneticlayerofµ /µ = 100withprofileζ = ζ +acoskxwithk = 2π,ζ = 0.1, 2 1 0 0 a = 0.01. Themagneticfield forthiscasewas calculated both • with direct numerical simulation, employing a finite volume method and the mesh size dx = dz = 0.01,and • analyticallyin Fourierspacewithintheframeworkofthelinear-in-defect approximation. 9 A.V.Pimenova, D.S. Goldobin,etal. MagneticFlux Leakage: AnalyticalTheory Linear in (ζ − ζ ) solution to the problem can be found analytically in Fourier space. Non- 0 divergingforz → ∞solutionto theproblemreads coshk|z| −Hx+A sinkx, |z| < ζ; coshkζ Φ(x,z) = 0 (3.1)  −Hx+Ae−k(|z|−ζ0)sinkx, |z| > ζ. Here  (µ −µ )Ha 1 2 A = . µ1tanhkζ0 +µ2e−kζ0 Onecan differentiatethelattersolutiontofind thecomponentsofthemagneticfield at heighth; H | = H−kAe−k(h−ζ0)coskx, x z=h (3.2) H | = kAe−k(h−ζ0)sinkx. z z=h In Figure 3, with synthetic data from direct numerical simulation, one can see the surface profilecanbewellreconstructedwithEq.(2.16). Theaccuracy ofanalyticalsolution(3.1)andthe roleofcorrection ζ (compare Eqs.(2.14), (2.15)and(2.16))can bejudgedfrom Figure3(b). 2 ~ 3.2. 3D layer: measurable H-field For the case of sensors measuring the H~-field at certain elevation above the layer with a dense enough grid of measurements points (e.g., [4]), one can directly employ Eqs.(2.25) and (2.26) with H~-field derivatives approximated by finite differences. If the elevation height is of the same order of magnitude as the characteristic defect width, one needs first to calculate H~-field on the “imaginary”undamagedsurfaceandthenusethisfieldforcalculationofthelayerthicknessprofile. For this calculation one can use the Taylor expansion of H~-field. Indeed, all the z-derivatives of H~-field can becalculatedfrom x-and y-derivativesofmeasuredfields, becauseH~ is agradientof aharmonicfunction(see Eq.(2.1)); ∂H ∂H ∂H z x y = − + , ∂z ∂x ∂y (cid:18) (cid:19) ∂2H ∂2 ∂2 z = − + H , z ∂z2 ∂x2 ∂y2 (cid:18) (cid:19) ∂n+2H ∂2 ∂2 ∂nH z z = − + , n = 1,2,3,... , ∂zn+2 ∂x2 ∂y2 ∂zn (cid:18) (cid:19) ∂H ∂H x z = , ∂z ∂x ∂n+2H ∂2 ∂2 ∂nH x x = − + , n = 0,1,2,... , ∂zn+2 ∂x2 ∂y2 ∂zn (cid:18) (cid:19) 10