METHOD OF AND APPARATUS FOR DETERMINING VARIATIONS IN WALL THICKNESS IN FERROMAGNETIC TUBES

Abstract

A method can include energizing a tube with a longitudinally extending magnetic field generated inside the tube, using a magnetic field-detecting logging tool to generate magnetic flux signals generated inside the tube externally of the material of the tube wall resulting from such energizing at circumferential locations on the inner surface of the tube and at distances along the tube, iteratively using a model of the relationship between the generated magnetic flux signals and the thickness of the tube wall to derive a thickness profile of the tube wall by using (i) the magnetic permeability of the tube material deduced from the magnetic flux signals and (ii) a defect-free flux parameter representative of any non-linearity between the magnetic field strength and flux density in the tube, the iteration including using the model to calculate an initial approximate wall thickness profile using an initial estimate of the defect-free flux parameter.

Claims

1. A method of determining variations in wall thickness in an elongate, cylindrical, hollow, ferromagnetic tube defining a tube wall, comprising the steps of: a) energizing the tube with an at least longitudinally extending magnetic field generated inside the tube that gives rise to near- or over-saturation of the tube wall material; b) using a magnetic field-detecting logging tool to (i) detect two or more magnetic flux leakage signals generated inside the tube other than in the material of the tube wall, resulting from such energizing, at plural circumferential locations on the inner and/or outer surfaces of the tube and at a plurality of distances along the tube and (ii) generate two or more magnetic flux leakage data signals indicative thereof; c) iteratively, one or more times, using a model of the relationship between the two or more magnetic flux leakage data signals generated in Step b) and the thickness of the tube wall to derive the thickness profile of the tube wall by relating a defect-free flux parameter representing a field strength defect offset in a magnetization plot of the tube averaged across two or more sensors of the logging tool and a maximum flux response within the defect measured by the logging tool to give rise to a first approximation ratio, forming part of the model, that is proportional to defect penetration; d) inverting in accordance with the model to produce a defect profile which depends on the magnetic flux leakage signals without any external parameters; e) determining the defect penetration by determining the maximum of the defect profile; and f) generating one or more signals representing a wall thickness profile based on the defect penetration.

2. A method according to claim 1 wherein Step a) of energizing the tube with an at least longitudinally extending magnetic field generated inside the tube includes operating, or permitting to operate, a source of an at least longitudinally extending magnetic field inside the tube supported in or by a magnetic field-generating logging tool.

3. A method according to claim 1 wherein Step a) of energizing the tube with an at least longitudinally extending magnetic field generated inside the tube includes operating, or permitting to operate, a source of an at least longitudinally extending magnetic field inside the tube supported in or by a magnetic field-generating logging tool and wherein the magnetic field-generating logging tool is or is operatively connected to the magnetic field-detecting logging tool.

4. A method according to claim 1 including the step of causing conveyance of at least the magnetic field-detecting logging tool from a position remote from the tube to two or more locations, inside the tube, that are spaced from one another along the tube; and carrying out Steps a) and b) in respect of the respective two or more locations.

5. A method according to claim 1 including the step of causing conveyance of at least the magnetic field-detecting logging tool from a position remote from the tube to two or more locations, inside the tube, that are spaced from one another along the tube; and carrying out Steps a) and b) in respect of the respective two or more locations, the method further including supporting at least the magnetic field-detecting logging tool on wireline, that connects the logging tool to one or more items of equipment that are remote from the tube, during such conveyance.

6. A method according to claim 1 including the step of causing conveyance of at least the magnetic field-detecting logging tool from a position remote from the tube to two or more locations, inside the tube, that are spaced from one another along the tube; and carrying out Steps a) and b) in respect of the respective two or more locations, the method further including supporting at least the magnetic field-detecting logging tool on wireline, that connects the logging tool to one or more items of equipment that are remote from the tube, during such conveyance; and further including supporting at least the magnetic field-detecting logging tool on wireline, that connects the magnetic field-detecting logging tool to processing apparatus that is remote from the tube, during carrying out of at least steps a) and b).

7. A method according to claim 1 wherein the tube is or includes wellbore casing and/or liner.

8. A method according to claim 1 including the step of causing conveyance of at least the magnetic field-detecting logging tool from a position remote from the tube to two or more locations, inside the tube, that are spaced from one another along the tube; and carrying out Steps a) and b) in respect of the respective two or more locations; wherein the tube is or includes wellbore casing and/or liner; and wherein the at least two locations inside the tube are non-coincident with any casing collars (if present) forming part of the tube.

9. A method according to claim 1 wherein the tube is one of a plurality of serially interconnected tubes fixed in a subterranean location and defining a hollow column communicating with a surface location or communicating with a further hollow column that is connected to a surface location.

10. A method according to claim 1 wherein the magnetic field-detecting logging tool includes a plurality of Hall-effect detectors of magnetic energy.

11. A method according to claim 1 wherein the magnetic field-detecting logging tool includes a plurality of Hall-effect detectors of magnetic energy and wherein the plurality of Hall-effect detectors of magnetic flux are arrayed in a circular pattern defined circumferentially with respect to the magnetic field-detecting logging tool.

12. A method according to claim 1 wherein the magnetic field-detecting logging tool includes a plurality of Hall-effect detectors of magnetic energy and further includes one or more arms supporting one or more pads mounting at least one pad-mounted said Hall-effect sensor, the one or more arms being moveable from a retracted position in which the at least one pad-mounted Hall-effect sensor is spaced from the material of the tube; and a deployed position in which the pad contacts the material of the tube, the method including causing movement of the one nor more arms between the retracted and deployed positions.

13. A method according to claim 1 wherein when the applicable magnetisation law is non-linear, the model of the relationship between the two or more magnetic flux signals generated in Step b) and the tube wall thickness is of the form:
∇.sub.2.Math.(μ(H.sub.1){right arrow over (H)}.sub.1|.sub.z=ζζ)=0  (1) wherein μ(H.sub.1) is magnetic permeability of the ferromagnetic material of the tube; {right arrow over (H)}.sub.1 is the magnetic field within the ferromagnetic material of the wall of the tube; ζ represents the nominal thickness of the tube wall; z is the thickness direction of the tube wall; and ∇.sub.2 is the gradient operator with respect to the ferromagnetic material of the wall of the tube.

14. A method according to claim 1 wherein when the applicable magnetisation law is non-linear, the model of the relationship between the two or more magnetic flux signals generated in Step b) and the tube wall thickness is of the form:
∇.sub.2.Math.(μ(H.sub.1){right arrow over (H)}.sub.1|.sub.z=ζζ)=0  (1) wherein μ(H.sub.1) is magnetic permeability of the ferromagnetic material of the tube; {right arrow over (H)}.sub.1 is the magnetic field within the ferromagnetic material of the wall of the tube; ζ represents the nominal thickness of the tube wall; z is the thickness direction of the tube wall; and ∇.sub.2 is the gradient operator with respect to the ferromagnetic material of the wall of the tube and wherein the magnetic permeability μ(H.sub.1) is approximated in accordance with the expression $\mu \left(H\right)=\alpha \left(1+\frac{H_0}{H}\right)$ in which H.sub.0 is a defect-free flux parameter representing a field strength defect offset in a magnetization plot of the tube averaged across two or more sensors of the logging tool and H is a maximum flux response within the defect measured by the logging tool.

15. A method according to claim 1 wherein when the applicable magnetisation law is non-linear, the model of the relationship between the two or more magnetic flux signals generated in Step b) and the tube wall thickness is of the form:
∇.sub.2.Math.(μ(H.sub.1){right arrow over (H)}.sub.1|.sub.z=ζζ)=0  (1) wherein μ(H.sub.1) is magnetic permeability of the ferromagnetic material of the tube; {right arrow over (H)}.sub.1 is the magnetic field within the ferromagnetic material of the wall of the tube; ζ represents the nominal thickness of the tube wall; z is the thickness direction of the tube wall; and ∇.sub.2 is the gradient operator with respect to the ferromagnetic material of the wall of the tube; and wherein Step c) includes the step g) of integrating the expression (1) with a magnetic field H.sub.2 derived from the magnetic flux signals generated in Step b) and the magnetic permeability μ(H.sub.2) of the interior of the tube other than the material of the tube wall.

16. A method according to claim 1 at least Steps c) to e) of which are carried out using a programmed processing apparatus.

17. A method according to claim 1 including the step of h) generating one or more images of the tube wall thickness profile from the one or more signals generated in Step f).

18. A method according to claim 1 including the step of i) transmitting, storing, saving or processing the one or more signals generated in Step f) or data representative thereof.

19. A method according to claim 1 including the step of h) generating one or more images of the tube wall thickness profile from the one or more signals generated in Step f); and further including the step j) of transmitting, storing, saving, processing or printing one or more said images of the tube wall thickness.

20. Apparatus for carrying out a method according to claim 1 including a source of a magnetic field that is capable of extending at least longitudinally along the interior of the tube, the source being moveable along the interior of the tube; a magnetic field-detecting logging tool that is moveable along the interior of the tube and that is capable of carrying out Step b); and processing apparatus to which the magnetic field-detecting logging tool is operatively connected or connectable and that is capable of carrying out at least Steps c) to e).

21. Apparatus according to claim 20 the processing apparatus of which also is capable of carrying out Step f).

22. Apparatus according to claim 20 wherein the magnetic field-detecting logging tool includes a plurality of Hall-effect detectors of magnetic energy.

23. Apparatus according to claim 20 wherein the magnetic field-detecting logging tool includes a plurality of Hall-effect detectors of magnetic energy and wherein the plurality of Hall-effect detectors of magnetic energy are arrayed in a circular pattern defined circumferentially with respect to the magnetic field-detecting logging tool.

24. Apparatus according to claim 20 wherein the magnetic field-detecting logging tool includes a plurality of Hall-effect detectors of magnetic energy and the apparatus including one or more arms supporting one or more pads mounting at least one pad-mounted said Hall-effect sensor, the one or more arms being moveable from a retracted position in which the pad-mounted Hall-effect sensor is spaced from the material of the tube; and a deployed position in which the pad contacts the material of the tube, the method including causing movement of the one or more arms between the retracted and deployed positions.

25. Apparatus according to claim 20 the processing apparatus of which is programmable.

26. Apparatus according to claim 20 wherein Step a) of energizing the tube with an at least longitudinally extending magnetic field generated inside the tube includes operating, or permitting to operate, a source of an at least longitudinally extending magnetic field inside the tube supported in or by a magnetic field-generating logging tool, and wherein the source is supported in or by the magnetic field-generating logging tool.

27. Apparatus according to claim 20 wherein Step a) of energizing the tube with an at least longitudinally extending magnetic field generated inside the tube includes operating, or permitting to operate, a source of an at least longitudinally extending magnetic field inside the tube supported in or by a magnetic field-generating logging tool, wherein the source is supported in or by the magnetic field-generating logging tool, and wherein the magnetic field-generating logging tool and the magnetic field-detecting logging tool are common one to the other or are secured one to the other.

28. Apparatus according to claim 20 including at least one logging tool supported by and operatively connected using wireline.

29. Apparatus according to claim 20 including at least one logging tool supported by and operatively connected using wireline, wherein the wireline interconnects the magnetic field-detecting logging tool and the processing apparatus.

Description

[0084] There now follows a description of preferred embodiments, by way of non-limiting example, with reference being made to the accompanying drawings in which:

[0085] FIG. 1 is a schematic, partly sectioned view of a borehole-casing system including inserted therein a logging tool that is suitable for carrying out the steps of the method disclosed herein;

[0086] FIG. 2 is a photograph of an example in a horizontally sectioned length of casing of wall depletion, outlined in superimposed dotted lines, caused by corrosion;

[0087] FIG. 3 shows in schematic form another logging tool that is suitable for carrying out the method steps disclosed herein and illustrating the nature of magnetic flux in the vicinity of a region of wall depletion as illustrated in FIG. 2;

[0088] FIG. 4 shows in schematic form the mirror-symmetric modelling of the wall material of casing or liner as used in the method;

[0089] FIG. 5 is a plot of flux density against field strength in a ferromagnetic material such as the wall material of casing or liner, and illustrating the non-linear nature of the magnetization law of such a material;

[0090] FIGS. 6a and 6b respectively illustrate two synthesized wall material defects in the form of an isolated segment of a sphere (FIG. 6a) and two overlapping semi-sphere shape (FIG. 6b);

[0091] FIGS. 7a, 7b, 7c, 7d, 7e and 7f respectively illustrate calculated tube wall profiles for the synthesized wall material defects of FIGS. 6a and 6b, using differing values of a defect-free flux parameter H.sub.0 explained herein and illustrating a phenomenon of impermissible profiles; and

[0092] FIGS. 8a and 8b are versions of the FIG. 6 profiles following constraining of the model to eliminate inadmissible profiles, and showing the accuracy of the method.

NON-LIMITING APPARATUS AND DEPLOYMENT

[0093] Referring to the figures and especially FIG. 1, one form of borehole-casing system 10 is schematically illustrated in which a borehole penetrates a formation 11 including a reservoir 12 containing liquid hydrocarbons 13 and, overlying it, a volume of gas 14.

[0094] The wall 15 of the borehole is for clarity omitted from FIG. 1 over most of the length of the illustrated system, but can readily be envisaged.

[0095] FIG. 1 shows the borehole-casing system 10 in an exaggeratedly shortened form, for convenience. In practice the length of the system 10 from a surface location 16 (that may be e.g. the ground or an ocean floor) to the downhole end of production tubing 27 may be several hundred or several thousand feet.

[0096] At its most uphole end the borehole is lined by one or more relatively short annuli 17, 18 of cement that encircle a (non-limiting) nominal 30-inch diameter length 21 of hollow, cylindrical casing between the outer wall of the casing 21 and the borehole wall 15.

[0097] A (non-limiting) nominal 16-inch diameter casing length 22 is hung off the downhole end of the 30-inch casing, overlapping it fora short length and extending in a downhole direction. A cement annulus 23 lies externally of the 16-inch casing 22 and encircles it.

[0098] A further casing length 24 of (non-limiting) nominal 13⅜-inch diameter is similarly hung from the downhole end of the 16-inch casing and surrounded by a cement annulus 26. Production tubing 27 that in the non-limiting example shown is of nominal 9⅝-inch diameter is hung off the downhole end of the 13⅜-inch diameter casing and is perforated in any of a variety of ways to permit the flow of fluids from the reservoir 12 in a per se known manner.

[0099] Various forms of casing collar 19 that are illustrated schematically in FIG. 1 serve to contain the cement as it is pumped downhole during construction of the borehole-casing system and (depending on the precise design) to secure the casing lengths one to another as a series.

[0100] The borehole-casing system of FIG. 1 as noted is shown highly schematically, for purposes of exemplifying environments in which the method and apparatus of the disclosure may be used. A great number of variations on the arrangement of FIG. 1 are possible within the scope of the disclosure and, as mentioned, use of the method and apparatus is not confined to subterranean borehole environments as illustrated.

[0101] An elongate magnetic flux leakage logging tool 28 is shown in schematic form in FIG. 1 suspended on a length of wireline 29 having been deployed from a surface location to deep inside the borehole 15. The wireline extends from a surface location where it is dispensed from a storage drum (that is omitted for simplicity) and connected (or at least connectable) to processing apparatus that processes signals generated by the logging tool 28. The wireline is to this end connected in a manner capable of transmitting signals generated by the logging tool 28. Means of achieving this will be known to the person of skill in the art.

[0102] The wireline 29 also preferably is connected in a manner permitting the transmission of surface-generated operational commands to the logging tool 28 and, in at least some embodiments, operational power for powering the tool 28.

[0103] The logging tool 28 is shown in position surveying the inner cylindrical wall of the casing (tube) length 24 in accordance with the method disclosed herein but may be deployed for example in the production tubing 27 the internal diameter of which is less than that of the casing 24. Features of the logging tool 28 permitting such operation are known per se and are described herein to the extent needed for an understanding.

[0104] The logging tool 28 includes spaced from one another at respective in-use uphole and downhole ends respective opposite magnet poles 31, 32. The poles 31, 32 are such as to cause a magnetic field to be emitted externally of the logging tool in a manner extending longitudinally along the casing 24 that is under inspection. The method of the invention relies on analysis of the longitudinal components of such a field and disregards any non-longitudinal components in a manner rendering the model used in the method solvable using the data signal outputs of the logging tool 28.

[0105] In the illustrated FIG. 1 embodiment the poles 31, 32 are interconnected by a rigid, elongate cylindrical body 33 that depending on the precise logging tool design may contain e.g. processing or signal conditioning electronics, one or more power sources or power convertors, one or more motors for causing deployment of support arms described below, and shielding preventing the creation of a magnetic short circuit between the poles 31, 32. Any power sources or power converters if present may for example be powered by power signals transmitted using the wireline 29.

[0106] The poles 31, 32 may be permanent magnets or may be electromagnets powered by a power source as aforesaid. Regardless of the precise arrangement the poles supported in the logging tool 28 amount to a magnetic field-generating logging tool as referred to herein.

[0107] The illustrated logging tool 28 includes pairs of extensible and retractile arms 34, 36, 37, 38. Each arm of each respective pair 34, 36; 37, 38 is pivotably fixed at one end to the cylindrical body 33 and at the other end to one of a plurality of pads 39, 41. The arrangement of the arms and the pads is such that each arm 34, 36, 37, 38 is moveable between a retracted position in which it lies close to or recessed within the cylindrical body 33 and an extended position protruding therefrom. As a result the pads 39, 41 also are moveable between retracted positions lying close to or recessed within the cylindrical body 33 and extended positions spaced radially outwardly therefrom as illustrated.

[0108] This permits the pads 39, 41 in a per se known manner to be caused to engage or at least lie closely adjacent the wall of the casing regardless, within a range of adjustment permitted by the arm and cylinder dimensions, of the internal diameter of the casing.

[0109] Each pad supports at least one detector of magnetism that in the preferred embodiment is a Hall-effect sensor. Such a sensor generates electrical signals the amplitude of which is in proportion to an incident magnetic field. The sensors are connected or connectable so that their output signals may be transmitted to the remainder of the logging tool 28 and thence, as necessary following processing in the logging tool 28, to the surface location via the wireline 29.

[0110] The arms 34, 36, 37, 38 may be powered to move between the described retracted and extended positions through operation of one or more springs or other force-generating means supported within the cylinder 33.

[0111] The arms 34, 36, 37, 38, the pads 39, 41, the Hall-effect sensors and any springs, etc., supported in the cylinder amount to a magnetic field-detecting logging tool. It will be apparent that the Hall-effect sensors in such an arrangement may be maintained in contact with the casing wall, regardless of its internal diameter, during deployment or the logging tool 28. In consequence when the logging tool 28 reaches a depth along the borehole 15 at which surveying of the integrity of the casing wall is to take place the arms 34, 36, 37, 38 are in appropriately extended positions as shown. In other words the Hall-effect sensors supported on the pads 39, 41 are arranged to be deployed in the magnetic field H.sub.2 inside the casing that results from interaction of the emitted magnetic field with the casing wall material and in particular any defects in it.

[0112] In the illustrated arrangement the magnetic field-detecting logging tool and the magnetic field-generating logging tool are formed as parts of one and the same component and therefore are secured one to the other. However this need not be the case, and it is possible to devise arrangements in which a magnetic field-generating device, in the form of a sub supporting permanent magnet poles, may be deployed separately from magnetic field-detecting logging apparatus (albeit that the detecting logging apparatus needs to be operated before the magnetic field generated by the generating device diminishes such that useful signal measurements could not be recorded).

[0113] In some magnetic-field detecting logging tools two field-generating tools or sources are provided. One source is a permanent magnet which is formed by multiple pieces that preferably define a cone shape. The other magnetic field source is a local magnet mounted on the Hall-effect sensors and designed to help discriminate an internal defect (ID) from an external defect (OD). In an embodiment the tool may have two sets of Hall effect sensors: corrosion sensors and discriminator sensors. In some embodiments the latter have the local magnet built on top of them. Such arrangements are omitted from FIG. 1, which shows generic magnets 31, 32 in a schematic form. FIG. 3, described hereinbelow, illustrates one of numerous possible alternative magnet and sensor arrangements.

[0114] In the embodiment shown in FIG. 1 the pads 39, 41, and hence the Hall-effect sensors, are located at longitudinally spaced locations on the respective pairs of arms 34, 36 and 37, 38. However this need not be the case, and instead the Hall-effect sensors could be located at the same longitudinal position along the logging tool 28.

[0115] Two sets of arms 34, 36, 37, 38 and associated pads 39, 41 are shown in the illustrated embodiment, but other numbers of pads, arms and Hall-effect sensors may be provided in other embodiments. If three or more pad/sensor combinations are provided these could be arranged in a circular pattern that is defined by or related to a circumference of the logging tool 28. This may be so regardless of whether the pads are all at the same longitudinal position along the logging tool 28.

[0116] Moreover it is not necessary for the magnetic field detectors to be deployable and retractile. On the contrary it is possible to devise embodiments in which the diameter at which the magnetic field detectors such as Hall-effect sensors are mounted is a fixed distance relative to the remainder of the logging tool 28, with such a logging tool then being dedicated to the surveying of a chosen size of casing. An example of such an arrangement is illustrated in FIG. 3.

[0117] As shown in FIG. 2 by a dotted outline, a region 42 of reduced thickness of a casing wall 43 caused by corrosion in practice will never adopt the assumedly circular shape used in prior art casing defect assessment methods. On the contrary such regions 42 will adopt highly randomized peripheral shapes and will not be of constant material depletion depth.

[0118] FIG. 3 illustrates in a schematic form another design of magnetic flux leakage logging tool 28′. Tool 28′ differs from tool 28 of FIG. 1 in that firstly the Hall-effect sensors are not mounted on moveable pads which therefore are dispensed with. Instead the Hall-effect sensors 44, 46 are secured on the cylindrical body 33′ of the logging tool 28′. As a result the Hall-effect sensors 44, 46 are positioned in an essentially fixed relationship relative to the casing 21 within which the logging tool 28′ is deployed.

[0119] FIG. 3 shows variant forms of the magnet poles 31′, 32′ that each adopt a frusto-conical shape. Other variants on the designs of the magnet poles are also possible.

[0120] FIG. 3 illustrates in a non-limiting way the effect of depletion of the material of the casing on the lines 47 of magnetic flux that extend between the magnet poles 31′, 32′. As is visible in FIG. 3 on a side 21a of the casing 21 where the material of the casing wall is intact the flux lines 47 extend smoothly along the casing wall metal without significant perturbation. As a consequence the Hall-effect sensor 44 adjacent wall side 21a detects no magnetic flux.

[0121] In contrast the material of casing wall side 21b has been depleted and gives rise to a region 42′ of reduced metal thickness. As illustrated this causes distortion of the flux lines 47 in the region 48. The flux leaks into the hollow interior of the casing in a manner that causes activation of the Hall-effect sensor 46 so as to generate a signal. Such a signal following treatment in accordance with the method disclosed herein is accurately indicative of the extent of depletion of the metal of the casing wall 21b.

[0122] FIG. 3 is intended to be merely illustrative and is not limiting. In embodiments the shapes and positions of the magnet poles, the Hall-effect (or other) sensors of magnetic flux, the nature of the casing wall 21, the shape and depth of any depleted zone 42′ and the density of the flux lines 47, 48 may vary from the exemplary versions shown.

[0123] Combinations of embodiments as would occur to the person of skill in the art are possible. As one example in this regard the logging tool of FIG. 1 may for instance include magnet poles of the type shown in FIG. 3. Another possibility is for the fixed flux sensors 44, 46 of FIG. 3 to be provided in a tool design that is otherwise generally similar to that of FIG. 1.

[0124] Notwithstanding the possibilities for variation of the hardware using which the method of the invention may be practised, the method in its basic form involves steps of, following deployment of the logging tool 28 to a position such as that shown in FIG. 1 at which logging of the casing wall material is required: [0125] a) energizing the tube with an at least longitudinally extending magnetic field generated inside the tube; [0126] b) using a magnetic field-detecting logging tool to generate two or more magnetic flux signals generated inside the tube other than in the material of the tube wall resulting from such energizing at plural circumferential locations on the inner surface of the tube and at a plurality of distances along the tube; [0127] c) iteratively, one or more times, using a model of the relationship between the two or more magnetic flux signals generated in Step b) and the thickness of the tube wall to derive the thickness profile of the tube wall by using the defect-free flux averaged across the sensors relating this defect offset to the maximum flux response within the defect as first approximation ratio proportional to the defect penetration, the iteration including: [0128] d) using the estimate of the defect-free flux parameter; [0129] e) inverting in accordance with Equation (6) hereof for the defect profile which depends only on the available flux data without any external parameters. Such action may include calculating and taking account of a metric that is representative of the magnitude of one or more physically inadmissible features in or forming part of the thickness profile; [0130] f) determining the defect penetration by taking the maximum of the reconstructed defect profile; and [0131] g) generating one or more signals representing the thickness profile resulting from use of selected defect-free flux parameter as the thickness profile of the wall of the tube.

Modelling and Simplifying Assumptions

[0132] The following explains such steps in more detail.

[0133] The hall-effect (or other magnetic field detecting) sensors exemplified in FIG. 1 generate electrical signals in proportion to the magnetic field detected inside the casing. The longitudinally extending field components (i.e. those extending in the longitudinal direction along the casing) are processed to generate signals indicative of the casing wall thickness profile ζ by practising the following steps:

[0134] Making use of a large-scale defect assumption explained below, the magnetic field is sensitive to the casing thickness profile but not to the inner and outer surface profiles independently. The method therefore relates the analytical consideration to the case of mirror-symmetric layers as shown in FIG. 4 in which a ferromagnetic layer is 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 half-thickness of the undamaged layer is ζ.sub.0. The uniform external magnetic field is applied along the x-axis. The long-wavelength (or large-scale) approximation conditions have the following form:

L>>ζ,|∇.sub.2ζ|<<1,

where L is the characteristic lateral scale (width) of a defect. ∇.sub.2 and Δ.sub.2 are gradient and Laplace operators respectively, and the index 2 indicates the two-dimensional versions of them calculated with respect to x and y coordinates.

[0135] The Maxwell equations for this system are:

[00002]$\{\begin{array}{cc}\nabla \times {\stackrel{.fwdarw.}{H}}_j& =0,\\ \nabla .Math.{\stackrel{.fwdarw.}{B}}_j& =0,\end{array}$

with boundary conditions

{right arrow over (H)}.sub.1τ={right arrow over (H)}.sub.2τ,B.sub.1n=B.sub.2n.

[0136] Here the subscripts j={1,2} indicate the ferromagnetic layer and the surrounding material respectively (FIG. 2); {right arrow over (B)}.sub.j is the magnetic flux density,

{right arrow over (B)}.sub.j=μ.sub.j{right arrow over (H)}.sub.j;  (1)

and μ.sub.j is magnetic permeability. For ferromagnets μ=μ(H), where H=|{right arrow over (H)}|. Since the magnetic permeability of a ferromagnet is considerably larger than that of the surrounding material,

[00003]$\frac{{\mu }_1}{{\mu }_2}\gg 1$

[0137] To the leading order of approximation, the flux of the magnetic field is captured within the ferromagnetic layer, i.e.

∇.sub.2.Math.(∫.sub.−ζ.sup.ζ{right arrow over (B)}.sub.1dz)=0  (2)

and

∇.sub.2.Math.(∫.sub.−ζ.sup.ζμ(H.sub.1){right arrow over (H)}.sub.1dz)=0  (3)

[0138] To the leading order, expression (3) is equivalent to

∇.sub.2.Math.(μ(H.sub.1){right arrow over (H)}.sub.1|.sub.z=ζζ)=0  (4)

[0139] Integration of Equation (4) with a given field {right arrow over (H)}.sub.2 (which is to be measured) and magnetic permeability μ(H.sub.2) allows modification of the signals to a form reconstructing the surface profile. If the magnetization law is linear (μ=const), the expression can be simplified:

∇.sub.2.Math.({right arrow over (H)}.sub.1|.sub.z=|ζζ)=0

[0140] However, in most real systems, the magnetization law is non-linear as mentioned. This would lead to a considerable increase in the complexity of the integration of Equation (4) in the absence of the method steps disclosed herein.

Non-Linear Magnetisation Law

[0141] A characteristic feature of ferromagnetic materials is hysteresis, illustrated by the solid magnetization curve in FIG. 5. This can lead to a loss of uniqueness in the solution of the profile-reconstruction problem. To minimize the impact of non-uniqueness, MFL tools are designed to operate where the magnetization is at or just below saturation. Hysteresis does not affect the signal here, but the magnetization law may still be significantly non-linear in this region.

[0142] Nonetheless, non-linear-magnetization problems can in accordance with the method be addressed for large-scale defects because the magnetic field ii within the ferromagnetic layer deviates only slightly from the field of the undamaged layer . For H close to , one can approximate the magnetization law as:

B(H)=α(H+H.sub.0),

where α and H.sub.0 are parameters of the approximation. The expression for the magnetic permeability can therefore be cast as:

[00004]$\begin{array}{cc}\mu \left(H\right)=\alpha \left(1+\frac{H_0}{H}\right)& \left(5\right)\end{array}$

H.sub.0 may be referred to as the Defect Free Flux (DFF). This can be inferred from the MFL scan data, as explained in the following section.

Numerical Determination of the Approximation Parameters

[0143] In MFL surveys the exact dependence B(H) is generally unknown. The characteristic magnetization law for wellbore casing depends on the grade of steel used for its manufacture, the manufacturing process and its stress state which is unknown. Parameters of the magnetization law can be found experimentally, but that would require the DFF to be determined for the specific type of casing prior to inspection.

[0144] In embodiments of the approach disclosed herein the parameters of magnetization are determined within an inversion process which reconstructs the thickness profile by minimizing physically inadmissible features of the calculated profile. Inadmissible features resulting from an incorrect magnetization law disappear when the correct law is used. Minimizing the total volume of positive defects (metal gains) produces good results, although casing connectors (such as collars) must be excluded from this part of the process.

[0145] Re-writing Equation (4) taking account of Equation (5) gives:

[00005]$\begin{array}{cc}{{\nabla }_2(\alpha \left(1+\frac{H_0}{H}\right)\stackrel{.fwdarw.}{H}.Math.}_{z=\zeta }\zeta )=0& \left(6\right)\end{array}$

[0146] Since α is constant, this may be simplified as:

[00006]${{\nabla }_2(\left(1+\frac{H_0}{H}\right)\stackrel{.fwdarw.}{H}.Math.}_{z=\zeta }\zeta )=0$

[0147] Only the DFF parameter (H.sub.0) has a significant effect on the results. If H.sub.0 is chosen incorrectly, the calculated profile {tilde over (ζ)} will not match the actual surface profile ζ. Moreover, the cumulative integration error δ={tilde over (ζ)}−ζ in the area near the defect can result in the uplift of the calculated surface profile above the undamaged level ζ.sub.0, which is physically inadmissible. Based on the foregoing, H.sub.0 is determined by:

[0148] 1. Selecting an initial arbitrary value of H.sub.0.

[0149] 2. Calculating the surface profile.

[0150] 3. Finding the quantitative norm of inadmissible features (surface uplifts),

∫dx∫dy({tilde over (ζ)}|.sub.z>ζ0−ζ.sub.0+|{tilde over (ζ)}|.sub.z>ζ0−ζ.sub.0|)  (8)

[0151] 4. Changing the value of H.sub.0 in a way that leads to the minimization of norm (8).

Derivation of H.SUB.0

[0152] The inversion algorithm was validated in the first instance using flux data from a numerical model independently of the analytical model used in the inversion. It was used to simulate the MFL response to a series of sphere segment defects with different values for the H field. A non-linear magnetization law and an applied U-field were specified. FIG. 6 shows two examples: an isolated sphere segment and two overlapping semi-sphere defects.

[0153] The results of the inversion procedure are shown in FIG. 7 for different values of the parameter H.sub.0 as indicated in the figure (i.e. H.sub.0={tilde over (H)}.sub.0 (top, FIGS. 7a and 7b), {tilde over (H)}.sub.0=2 {tilde over (H)}.sub.0 (middle, FIGS. 7c and 7d) and H.sub.0=0.5{tilde over (H)}.sub.0 (bottom, FIGS. 7e and 7f), where {tilde over (H)}.sub.0 is the reference value). Parameter values that do not correspond to the real magnetization law produce implausible metal-gain in the reconstructed profiles around the defect area. When H.sub.0 is overestimated the uplifts are on the sides of the defect area (displaced from the defect in the y-direction), and for underestimated H.sub.0 the uplifts are before and after the defect along the casing (displaced in the x-direction). Any 1D minimization algorithm can be used to find the optimal H.sub.0; the Nelder-Mead or Broyden-Fletcher-Goldfarb-Shanno algorithms were found to be adequate, for example.

[0154] FIG. 7 in summary shows the effect of constraining the model so that inadmissible results are excluded from the output.

[0155] The final reconstruction results and H.sub.0 values are shown in FIG. 8. The error in reconstructed penetration is less than 5% absolute.

[0156] The FIG. 8 results are presented in graphical form but may just as readily be made available as e.g. further electrical signals.

[0157] These results for wall thickness ζ therefore may be transmitted, plotted, displayed, saved, interpreted (either by eye or using e.g. a machine recognition technique) and/or processed in numerous ways.

[0158] It is clear from the foregoing therefore that following operation of a logging tool such as tool 28 or 28′ as described and processing of the signals output by the Hall-effect sensors in the manner explained above the result is a modelled wall thickness profile that is highly realistic. As noted the signal processing simplifications, to limit to longitudinal components of the detected magnetic field; and the constraint to remove impermissible results, make the method viable and considerably more accurate than prior art methods. The method therefore represents a significant advance in the effort to avoid the deleterious effects of casing wall corrosion damage and metal loss

[0159] Moreover as mentioned the described method can be employed in other forms of tube albeit with modification as necessary of the deployment and signal transmission aspects to suit the different environments of such tube types.

[0160] The listing or discussion of an apparently prior-published document in this specification should not necessarily be taken as an acknowledgement that the document is part of the state of the art or is common general knowledge.

[0161] Preferences and options for a given aspect, feature or parameter of the invention should, unless the context indicates otherwise, be regarded as having been disclosed in combination with any and all preferences and options for all other aspects, features and parameters of the invention.